Contributions to the Theory of Generalized Ordinary Differential

Charles University in Prague Faculty of Mathematics and Physics

Contributions to the Theory of Generalized Ordinary Diﬀerential Equations

Habilitation thesis

Anton´ınSlav´ık

July 1, 2014 Preface

This habilitation thesis is based on the following nine papers:

• A. Slav´ık, Dynamic equations on time scales and generalized ordinary diﬀerential equations, J. Math. Anal. Appl. 385 (2012), 534–550. • J. G. Mesquita, A. Slav´ık, Periodic averaging theorems for various types of equations, J. Math. Anal. Appl. 387 (2012), 862–877.

• M. Federson, J. G. Mesquita, A. Slav´ık, Measure functional differential equations and functional dynamic equations on time scales, J. Differential Equations 252 (2012), 3816–3847. • M. Federson, J. G. Mesquita, A. Slav´ık, Basic results for functional differential and dynamic equations involving impulses, Math. Nachr. 286 (2013), no. 2–3, 181–204.

• A. Slav´ık, Measure functional differential equations with infinite delay, Nonlinear Anal. 79 (2013), 140–155. • A. Slav´ık, Generalized differential equations: Differentiability of solutions with respect to initial conditions and parameters, J. Math. Anal. Appl. 402 (2013), 261–274. • G. A. Monteiro, A. Slav´ık, Linear measure functional differential equations with infinite delay, Math. Nachr., in press. DOI: 10.1002/mana.201300048. • A. Slav´ık, Well-posedness results for abstract generalized differential equations and measure functional differential equations, submitted for publication. • G. A. Monteiro, A. Slav´ık, Generalized elementary functions, J. Math. Anal. Appl. 411 (2014), 838–852. Up to small details, the contents of these papers are essentially identical to Chapters 2–10 of the thesis; all references are collected at the end of the text. Chapter 1 provides some introductory remarks on generalized ordinary differential equations and Kurzweil integration theory. I am indebted to Stefanˇ Schwabik, Milan Tvrdý,and Jaroslav Kurzweil for introducing me to the beautiful subject of Kurzweil integration and generalized differential equations. I appreciate their valuable feedback, as well as the friendly atmosphere at the seminar on differential equations and integration theory held at the Institute of Mathematics, Academy of Sciences of the Czech Republic. I thank Jaqueline Godoy Mesquita, MárciaFederson, Giselle Antunes Monteiro (coauthors of several chapters of this thesis), Petr Stehl´ık,and Stan Wagon (my other coauthors) for a fruitful collaboration. Finally, I am grateful to my friends and colleagues at Charles University in Prague, above all to Jindˇrich Beˇcváˇr;it was a pleasure to serve as an assistant professor under his direction.

2 Contents

1 Generalized ordinary diﬀerential equations 5

2 Dynamic equations on time scales and their relation to generalized ODEs 7 2.1 Introduction ...... 7 2.2 Kurzweil-Stieltjes integrals and ∆-integrals ...... 8 2.3 Main result ...... 11 2.4 Linear equations ...... 16 2.5 Continuous dependence on a parameter ...... 17 2.6 Stability ...... 19

3 Periodic averaging theorems for various types of equations 23 3.1 Introduction ...... 23 3.2 Averaging for generalized ordinary differential equations ...... 23 3.3 Ordinary differential equations with impulses ...... 28 3.4 Dynamic equations on time scales ...... 31 3.5 Functional differential equations ...... 33

4 Measure functional differential equations and functional dynamic equations 38 4.1 Introduction ...... 38 4.2 Preliminaries ...... 39 4.3 Measure functional differential equations and generalized ordinary differential equations . . 39 4.4 Functional dynamic equations on time scales ...... 49 4.5 Existence-uniqueness theorems ...... 52 4.6 Continuous dependence results ...... 55 4.7 Periodic averaging theorems ...... 59

5 Basic results for functional diﬀerential and dynamic equations involving impulses 62 5.1 Introduction ...... 62 5.2 Preliminaries ...... 63 5.3 Impulsive measure functional diﬀerential equations ...... 65 5.4 Integration on time scales ...... 68 5.5 Impulsive functional dynamic equations on time scales ...... 71 5.6 Existence-uniqueness theorems ...... 74 5.7 Continuous dependence results ...... 76 5.8 Periodic averaging theorems ...... 80

6 Measure functional differential equations with infinite delay 84 6.1 Introduction ...... 84 6.2 Phase space description ...... 85 6.3 Relation to generalized ordinary differential equations ...... 88

3 7 Diﬀerentiability of solutions with respect to initial conditions and parameters 101 7.1 Introduction ...... 101 7.2 Generalized diﬀerential equations ...... 101 7.3 Linear equations ...... 103 7.4 Main results ...... 106 7.5 Relation to other types of equations ...... 112

8 Linear measure functional differential equations with infinite delay 117 8.1 Introduction ...... 117 8.2 Axiomatic description of the phase space ...... 118 8.3 Kurzweil integration ...... 121 8.4 Linear measure functional differential equations and generalized linear ODEs ...... 122 8.5 Existence and uniqueness of solutions ...... 126 8.6 Continuous dependence theorems ...... 127 8.7 Application to functional differential equations with impulses ...... 130

9 Well-posedness results for abstract generalized differential equations 135 9.1 Introduction ...... 135 9.2 Preliminaries ...... 136 9.3 An Osgood-type existence theorem ...... 142 9.4 Continuous dependence ...... 146 9.5 Application to measure functional differential equations ...... 152 9.6 Application to functional differential equations with impulses ...... 158

10 Generalized elementary functions 162 10.1 Introduction ...... 162 10.2 Preliminaries ...... 163 10.3 Exponential function ...... 165 10.4 Hyperbolic and trigonometric functions ...... 170 10.5 Time scale elementary functions ...... 172 10.6 Conclusion ...... 176

4 Chapter 1

Generalized ordinary diﬀerential equations

The concept of a generalized ordinary differential equation was originally introduced by J. Kurzweil in [51] as a tool in the study of continuous dependence of solutions to ordinary differential equations of the usual form x0(t) = f(x(t), t). He observed that instead of dealing directly with the right-hand side f, it might be advantageous to work with the function F (x, t) = R t f(x, s) ds, i.e., the primitive to f. In this connection, t0 he also introduced an integral whose special case is the Kurzweil-Henstock integral. Gauge-type integrals are well known to specialists in integration theory, but they are also becoming more popular in the field of differential equations (see e.g. [10]). We start by recalling J. Kurzweil’s definition of the integral, and refer the reader to [52, 67] for more information about its properties. Given a function δ :[a, b] → R+ (called a gauge on [a, b]), a tagged partition of the interval [a, b] with division points a = t0 ≤ t1 ≤ · · · ≤ tk = b and tags τi ∈ [ti−1, ti], i ∈ {1, . . . , k}, is called δ-fine if

[ti−1, ti] ⊂ (τi − δ(τi), τi + δ(τi)), i ∈ {1, . . . , k}. Let X be a Banach space. A function U :[a, b] × [a, b] → X is called Kurzweil integrable on [a, b], if there is an element I ∈ X such that for every ε > 0, there is a function δ :[a, b] → R+ such that

k X (U(τi, ti) − U(τi, ti−1)) − I < ε i=1 R b for every δ-fine tagged partition of [a, b]. In this case, we define a DU(τ, t) = I. If a < b, we let R a R b R b b DU(τ, t) = − a DU(τ, t) provided the right-hand side exists; we also set a DU(τ, t) = 0 when a = b. An important special case of this definition is the Kurzweil-Stieltjes integral (also known as the Perron- Stieltjes integral) of a function f :[a, b] → X with respect to a function g :[a, b] → R, which corresponds R b to the choice U(τ, t) = f(τ)g(t). This integral will be denoted by a f(t) dg(t). Finally, if g(t) = t, R b i.e. for U(τ, t) = f(τ)t, we obtain the well-known Kurzweil-Henstock integral a f(t) dt, which generalizes the integrals of Riemann, Lebesgue and Newton. For more information about the Kurzweil-Stieltjes and Kurzweil-Henstock integrals, see [31, 55, 67, 74, 82]. We can now proceed to the definition of a generalized ordinary differential equation. Consider a set B ⊂ X, an interval I ⊂ R, and a function F : B × I → X. A generalized ordinary differential equation with the right-hand side F has the form dx = DF (x, t), t ∈ I, (1.0.1) dτ which is a shorthand notation for the integral equation

Z s2 x(s2) − x(s1) = DF (x(τ), t), [s1, s2] ⊂ I. (1.0.2) s1

5 In other words, a function x : I → B is a solution of (1.0.1) if and only if (1.0.2) is satisfied. The reader should keep in mind that (1.0.1) is a symbolic notation only and does not necessarily mean that x is differentiable. The next definition introduces classes of functions which occur frequently in the theory of generalized differential equations.

Deﬁnition 1.0.1. Assume that B ⊂ X, G = B × I, h1 : I → R, h2 : I → R are nondecreasing functions, and ω : [0, ∞) → R is a continuous increasing function with ω(0) = 0. The class F(G, h1, h2, ω) consists of all functions F : G → X satisfying the following conditions:

kF (x, t2) − F (x, t1)k ≤ h1(t2) − h1(t1), x ∈ B, [t1, t2] ⊂ I,

kF (x, t2) − F (x, t1) − F (y, t2) + F (y, t1)k ≤ ω(kx − yk)(h2(t2) − h2(t1)), x, y ∈ B, [t1, t2] ⊂ I.

For the rest of the thesis, let us make the following agreement: Whenever we write F ∈ F(G, h1, h2, ω), we assume that h1, h2, ω satisfy the assumptions listed in Deﬁnition 1.0.1, without mentioning all of them explicitly. In the special case when h1 and h2 are equal to the same function h, we write F(G, h, ω) instead of F(G, h1, h2, ω). Many authors focus solely on this special case. Indeed, in many situations, the important thing is the existence of a pair of functions such that F ∈ F(G, h1, h2, ω), while the particular values of h1 and h2 play no role. In this case, the assumption h1 = h2 presents no loss of generality, because we always have F(G, h1, h2, ω) ⊂ F(G, h1 + h2, h1 + h2, ω) = F(G, h1 + h2, ω).

Still, for certain purposes, it is useful to distinguish h1 and h2. In Chapter 9, this will enable us to provide a more reasonable lower bound for the length of the interval where a local solution is guaranteed to exist. Finally, in the special case when ω(r) = r, we use the simpler notation F(G, h) instead of F(G, h, ω). Under certain assumptions on the right-hand side (see [67, Theorem 5.14]), a classical ordinary differential equation of the form x0(t) = f(x(t), t), t ∈ I, is equivalent to the generalized ordinary differential equation dx = DF (x, t), t ∈ I, dτ where Z t F (x, t) = f(x, s) ds, t ∈ I, t0 and t0 is an arbitrary point in I. In this situation, the two conditions from Definition 1.0.1 reduce to

Z t2

f(x, s) ds ≤ h1(t2) − h1(t1), x ∈ B, [t1, t2] ⊂ I, (1.0.3) t1

Z t2

(f(x, s) − f(y, s)) ds ≤ ω(kx − yk)(h2(t2) − h2(t1)), x, y ∈ B, [t1, t2] ⊂ I. (1.0.4) t1 However, we emphasize that not every generalized differential equation is equivalent to a classical one; different choices of F might lead to other types of equations. It turns out that the theory of generalized differential equations is not only useful in the study of classical nonautonomous differential equations (see e.g. [3]), but also represents a suitable tool for the investigation of equations with discontinuous solutions. (For other approaches to equations with discontinuous solutions or right-hand sides, such as measure differential equations, equations in Filippov’s sense, distributional differential equations, or impulsive differential equations, see e.g. [13, 28, 37, 53]). In particular, generalized differential equations encompass other types of equations, such as equations with impulses, dynamic equations on time scales, functional differential equations with impulses, or measure functional differential equations. Hence, the existing theory of generalized equations is applicable to the above-mentioned types of equations. On the other hand, existing results for various types of equations provide inspiration for extending the theory of generalized differential equations. Both aspects are illustrated throughout this thesis.

6 Chapter 2

Dynamic equations on time scales and their relation to generalized ODEs

2.1 Introduction

In 1990, Stefan Hilger created the so-called time scale calculus in order to unify and extend the continuous as well as discrete calculus (see [38]). A time scale T is an arbitrary closed nonempty subset of the real line. For every point t ∈ T, t > inf T, we introduce the backward-jump operator ρ(t) = sup{s < t; s ∈ T}. If ρ(t) = t, we say that t is a left-dense point; otherwise, t is a left-scattered point. Similarly, for every t ∈ T, t < sup T, we define the forward-jump operator σ(t) = inf{s > t; s ∈ T}. If σ(t) = t, we say that t is a right-dense point; otherwise, t is a right scattered point. We also define the graininess µ(t) = σ(t) − t. A function f : T → R is called rd-continuous if it is regulated and continuous at all right-dense points. 0 R b Given a function f : T → R, the classical derivative f and integral a f(t) dt are replaced by the ∆ R b so-called ∆-derivative f and ∆-integral a f(t)∆t. The precise definitions of these concepts can be found in [8, 9, 38]. The important fact is that

 lim f(t)−f(s) if t is right-dense,  s→t t−s f ∆(t) =  f(σ(t))−f(t) σ(t)−t if t is right-scattered, provided that the ∆-derivative f ∆(t) exists. As in the classical calculus, the ∆-derivative and ∆-integral are in a certain sense inverse operations to each other. More precisely, we have the following analogue of the fundamental theorem of calculus:

• If f : T → R is a rd-continuous function, then the function Z t F (t) = f(s)∆s, t ∈ T a

is ∆-diﬀerentiable and F ∆ = f.

• If F : T → R is a ∆-diﬀerentiable function and f = F ∆ is rd-continuous, then Z b f(t)∆t = F (b) − F (a). a

7 When T = R, the ∆-derivative and ∆-integral coincide with the classical derivative and integral. For ∆ R b Pb−1 T = Z, we obtain the discrete calculus where f (t) = f(t + 1) − f(t) and a f(t)∆t = i=a f(i). Another interesting case is T = qZ = {qn; n ∈ Z}, where q ∈ (1, ∞); this choice leads to the quantum calculus. For other examples of time scales, see e.g. [8]. Equations of the form ∆ x (t) = f(x(t), t), t ∈ T, where x takes values in Rn (or in a general Banach space), are called dynamic equations. Both differential and difference equations represent special cases of dynamic equations. The main goal of this chapter is to show that under certain assumptions, dynamic equations on time scales can be transformed to generalized ordinary differential equations. We then illustrate the usefulness of this procedure and obtain some new results concerning the stability and continuous dependence on parameters for dynamic equations on time scales as corollaries of known results for generalized differential equations. This chapter extends the results from Stefanˇ Schwabik’s paper [68], where he showed that discrete systems of the form xk+1 = f(xk), k ∈ N, might be rewritten as generalized differential equations. We assume some basic familiarity with the time scale calculus and dynamic equations; for more information on this subject, see [8, 9, 38].

2.2 Kurzweil-Stieltjes integrals and ∆-integrals

We start by summarizing some basic properties of the Kurzweil-Stieltjes integral. For the next result, see Corollary 1.34 in [67]; the inequality follows easily from the deﬁnition of the integral.

Theorem 2.2.1. If f :[a, b] → Rn is a regulated function and g :[a, b] → R is a nondecreasing function, R b then the integral a f(s) dg(s) exists. Moreover, when kf(s)k ≤ C for every s ∈ [a, b], then

Z b

f(s) dg(s) ≤ C(g(b) − g(a)). a The following statement is a special case of Theorem 1.16 in [67].

Theorem 2.2.2. Let f :[a, b] → Rn and g :[a, b] → R be a pair of functions such that g is regulated and R b a f(s) dg(s) exists. Then the function Z t h(t) = f(s) dg(s), t ∈ [a, b], a is regulated and satisﬁes

h(t+) = h(t) + f(t)∆+g(t), t ∈ [a, b), h(t−) = h(t) − f(t)∆−g(t), t ∈ (a, b], where ∆+g(t) = g(t+) − g(t) and ∆−g(t) = g(t) − g(t−). The following dominated convergence theorem is a consequence of Corollary 1.32 in [67].

Theorem 2.2.3. Let g :[a, b] → R be a nondecreasing function. Consider a sequence of functions n R b fk :[a, b] → R , k ∈ N, such that a fk(t) dg(t) exists for every k ∈ N. Assume there exists a function R b m :[a, b] → R such that the integral a m(t) dg(t) exists, and such that

kfk(t)k ≤ m(t), t ∈ [a, b], k ∈ N.

8 R b If lim fk(t) = f(t) for every t ∈ [a, b], then f(t) dg(t) exists and k→∞ a

Z b Z b f(t) dg(t) = lim fk(t) dg(t). a k→∞ a Our next goal in this section is to show that the Riemann ∆-integral, which represents a time scale version of the classical Riemann integral, is in fact a special case of the Kurzweil-Stieltjes integral. We assume that the reader is familiar with the Riemann ∆-integral as described in Chapter 5 of [9].

Given a time scale T and a pair of numbers a, b ∈ T, the symbol [a, b]T will be used throughout this chapter to denote a compact interval in T, i.e. [a, b]T = {t ∈ T; a ≤ t ≤ b}. The open and half-open intervals are deﬁned in an similar way. On the other hand, [a, b] will be used to denote intervals on the real line, i.e. [a, b] = {t ∈ R; a ≤ t ≤ b}. This notational convention should help the reader to distinguish between ordinary and time scale intervals.

Given a real number t ≤ sup T, we deﬁne

∗ t = inf{s ∈ T; s ≥ t}.

Since T is a closed set, we have t ∈ T. Further, let (−∞, sup ] if sup < ∞, ∗ = T T T (−∞, ∞) otherwise.

Given a function f : T → Rn, we deﬁne a function f ∗ : T∗ → Rn by

∗ ∗ ∗ f (t) = f(t ), t ∈ T .

Similarly, given a set B ⊂ Rn and a function f : B × T → Rn, we deﬁne

∗ ∗ ∗ f (x, t) = f(x, t ), x ∈ B, t ∈ T .

Lemma 2.2.4. If f : T → Rn is a regulated function, then f ∗ : T∗ → Rn is also regulated. If f is left- continuous on T, then f ∗ is left-continuous on T∗. If f is right-continuous on T, then f ∗ is right-continuous at right-dense points of T. ∗ ∗ Proof. Let us calculate limt→t0− f (t), where t0 ∈ T . If t0 ∈ T and it is left-dense, then lim f ∗(t) = lim f(t). t→t0− t→t0−

If t0 ∈ T and it is left-scattered, then

∗ ∗ lim f (t) = f(t0) = f (t0). t→t0−

Finally, if t0 ∈/ T, then ∗ ∗ ∗ lim f (t) = f(t0) = f (t0). t→t0− ∗ ∗ ∗ Now consider limt→t0+ f (t), where t0 ∈ T and t0 < sup T . If t0 ∈ T and it is right-dense, then lim f ∗(t) = lim f(t). t→t0+ t→t0+

If t0 ∈ T and it is right-scattered, then

∗ lim f (t) = f(σ(t0)). t→t0+

Finally, if t0 ∈/ T, then ∗ ∗ ∗ lim f (t) = f(t0) = f (t0). t→t0+

9 Theorem 2.2.5. Let f : T → Rn be an rd-continuous function. Choose an arbitrary a ∈ T and deﬁne Z t F1(t) = f(s) ∆s, t ∈ T, a Z t ∗ ∗ F2(t) = f (s) dg(s), t ∈ T , a ∗ ∗ ∗ where g(s) = s for every s ∈ T . Then F2 = F1 .

Proof. Note that the functions F1 and F2 are well-defined; indeed, the Riemann ∆-integral in the definition of F1 exists because f is rd-continuous, and the Kurzweil-Stieltjes integral in the definition of F2 exists because f ∗ is regulated (use Lemma 2.2.4 and the fact that every rd-continuous function is regulated) and g is nondecreasing. To complete the proof, it is sufficient to prove the following two statements:

(1) F1(t) = F2(t) for every t ∈ T.

(2) If t ∈ T and s = sup{u ∈ T; u < t}, then F2 is constant on (s, t]. We start with the second statement, which is easy to prove: If u, v ∈ (s, t] and u < v, then Z v ∗ F2(v) − F2(u) = f (s) dg(s) = 0, u where the last equality follows from the definition of the Kurzweil-Stieltjes integral and the fact that g is constant on [u, v]. To prove the first statement, we note that F1(a) = F2(a) = 0 and it is thus sufficient to show that ∆ ∆ F1 (t) = F2 (t) for every t ∈ T (any two functions with the same ∆-derivative differ only by a constant). ∆ It follows from the properties of the Riemann ∆-integral that F1 (t) = f(t), and it remains to calculate ∆ F2 . When t is a right-dense point, then f is continuous at t and lim f ∗(s) = f ∗(t) = f(t) s→t (see Lemma 2.2.4). Therefore, given an arbitrary ε > 0, there is a δ > 0 such that kf ∗(s) − f(t)k < ε ∞ whenever |s − t| < δ. Now, consider a sequence of time scale points {tk}k=1 such that limk→∞ tk = t. We can find a k0 ∈ N such that |tk − t| < δ whenever k ≥ k0. Thus for every k ≥ k0 we have Z tk F2(tk) − F2(t) 1 ∗ − f(t) = f (s) dg(s) − f(t) tk − t tk − t t

Z tk 1 ∗ g(tk) − g(t) = (f (s) − f(t)) dg(s) ≤ ε = ε, tk − t t tk − t since g(tk) = tk and g(t) = t. It follows that F (t ) − F (t) lim 2 k 2 = f(t), k→∞ tk − t ∆ i.e. F2 (t) = f(t). On the contrary, when t is a right-scattered point, we have + F2(σ(t)) = F2(t+) = F2(t) + f(t)∆ g(t), where the ﬁrst equality follows from the fact that F2 is constant on (t, σ(t)] and the second equality is a consequence of Theorem 2.2.2. But ∆+g(t) = g(t+) − g(t) = σ(t) − t = µ(t), and it follows that F (σ(t)) − F (t) F ∆(t) = 2 2 = f(t). 2 µ(t)

R t ∗ Note that the integral a f (s) dg(s) in the deﬁnition of F2 also exists as the Lebesgue-Stieltjes integral. We have chosen the Kurzweil-Stieltjes integral simply because it seems to be more natural in the context of generalized diﬀerential equations. On the other hand, the integral need not exist as the Riemann-Stieltjes integral, because there might be points where both f ∗ and g are discontinuous (this is in fact a typical behavior at right-scattered points).

10 2.3 Main result

This section describes the correspondence between dynamic equations on time scales and generalized ordinary diﬀerential equations. To obtain a reasonable theory, we restrict ourselves to dynamic equations whose right-hand sides are functions satisfying the conditions given below. Consider a set B ⊂ Rn and a function f : B ×T → Rn. Let us introduce the following three conditions:

(C1) f is rd-continuous, i.e. the function t 7→ f(x(t), t) is rd-continuous whenever x : T → B is a continuous function.

(C2) There exists a regulated function m : T → R such that kf(x, t)k ≤ m(t) for every x ∈ B and t ∈ T.

(C3) There exists a continuous increasing function ω : [0, ∞) → R with ω(0) = 0 and a regulated function l : T → R such that kf(x, t) − f(y, t)k ≤ l(t)ω(kx − yk)

for every x, y ∈ B and t ∈ T.

Lemma 2.3.1. Consider a set B ⊂ Rn. Assume that f : B × T → Rn satisﬁes conditions (C1)–(C3). ∗ ∗ Deﬁne g(s) = s for every s ∈ T . Then for arbitrary t0 ∈ T, the function Z t ∗ ∗ F (x, t) = f (x, s) dg(s), x ∈ B, t ∈ T , t0 is an element of F(G, h, ω), where G = B × T∗ and Z t h(t) = (l∗(s) + m∗(s)) dg(s). t0

Proof. For a fixed x ∈ B, condition (C1) implies that the function t 7→ f(x, t) is rd-continuous on T, and therefore t 7→ f ∗(x, t) is regulated on T∗. The function g is nondecreasing, and thus the Kurzweil-Stieltjes integral exists and F is well-defined. Similarly, the functions l∗ and m∗ are regulated, and thus the integral in the definition of h exists. When t1 ≤ t2, we have

Z t2 Z t2 Z t2 ∗ ∗ ∗ kF (x, t2) − F (x, t1)k = f (x, s) dg(s) ≤ kf (x, s)k dg(s) ≤ m (s) dg(s) ≤ h(t2) − h(t1) t1 t1 t1 and kF (x, t2) − F (x, t1) − F (y, t2) + F (y, t1)k

Z t2 Z t2 Z t2 ∗ ∗ ∗ ∗ = f (x, s) dg(s) − f (y, s) dg(s) = (f (x, s) − f (y, s)) dg(s) t1 t1 t1

Z t2 Z t2 ∗ ∗ ∗ ≤ kf (x, s) − f (y, s)k dg(s) ≤ ω(kx − yk) l (s) dg(s) ≤ ω(kx − yk)(h(t2) − h(t1)). t1 t1

The case t1 > t2 is similar and is left to the reader.

Before proceeding to the main result, we need the following auxiliary lemmas.

Lemma 2.3.2. Let G = B ×[α, β], where B ⊂ Rn. Consider a function F : G → Rn such that t 7→ F (x, t) is regulated on I for every x ∈ B. If x :[α, β] → B is a step function, i.e. if there exists a partition

α = s0 < s1 < ··· < sk = β

n and vectors c1, . . . , ck ∈ R such that

x(s) = ci for every s ∈ (si−1, si),

11 then k Z β X DF (x(τ), t) = F (cj, sj−) − F (cj, sj−1+) α j=1 ! +F (x(sj−1), sj−1+) − F (x(sj−1), sj−1) + F (x(sj), sj) − F (x(sj), sj−) .

Proof. See the proof of Corollary 3.15 in [67].

Lemma 2.3.3. Consider a set B ⊂ Rn and a function f : B × T → Rn such that t 7→ f(x, t) is regulated ∗ ∗ on T for every x ∈ B. Deﬁne g(t) = t for every t ∈ T , choose an arbitrary t0 ∈ T and let Z t ∗ ∗ F (x, t) = f (x, s) dg(s), x ∈ B, t ∈ T . t0

If [α, β] ⊂ T∗ and x :[α, β] → B is a step function, then Z β Z β DF (x(τ), t) = f ∗(x(t), t) dg(t). α α

Proof. By Lemma 2.2.4, the function t 7→ f ∗(x, t) is regulated on T∗ for every x ∈ B, and thus the integral in the deﬁnition of F exists. Given a step function x :[α, β] → B, there exists a partition

α = s0 < s1 < ··· < sk = β

n and vectors c1, . . . , ck ∈ R such that

x(s) = ci for every s ∈ (si−1, si).

The function t 7→ F (x, t) is regulated by Theorem 2.2.2 and we may use Lemma 2.3.2 to obtain

Z β k X DF (x(τ), t) = lim F (cj, sj − ε) − F (cj, sj−1 + ε) (2.3.1) ε→0+ α j=1 k X + lim F (x(sj−1), sj−1 + ε) − F (x(sj−1), sj−1) (2.3.2) ε→0+ j=1 k X + lim F (x(sj), sj) − F (x(sj), sj − ε) . (2.3.3) ε→0+ j=1

Now, since x is a step function, it is not diﬃcult to see that t 7→ f ∗(x(t), t) is regulated, and thus the R β ∗ integral α f (x(t), t) dg(t) exists. In this case, we obtain

k Z β X Z sj f ∗(x(t), t) dg(t) = f ∗(x(s), s) dg(s) α j=1 sj−1

k X Z sj−1+ε = lim f ∗(x(s), s) dg(s) (2.3.4) ε→0+ j=1 sj−1 k X Z sj −ε + lim f ∗(x(s), s) dg(s) (2.3.5) ε→0+ j=1 sj−1+ε k X Z sj + lim f ∗(x(s), s) dg(s). (2.3.6) ε→0+ j=1 sj −ε

12 Obviously, for every i ∈ {1, . . . , k} we have

Z sj −ε ∗ F (cj, sj − ε) − F (cj, sj−1 + ε) = f (x(t), t) dg(t), sj−1+ε and thus (2.3.1) equals (2.3.5). Theorem 2.2.2 gives

lim F (x(sj−1), sj−1 + ε) − F (x(sj−1), sj−1) ε→0+

Z sj−1+ε ∗ ∗ + = lim f (x(sj−1), s) dg(s) = f (x(sj−1), sj−1)∆ g(sj−1) ε→0+ sj−1 and Z sj−1+ε ∗ ∗ + lim f (x(s), s) dg(s) = f (x(sj−1), sj−1)∆ g(sj−1) ε→0+ sj−1 and thus (2.3.2) equals (2.3.4). Finally,

Z sj ∗ ∗ − lim F (x(sj), sj) − F (x(sj), sj − ε) = lim f (x(sj), s) dg(s) = f (x(sj), sj)∆ g(sj) ε→0+ ε→0+ sj −ε and Z sj ∗ ∗ − lim f (x(s), s) dg(s) = f (x(sj), sj)∆ g(sj) ε→0+ sj −ε and thus (2.3.3) equals (2.3.6).

Lemma 2.3.4. Let G = B × [α, β], where B ⊂ Rn. Assume that F : G → Rn is an element of F(G, h, ω) for some h and ω. If x :[α, β] → B is a pointwise limit of step functions xk :[α, β] → B, then Z β Z β DF (x(τ), t) = lim DF (xk(τ), t) α k→∞ α Proof. See Corollary 3.15 in [67].

It is a known fact that given a regulated function x :[α, β] → Rn and a number ε > 0, there is a step function ϕ :[α, β] → Rn such that kx(t) − ϕ(t)k < ε for every t ∈ [α, β]. In other words, every regulated function is the uniform limit of step functions. Now suppose there is a set B ⊂ Rn such that x(t) ∈ B for every t ∈ [α, β]; we wish to show that x can be uniformly approximated by step functions with values in B. Assume that the above mentioned step function ϕ is constant on intervals (si−1, si), where α = s0 < s1 < ··· < sk = β is a partition of [α, β]. Now, choose a ti ∈ (si−1, si) for every i ∈ {1, . . . , k} and construct a function ψ :[α, β] → B as follows: x(s ) for s = s , ψ(s) = i i x(ti) for s ∈ (si−1, si).

It is clear that ψ is a step function. Moreover, when s ∈ (si−1, si), then

kx(s) − ψ(s)k ≤ kx(s) − ϕ(s)k + kϕ(s) − ψ(s)k=kx(s) − ϕ(s)k + kϕ(ti) − x(ti)k< 2ε. It follows that kx(t) − ψ(t)k < 2ε for every t ∈ [α, β]. This means that x can be uniformly approximated by step functions with values in B. Lemma 2.3.5. Let B ⊂ Rn and assume that f : B × T → Rn satisﬁes conditions (C1)–(C3). Deﬁne ∗ ∗ g(t) = t for every t ∈ T , choose an arbitrary t0 ∈ T and let Z t ∗ ∗ F (x, t) = f (x, s) dg(s), x ∈ B, t ∈ T . t0 If [α, β] ⊂ T∗ and x :[α, β] → B is a regulated function, then Z β Z β DF (x(τ), t) = f ∗(x(t), t) dg(t). α α

13 Proof. Given a regulated function x :[α, β] → B, there is a sequence of step functions xk :[α, β] → B which converge uniformly to x on [α, β]. Condition (C3) implies

∗ ∗ ∗ kf (xk(t), t) − f (x(t), t)k ≤ l(t )ω(kxk(t) − x(t)k), k ∈ N, t ∈ [α, β], ∗ ∗ and thus limk→∞ f (xk(t), t) = f (x(t), t) for every t ∈ [α, β]. Using first Lemma 2.3.4 (the assumptions are satisfied by Lemma 2.3.1) and then Lemma 2.3.3, we obtain Z β Z β Z β Z β ∗ ∗ DF (x(τ), t) = lim DF (xk(τ), t) = lim f (xk(t), t) dg(t) = f (x(t), t) dg(t), α k→∞ α k→∞ α α where the last equality follows from Theorem 2.2.3 (note that kf ∗(x(t), t)k ≤ m∗(t), m∗ is regulated, and thus the assumptions are satisfied).

Lemma 2.3.6. If x :[α, β] → Rn is a solution of the generalized ordinary diﬀerential equation dx = DF (x, t), dτ then lim(x(u) − F (x(t), u) + F (x(t), t)) = x(t) u→t for every t ∈ [α, β]. Proof. See Proposition 3.6 in [67]. Now we have all prerequisites necessary for the proof of the main result.

Theorem 2.3.7. Let X ⊂ Rn and assume that f : X × T → Rn is such that conditions (C1)–(C3) are satisfied on every set G = B × [α, β]T, where B ⊂ X is bounded. If x : T → X is a solution of ∆ x (t) = f(x(t), t), t ∈ T, (2.3.7) then x∗ : T∗ → X is a solution of dx = DF (x, t), t ∈ ∗, (2.3.8) dτ T where Z t ∗ ∗ F (x, t) = f (x, s) dg(s), x ∈ X, t ∈ T , t0 ∗ ∗ ∗ t0 ∈ T, and g(s) = s for every s ∈ T . Moreover, every solution y : T → X of (2.3.8) has the form y = x∗, where x : T → X is a solution of (2.3.7). Proof. Choose an arbitrary a ∈ T. If x : T → Rn is a solution of (2.3.7), then Z s x(s) = x(a) + f(x(t), t)∆t, s ∈ T. a It follows that Z s∗ ∗ ∗ x(s ) = x(a) + f(x(t), t)∆t, s ∈ T . a Using Theorem 2.2.5, we rewrite the last equation as Z s ∗ ∗ ∗ ∗ ∗ x (s) = x (a) + f (x (t), t) dg(t), s ∈ T . (2.3.9) a Let I be a compact interval in T containing both a and s∗. Since x is continuous, it is bounded on I. Therefore it is possible to find a bounded set B ⊂ X such that x(t) ∈ B for every t ∈ I. The function f satisfies conditions (C1)–(C3) on B × I and we may use Lemma 2.3.5 to replace the last equality by Z s ∗ ∗ ∗ ∗ x (s) = x (a) + DF (x (τ), t), s ∈ T , a

14 which means that x∗ is a solution of the generalized equation (2.3.8). To prove the second assertion, let y : T∗ → X be a solution of (2.3.8). Then Z s ∗ y(s) = y(a) + DF (y(τ), t), s ∈ T . a

∗ Fix an arbitrary s ∈ T and let [α, β]T be a time scale interval such that a, s ∈ [α, β]. For every τ ∈ [α, β), Lemma 2.3.6 implies that

y(τ) = lim (y(u) − F (y(τ), u) + F (y(τ), τ)) = u→τ+ Z u = lim y(u) − f ∗(y(τ), s) dg(s) = lim y(u) − f ∗(y(τ), τ)∆+g(τ) , u→τ+ τ u→τ+ and therefore limu→τ+ y(u) exists. Similarly, for every τ ∈ (α, β], we have Z u y(τ) = lim (y(u) − F (y(τ), u) + F (y(τ), τ)) = lim y(u) − f ∗(y(τ), s) dg(s) = u→τ− u→τ− τ

= lim y(u) + f ∗(y(τ), τ)∆−g(τ) = lim y(u), u→τ− u→τ− because g is a left-continuous function. Since y is regulated and therefore bounded on [α, β], it is possible to ﬁnd a bounded set B ⊂ X such that y(t) ∈ B for every t ∈ [α, β]. The function f satisﬁes conditions (C1)–

(C3) on B × [α, β]T and Lemma 2.3.1 guarantees that the function F is an element of F(B × [α, β], h, ω). Using Lemma 2.3.5 again, we obtain Z s ∗ ∗ y(s) = y(a) + f (y(t), t) dg(t), s ∈ T . a

But the right-hand side is constant on every interval (s, t], where t ∈ T and s = sup{u ∈ T; u < t} (see the argument in the proof of Theorem 2.2.5). Thus y = x∗, where x : T → B is the restriction of y to T. This implies (2.3.9), and consequently also (2.3.7) (note that, according to Theorem 2.2.2, x is a rd-continuous function). From now on, the letter g will always denote the function g(s) = s∗. Let us pause for a moment to discuss conditions (C1)–(C3). Condition (C1) is fairly common in the theory of dynamic equations; its purpose is to ensure that the integral equation Z s x(s) = x(a) + f(x(t), t)∆t a can be diﬀerentiated to obtain x∆(t) = f(x(t), t). In a more general setting, we could focus our interest on the integral equation only; in this case, it would be suﬃcient to assume that t 7→ f(x(t), t) is regulated whenever x : T → X is a regulated function. Conditions (C2)–(C3) were used to prove that the function

Z t ∗ ∗ F (x, t) = f (x, s) dg(s), x ∈ B, t ∈ T , t0 is an element of F(G, h, ω). Condition (C3) represents a generalization of Lipschitz-continuity with respect to x, which corresponds to the special case ω(r) = r and l(t) = L. Again, this is a fairly standard condition. In many cases, the function f is defined on Rn × T and has continuous partial derivatives with respect to x1, . . . , xn. Since we require the conditions to be satisfied only for sets of the form G = B × [α, β]T with B ⊂ X bounded, it is easy to see that both (C2) and (C3) are satisfied. Moreover, if B = {x ∈ Rn; kxk ≤ r}, condition (C3) can be weakened; in this case, it is sufficient to assume that x 7→ f(x, t) is continuous for every t ∈ T (see Chapter 5 of [67], which describes the case T = R, but the same reasoning can be used for a general time scale).

15 2.4 Linear equations

To illustrate Theorem 2.3.7 on a simple example, consider the linear dynamic equation

∆ x (t) = a(t)x(t) + h(t), t ∈ T, (2.4.1) where a : T → Rn×n and h : T → Rn are rd-continuous functions (we use the symbol Rn×n to denote the set of all n × n matrices). It is easy to see that the function f(x, t) = a(t)x + h(t) satisﬁes conditions

(C1)–(C3) on every set G = B × [α, β]T, where B ⊂ X is bounded. To obtain the corresponding generalized diﬀerential equation, we choose an arbitrary τ0 ∈ T and let Z t F (x, t) = (a∗(s)x + h∗(s)) dg(s) = A(t)x + H(t), τ0 where A(t) = R t a∗(s) dg(s) and H(t) = R t h∗(s) dg(s). Now, Theorem 2.3.7 says that if x : → X is a τ0 τ0 T solution of (2.4.1), then the function x∗ is a solution of the linear generalized diﬀerential equation

dx = DA(t)x + H(t), t ∈ ∗. (2.4.2) dτ T

Conversely, every solution of this generalized equation has the form x∗, where x : T → Rn is a solution of the dynamic equation (2.4.1). The monograph [67] contains a fairly complete theory of linear generalized equations. For example, the equation (2.4.2) is known to have a unique solution satisfying x(t0) = x0, whenever

I − (A(t) − A(t−)) and I + A(t+) − A(t) are regular for every t. (2.4.3)

Let us rephrase this condition in the language of equation (2.4.1); Theorem 2.2.2 gives

A(t+) = A(t) + a∗(t)∆+g(t), A(t−) = A(t) − a∗(t)∆−g(t).

First, if t ∈ T∗\T, then ∆+g(t) = ∆−g(t) = 0 and (2.4.3) is satisﬁed. Next, assume t ∈ T. Since g is a left- continuous function, we always have ∆−g(t) = 0 and therefore I − (A(t) − A(t−)) = I is regular. Finally, if t is a right-dense point, then ∆+g(t) = 0 and I + A(t+) − A(t) = I is regular; if t is right-scattered, then ∆+g(t) = µ(t) and I + A(t+) − A(t) is regular if and only if I + a(t)µ(t) is regular. The last condition is called regressivity and is well known in the theory of linear dynamic equations. Let us mention one more result: Under assumption (2.4.3), there exists a function U : T∗ × T∗ → Rn×n such that the function x(t) = U(t, t0)x0 represents the unique solution of the homogeneous equation

dx = D(A(t)x), t ∈ ∗, x(t ) = x . dτ T 0 0 The function U has the following properties:

1. U(t, t) = I for every t ∈ T∗,

2. U(t, s) = U(t, r)U(r, s) for every r, s, t ∈ T∗,

3. U(t+, s) = (I + A(t+) − A(t))U(t, s) for every s, t ∈ T∗, 4. U(t, s) is always a regular matrix and U(t, s)−1 = U(s, t).

We already know that for t ∈ T, the third condition might be written as U(t+, s) = (I +a(t)µ(t))U(t, s). It is easy to recognize that the restriction of U to T × T is the matrix exponential function, which is denoted by ea(t, t0) in the book [8].

16 In his paper [69], S.ˇ Schwabik presents the following interesting construction of the function U: He Qb n×n deﬁnes the Perron product integral a(I + dA(s)) as a matrix P ∈ R such that for every ε > 0, there is a function δ :[a, b] → R+ which satisﬁes

1 Y I + A(αj) − A(αj−1) − P < ε

j=k for every δ–ﬁne partition with division points a = α0 ≤ α1 ≤ · · · ≤ αk = b and tags τi ∈ [αi−1, αi], i = 1, . . . , k. Now, the function U is obtained by considering the product integral as a function of its upper bound, i.e. t Y U(t, t0) = (I + dA(s)).

t0 A similar result for linear systems on time scales is given in the paper [80], which shows that the matrix n×n exponential function ea(t, t0) corresponding to an rd-continuous function a : T → R can be written in the form t Y ea(t, t0) = (I + a(s)∆s),

t0 where the symbol on the right-hand side stands for the product ∆-integral. Thus our considerations imply that t t Y Y (I + a(s)∆s) = (I + dA(s)), t0, t ∈ T t0 t0 for every rd-continuous function a : → n×n and A(t) = R t a∗(s) dg(s). T R τ0

2.5 Continuous dependence on a parameter

In this section, we use two known results concerning continuous dependence of generalized equations on parameters to obtain new theorems about dynamic equations on time scales. The symbol Br will be used n to denote the open ball {x ∈ R ; kxk < r} and Br stands for the corresponding closed ball. n Theorem 2.5.1. Let c > 0, G = Bc × [α, β], and consider a sequence of functions Fk : G → R , k ∈ N0, such that lim Fk(x, t) = F0(x, t), x ∈ Bc, t ∈ [α, β]. k→∞

Assume there exist functions h and ω such that Fk ∈ F(G, h, ω) for every k ∈ N0. Finally, suppose there exist a function x :[α, β] → Bc and a sequence of functions xk :[α, β] → Bc such that dx k = DF (x , t), t ∈ [α, β], k ∈ , dτ k k N

lim xk(s) = x(s), s ∈ [α, β]. k→∞ Then dx = DF (x, t), t ∈ [α, β]. dτ 0 Proof. See Theorem 8.2 in [67].

n Theorem 2.5.2. Consider a sequence of functions fk : Bc × [α, β]T → R , k ∈ N0. Assume there exist functions l, m and ω such that each function fk, k ∈ N0, satisﬁes conditions (C1)–(C3). Suppose that Z t Z t lim fk(x, s)∆s = f0(x, s)∆s (2.5.1) k→∞ α α

17 for every x ∈ Bc and t ∈ [α, β]T. Finally, suppose there exist a function x :[α, β]T → Bc and a sequence of functions xk :[α, β]T → Bc, k ∈ N, such that ∆ xk (t) = fk(xk(t), t), t ∈ [α, β]T,

lim xk(s) = x(s), s ∈ [α, β] . k→∞ T Then ∆ x (t) = f0(x(t), t), t ∈ [α, β]T.

Proof. Let G = Bc × [α, β] and Z t ∗ Fk(x, t) = fk (x, s) dg(s), x ∈ Bc, t ∈ [α, β], k ∈ N0. α Equation (2.5.1) together with Theorem 2.2.5 imply

lim Fk(x, t) = F0(x, t), x ∈ Bc, t ∈ [α, β]. k→∞

It follows from Lemma 2.3.1 that Fk ∈ F(G, h, ω) for every k ∈ N0. It is clear that ∗ ∗ lim xk(s) = x (s), s ∈ [α, β]. k→∞ Theorem 2.3.7 implies dx∗ k = DF (x∗, t), t ∈ [α, β], k ∈ . dτ k k N Thus the assumptions of Theorem 2.5.1 are satisﬁed and dx∗ = DF (x∗, t), t ∈ [α, β]. dτ 0 The function x∗ is bounded and it follows from Theorem 2.3.7 that

∆ x (t) = f0(x(t), t), t ∈ [α, β]T. Given a function F : B × I → Rn and an interval [α, β] ⊂ I, a solution x :[α, β] → B of the generalized diﬀerential equation dx = DF (x, t) (2.5.2) dτ is said to be unique if every other solution y :[α, γ] → B of (2.5.2) such that x(α) = y(α) satisﬁes x(t) = y(t) for every t ∈ [α, γ] ∩ [α, β].

n Theorem 2.5.3. Let c > 0, G = Bc × [α, β], and consider a sequence of functions Fk : G → R , k ∈ N0, such that lim Fk(x, t) = F0(x, t), x ∈ Bc, t ∈ [α, β]. k→∞

Assume there exist a left-continuous function h and a function ω such that Fk ∈ F(G, h, ω) for every k ∈ N0. Let x :[α, β] → Bc be a unique solution of dx = DF (x, t). dτ 0

Finally, assume there exists a ρ > 0 such that ky − x(s)k < ρ implies y ∈ Bc whenever s ∈ [α, β] (i.e., a ρ-neighborhood of x is contained in Bc). Then, given an arbitrary sequence of n-dimensional vectors ∞ {yk}k=1 such that limk→∞ yk = x(α), there is a k0 ∈ N and a sequence of functions xk :[α, β] → Bc, k ≥ k0, which satisfy dx k = DF (x , t), t ∈ [α, β], x (α) = y , dτ k k k k

lim xk(s) = x(s), s ∈ [α, β]. k→∞

18 Proof. See Theorem 8.6 in [67].

In analogy with the previous case, we say that a solution x :[α, β]T → B of the dynamic equation x∆(t) = f(x(t), t) is unique if every other solution y :[α, γ] → B such that x(α) = y(α) satisﬁes x(t) = y(t) for every t ∈ [α, γ] ∩ [α, β].

n Theorem 2.5.4. Consider a sequence of functions fk : Bc × [α, β]T → R , k ∈ N0. Assume there exist functions l, m, ω such that each function fk, k ∈ N0, satisﬁes conditions (C1)–(C3). Suppose that Z t Z t lim fk(x, s)∆s = f0(x, s)∆s (2.5.3) k→∞ α α for every x ∈ Bc and t ∈ [α, β]T. Let x :[α, β]T → Bc be a unique solution of

∆ x (t) = f0(x(t), t).

Finally, assume there exists a ρ > 0 such that ky − x(s)k < ρ implies y ∈ Bc whenever s ∈ [α, β]T. ∞ Then, given an arbitrary sequence of n-dimensional vectors {yk}k=1 such that limk→∞ yk = x(α), there is a k0 ∈ N and a sequence of functions xk :[α, β]T → Bc, k ≥ k0, which satisfy

∆ xk (t) = fk(xk(t), t), t ∈ [α, β]T, xk(α) = yk,

lim xk(s) = x(s), s ∈ [α, β] . k→∞ T Proof. Using the same reasoning as in the proof of Theorem 2.5.2, we construct a sequence of functions ∞ R t ∗ {Fk} deﬁned on G = Bc × [α, β]. All these functions are elements of F(G, h, ω) with h(t) = (l (s) + k=0 t0 m∗(s)) dg(s); note that g(t) = t∗ is a left-continuous function, and thus h is left-continuous according to Theorem 2.2.2. It follows from Theorem 2.3.7 that x∗ is a unique solution of

dx∗ = DF (x∗, t), t ∈ [α, β]. dτ 0 The proof is ﬁnished by applying Theorem 2.5.3.

Let us note that Theorem 4.11 in [67] states that if the function F is an element of F(G, h, ω) with ω such that Z u dr lim = ∞ (2.5.4) v→0+ v ω(r) for every u > 0, then every solution x :[α, β] → Bc of the generalized equation dx = DF (x, t) dτ is unique. Now, it is often the case that the function f is Lipschitz-continuous with respect to x on Bc × [α, β], and thus f satisﬁes conditions (C1)–(C3) with l(t) = L and ω(r) = r. In this case, we see that (2.5.4) is true and therefore every solution is unique.

2.6 Stability

The dynamic equation x∆(t) = f(x(t), t) has the trivial solution x ≡ 0 if and only if f(0, t) = 0 for every t ∈ T. The present section is devoted to the investigation of stability of this trivial solution. The problem of stability has already been considered in a number of papers, see e.g. [40, 48, 49]. However, the theorem which will be obtained in this section describes two slightly diﬀerent types of stability.

19 Consider a set I ⊂ R and a function f : I → Rn. Given a ﬁnite set of points

D = {t0, t1, . . . , tk} ⊂ I such that t0 ≤ t1 ≤ · · · ≤ tk, let k X v(f, D) = kf(ti) − f(ti−1)k. i=1 The variation of f over I is deﬁned as

var f(t) = sup v(f, D), t∈I D where the supremum ranges over all finite subsets D of I. Note that when I is an interval on the real line, then we obtain the usual variation of a function over an interval, but our slightly more general definition permits us to calculate the variation of a function defined on a time scale interval [a, b]T. n Lemma 2.6.1. Given an arbitrary function f :[a, b]T → R , we have var f(t) = var f ∗(t). t∈[a,b] t∈[a,b] T

∗ ∗ ∗ ∗ Proof. The statement follows from the fact that if D = {t0, t1, . . . , tk} ⊂ [a, b], then D = {t0, t1, . . . , tk} ⊂ ∗ ∗ [a, b]T and v(f, D) = v(f ,D ). We start with a Lyapunov-type theorem for the generalized equation dx = DF (x, t). dτ

Note that this equation has the trivial solution x ≡ 0 on an interval I ⊂ R if and only if F (0, t1) = F (0, t2) for each pair t1, t2 ∈ I.

n Theorem 2.6.2. Let c > 0, t0 ∈ R and G = Bc × [t0, ∞). Consider a function F : G → R , which is an element of F(G, h, ω) and F (0, t1) = F (0, t2) for every t1, t2 ≥ t0. Assume there exists a number a ∈ (0, c) and a function V :[t0, ∞) × Ba → R with the following properties:

(V1) t 7→ V (t, x) is left-continuous for every x ∈ Ba.

(V2) There exists a continuous increasing function b : [0, ∞) → R such that b(ρ) = 0 if and only if ρ = 0 and V (t, x) ≥ b(kxk) for every t ∈ [t0, ∞) and x ∈ Ba.

(V3) V (t, 0) = 0 for every t ∈ [t0, ∞).

(V4) There exists a constant K > 0 such that kV (t, x) − V (t, y)k ≤ Kkx − yk for every t ∈ [t0, ∞) and x, y ∈ Ba. (V5) t 7→ V (t, x(t)) is nonincreasing along every solution x of the generalized equation dx = DF (x, t). dτ

Then the following statements are true:

(1) For every ε > 0, there is a δ > 0 such that if α ≥ t0 and y :[α, β] → Bc is a left-continuous function with bounded variation which satisﬁes ky(α)k < δ and Z s var y(s) − DF (y(τ), t) < δ, s∈[α,β] α then ky(t)k < ε for every t ∈ [α, β].

20 (2) For every ε > 0, there is a δ > 0 such that if P :[α, β] → Bc is a left-continuous function with

var P (s) < δ, s∈[α,β]

then an arbitrary function y :[α, β] → Bc which is a solution of dx = DF (x, t) + P (t) dτ and ky(α)k < δ satisﬁes ky(t)k < ε for every t ∈ [α, β].

Proof. See Theorem 10.8 and Theorem 10.13 in [67].

The statement (1) is called variational stability; it says that functions which are initially small, and which are “almost solutions” of the given generalized equation, are close to zero in the whole interval. The statement (2) is called stability with respect to perturbations; it says that functions which are initially small, and which are solutions of a generalized equation with a small perturbation term, are again close to zero in the whole interval. We now proceed to a similar theorem concerning dynamic equations on time scales.

n Theorem 2.6.3. Let c > 0 and t0 ∈ T. Consider a function f : Bc × [t0, ∞)T → R which satisﬁes conditions (C1)–(C3) and f(0, t) = 0 for every t ∈ [t0, ∞)T. Assume there exists a number a ∈ (0, c) and function V :[t0, ∞)T × Ba → R with the following properties:

(V1) t 7→ V (t, x) is left-continuous for every x ∈ Ba.

(V2) There exists a continuous increasing function b : [0, ∞) → R such that b(ρ) = 0 if and only if ρ = 0 and V (t, x) ≥ b(kxk) for every t ∈ [t0, ∞)T and x ∈ Ba.

(V3) V (t, 0) = 0 for every t ∈ [t0, ∞)T.

(V4) There exists a constant K > 0 such that kV (t, x) − V (t, y)k ≤ Kkx − yk for every t ∈ [t0, ∞)T and x, y ∈ Ba. (V5) t 7→ V (t, x(t)) is nonincreasing along every solution x of the dynamic equation

x∆(t) = f(x(t), t).

Then the following statements are true:

(1) For every ε > 0, there is a δ > 0 such that if α ≥ t0 and y :[α, β]T → Bc is a left-continuous function with bounded variation which satisﬁes ky(α)k < δ and

Z s var y(s) − f(y(t), t) ∆t < δ, s∈[α,β] T α

then ky(t)k < ε for every t ∈ [α, β]T.

(2) For every ε > 0, there is a δ > 0 such that if p :[α, β]T → Bc is an rd-continuous function and Z β kp(t)k ∆t < δ, α

then every function y :[α, β]T → Bc such that ky(α)k < δ and

∆ y (t) = f(y(t), t) + p(t), t ∈ [α, β]T

satisﬁes ky(t)k < ε for every t ∈ [α, β]T.

21 Proof. It is suﬃcient to apply Theorem 2.6.2 to the functions

Z t F (x, t) = f ∗(x, s) dg(s), t0 V ∗(t, x) = V (t∗, x). To prove (1), note that Z s var y(s) − f(y(t), t) ∆t < δ s∈[α,β] T α implies Z s var y∗(s) − DF (y∗(τ), t) < δ s∈[α,β] α (this follows from Lemma 2.6.1, Theorem 2.2.5, and Lemma 2.3.5). To prove (2), note that

∆ y (t) = f(y(t), t) + p(t), t ∈ [α, β]T implies dx = DF (x, t) + P (t) dτ R t ∗ with P (t) = α p (s) dg(s), and that Z β kp(t)k ∆t < δ α implies Z s var p(t) ∆t < δ s∈[α,β] T α (this is easy to see from the deﬁnition of variation) and consequently

Z s var P (s) = var p∗(t) dg(s) < δ s∈[α,β] s∈[α,β] α (this follows from Lemma 2.6.1 and Theorem 2.2.5).

22 Chapter 3

Periodic averaging theorems for various types of equations

3.1 Introduction

Classical averaging theorems for ordinary diﬀerential equations are concerned with the initial-value problem

0 2 x (t) = εf(t, x(t)) + ε g(t, x(t), ε), x(t0) = x0, where ε > 0 is a small parameter. Assume that f is T -periodic in the first argument. Then, according to the periodic averaging theorem, we can obtain a good approximation of this initial-value problem by neglecting the ε2-term and taking the average of f with respect to t. In other words, we consider the autonomous differential equation 0 y (t) = εf0(y(t)), y(t0) = x0, where 1 Z t0+T f0(y) = f(t, y) dt. T t0 Different proofs of the periodic averaging theorem can be found e.g. in [65, 66, 83]; these monographs also include many applications. In this chapter, we derive a periodic averaging theorem for generalized ordinary differential equations. We then show that the classical averaging theorem (even with the possibility of including impulses) is a simple corollary of our theorem. As a second application, we obtain a periodic averaging theorem for dynamic equations on time scales. In the final section, we derive a periodic averaging theorem for a large class of functional differential equations (these equations are studied in greater detail in Chapter 4, where they are called measure functional differential equations).

3.2 Averaging for generalized ordinary diﬀerential equations

We start by deriving a periodic averaging theorem for generalized ordinary diﬀerential equations. A basic source in the theory of generalized ODEs is the book [67]. It is known that an ordinary diﬀerential equation x0(t) = f(x(t), t) is equivalent to the generalized equation

dx = DF (x, t), dτ where F (x, t) = R t f(x, s) ds. However, generalized equations include many other types of equations such t0 as impulsive equations, functional diﬀerential equations, or dynamic equations on time scales.

23 Without loss of generality, we can always assume that the right-hand side of a generalized equation satisﬁes F (x, 0) = 0 for every x ∈ B. Otherwise, we let

F˜(x, t) = F (x, t) − F (x, 0), x ∈ B, t ∈ I, and consider the equation dx = DF˜(x, t). dτ Then we have F˜(x, 0) = 0 for every x ∈ B, and it follows from the deﬁnition of the Kurzweil integral that the new equation has the same set of solutions as the original one. The following existence theorem is proved in [67, Corollary 1.34]. The inequality follows easily from the deﬁnition of the Kurzweil-Stieltjes integral.

Theorem 3.2.1. If f :[a, b] → Rn is a regulated function and g :[a, b] → R is a nondecreasing function, R b then the integral a f(s) dg(s) exists. Moreover,

Z b Z b

f(s) dg(s) ≤ kf(s)k dg(s). a a The following lemma combines two statements from [67] (see Lemma 3.9 and Corollary 3.15).

Lemma 3.2.2. Let B ⊂ Rn, Ω = B × [a, b]. Assume that F :Ω → Rn belongs to the class F(Ω, h). If R b x :[a, b] → B is a regulated function, then the integral a DF (x(τ), t) exists and

Z b

DF (x(τ), t) ≤ h(b) − h(a). a We also need the following theorem, which can be found in [67, Lemma 3.12].

Lemma 3.2.3. Let B ⊂ Rn, Ω = B × [a, b]. Assume that F :Ω → Rn belongs to the class F(Ω, h). Then every solution x :[α, β] → B of the generalized ordinary diﬀerential equation dx = DF (x, t) dτ is a regulated function. The following inequality will be useful in the proof of the averaging theorem.

Lemma 3.2.4. Let B ⊂ Rn, Ω = B × [a, b]. Assume that F :Ω → Rn belongs to the class F(Ω, h). If x, y :[a, b] → B are regulated functions, then

Z b Z b

D[F (x(τ), t) − F (y(τ), t)] ≤ kx(t) − y(t)k dh(t). a a Proof. The Kurzweil-Stieltjes integral on the right-hand side exists, because h is nondecreasing and kx−yk m is regulated. For an arbitrary partition (τi, [si−1, si])i=1 of [a, b], we have

m X (F (x(τi), si) − F (x(τi), si−1) − F (y(τi), si) + F (y(τi), si−1)) ≤ i=1

m m X X kF (x(τi), si) − F (x(τi), si−1) − F (y(τi), si) + F (y(τi), si−1)k ≤ kx(τi) − y(τi)k (h(si) − h(si−1)). i=1 i=1 m Now, given an ε > 0, there is a partition (τi, [si−1, si])i=1 such that

Z b m X D[F (x(τ), t) − F (y(τ), t)] − (F (x(τi), si) − F (x(τi), si−1) − F (y(τi), si) + F (y(τi), si−1)) < ε a i=1

24 and

Z b m X kx(t) − y(t)k dh(t) − kx(τi) − y(τi)k (h(si) − h(si−1)) < ε. a i=1 It follows that

Z b m X D[F (x(τ), t) − F (y(τ), t)] ≤ (F (x(τi), si) − F (x(τi), si−1) − F (y(τi), si) + F (y(τi), si−1)) a i=1

Z b m X + D[F (x(τ), t) − F (y(τ), t)] − (F (x(τi), si) − F (x(τi), si−1) − F (y(τi), si) + F (y(τi), si−1)) a i=1

m X < kx(τi) − y(τi)k (h(si) − h(si−1)) + ε i=1

m Z b Z b X ≤ kx(τi) − y(τi)k (h(si) − h(si−1)) − kx(t) − y(t)k dh(t) + kx(t) − y(t)k dh(t) + ε i=1 a a

Z b < 2ε + kx(t) − y(t)k dh(t). a This proves the statement since ε can be arbitrarily small.

The following theorem represents an analogue of Gronwall’s inequality for the Kurzweil-Stieltjes integral; the proof can be found in [67, Corollary 1.43].

Theorem 3.2.5. Let h :[a, b] → [0, ∞) be a nondecreasing left-continuous function, k > 0, l ≥ 0. Assume that ψ :[a, b] → [0, ∞) is bounded and satisﬁes

Z ξ ψ(ξ) ≤ k + l ψ(τ) dh(τ), ξ ∈ [a, b]. a

Then ψ(ξ) ≤ kel(h(ξ)−h(a)) for every ξ ∈ [a, b].

We proceed to our main result, which is a periodic averaging theorem for generalized ordinary diﬀeren- tial equations. The proof is inspired by a proof of the classical averaging theorem for ordinary diﬀerential equations given in [66] (see Theorem 2.8.1 and Lemma 2.8.2 there).

n n Theorem 3.2.6. Let B ⊂ R , Ω = B × [0, ∞), ε0 > 0, L > 0. Consider functions F :Ω → R and n G :Ω × (0, ε0] → R which satisfy the following conditions:

1. There exist nondecreasing left-continuous functions h1, h2 : [0, ∞) → [0, ∞) such that F belongs to the class F(Ω, h1), and for every ﬁxed ε ∈ (0, ε0], the function (x, t) 7→ G(x, t, ε) belongs to the class F(Ω, h2).

2. F (x, 0) = 0 and G(x, 0, ε) = 0 for every x ∈ B, ε ∈ (0, ε0].

3. There exists a number T > 0 and a bounded Lipschitz-continuous function M : B → Rn such that F (x, t + T ) − F (x, t) = M(x) for every x ∈ B and t ∈ [0, ∞).

4. There exists a constant α > 0 such that h1(iT ) − h1((i − 1)T ) ≤ α for every i ∈ N.

5. There exists a constant β > 0 such that |h2(t)/t| ≤ β for every t ≥ L/ε0.

25 Let F (x, T ) F (x) = , x ∈ B. 0 T

Suppose that for every ε ∈ (0, ε0], the initial-value problems dx = D εF (x, t) + ε2G(x, t, ε) , x(0) = x (ε), dτ 0

0 y (t) = εF0(y(t)), y(0) = y0(ε) L have solutions xε, yε : 0, ε → B. If there is a constant J > 0 such that kx0(ε) − y0(ε)k ≤ Jε for every ε ∈ (0, ε0], then there exists a constant K > 0 such that

kxε(t) − yε(t)k ≤ Kε

L for every ε ∈ (0, ε0] and t ∈ 0, ε . Proof. If x ∈ B, then

F (x, T ) F (x, T ) − F (x, 0) kM(x)k m kF0(x)k = = = ≤ , T T T T where m is a bound for M. Let l be a Lipschitz constant for M. The function H : B × [0, ∞) → Rn given by F (x, T ) H(x, t) = F (x)t = t 0 T satisﬁes 1 1 m kH(x, s ) − H(x, s )k = kF (x, T )s − F (x, T )s k = kF (x, T )k(s − s ) ≤ (s − s ) 2 1 T 2 1 T 2 1 T 2 1 and 1 kH(x, s ) − H(x, s ) − H(y, s ) + H(y, s )k = kF (x, T )s − F (x, T )s − F (y, T )s + F (y, T )s k 2 1 2 1 T 2 1 2 1 1 1 l = kF (x, T ) − F (y, T )k(s − s ) = kM(x) − M(y)k(s − s ) ≤ kx − yk(s − s ) T 2 1 T 2 1 T 2 1 for every x, y ∈ B and every s1, s2 ∈ [0, ∞), s1 ≤ s2. It follows that H belongs to the class F(Ω, h3), where h3(t) = (m + l)t/T . For every t ∈ [0, L/ε], we have

Z t Z t 2 xε(t) = x0(ε) + ε DF (xε(τ), s) + ε DG(xε(τ), s, ε), 0 0 Z t Z t yε(t) = y0(ε) + ε F0(yε(τ)) dτ = yε(0) + ε D[F0(yε(τ))s]. 0 0 Consequently,

Z t Z t Z t 2 kxε(t) − yε(t)k = x0(ε) − y0(ε) + ε DF (xε(τ), s) + ε DG(xε(τ), s, ε) − ε D[F0(yε(τ))s] ≤ 0 0 0 Z t Z t Z t 2 ≤ Jε+ε D[F (xε(τ), s) − F (yε(τ), s)] +ε D[F (yε(τ), s) − F0(yε(τ))s] +ε DG(xε(τ), s, ε) . 0 0 0 According to Lemma 3.2.2, we have the estimate

Z t 2 2 2 h2(L/ε) ε DG(xε(τ), s, ε) ≤ ε (h2(t) − h2(0)) ≤ ε h2(L/ε) = εL ≤ εLβ. 0 L/ε

26 Also, it follows from Lemma 3.2.4 that Z t Z t

D[F (xε(τ), s) − F (yε(τ), s)] ≤ kxε(s) − yε(s)k dh1(s). 0 0 Let p be the largest integer such that pT ≤ t. Then

p Z t X Z iT Z t D[F (yε(τ), s) − F0(yε(τ))s] = D[F (yε(τ), s) − F0(yε(τ))s] + D[F (yε(τ), s) − F0(yε(τ))s] 0 i=1 (i−1)T pT For every i ∈ {1, . . . , p}, we obtain

Z iT Z iT D[F (yε(τ), s) − F0(yε(τ))s] = D[F (yε(τ), s) − F (yε(iT ), s)] (i−1)T (i−1)T

Z iT Z iT − D[F0(yε(τ))s − F0(yε(iT ))s] + D[F (yε(iT ), s) − F0(yε(iT ))s]. (i−1)T (i−1)T We estimate the ﬁrst integral as follows:

Z iT Z iT

D[F (yε(τ), s) − F (yε(iT ), s)] ≤ kyε(s) − yε(iT )k dh1(s) (i−1)T (i−1)T

0 Since yε satisﬁes yε(t) = εF0(yε(t)), the mean value theorem gives m ky (s) − y (iT )k ≤ ε (iT − s) ≤ εm, s ∈ [(i − 1)T, iT ], ε ε T and consequently Z iT kyε(s) − yε(iT )k dh1(s) ≤ εm(h1(iT ) − h1((i − 1)T )) ≤ εmα. (i−1)T The same procedure applied to the second integral gives

Z iT

D[F0(yε(τ))s − F0(yε(iT ))s] ≤ εm(h3(iT ) − h3((i − 1)T )) ≤ εm(m + l). (i−1)T The third integral is zero, because for an arbitrary y ∈ B, we have Z iT D[F (y, s) − F0(y)s] = F (y, iT ) − F (y, (i − 1)T ) − F0(y)T = M(y) − F (y, T ) = 0. (i−1)T Since pT ≤ L/ε, we obtain

p X Z iT Lmα m(m + l)L D[F (y (τ), s) − F (y (τ))s] ≤ pεmα + pεm(m + l) ≤ + . ε 0 ε T T i=1 (i−1)T Finally, the following estimate is a consequence of Lemma 3.2.2: Z t Z t Z t

D[F (yε(τ), s) − F0(yε(τ))s] ≤ DF (yε(τ), s) + D[F0(yε(τ))s] pT pT pT

≤ h1(t) − h1(pT ) + h3(t) − h3(pT ) ≤ h1(pT + T ) − h1(pT ) + h3(pT + T ) − h3(pT ) ≤ α + m + l By combining the previous inequalities, we obtain Z t

D[F (yε(τ), s) − F0(yε(τ))s] ≤ K, 0

27 where K is a certain constant. It follows that Z t kxε(t) − yε(t)k ≤ ε kxε(s) − yε(s)k dh1(s) + ε(J + K + Lβ). 0

Since xε is a regulated function (we have used Lemma 3.2.3) and yε is a continuous functions, both of them must be bounded and we can apply Gronwall’s inequality from Theorem 3.2.5 to obtain

ε(h1(t)−h1(0)) kxε(t) − yε(t)k ≤ e ε(J + K + Lβ). The proof is concluded by observing that

ε(h1(t) − h1(0)) ≤ ε(h1(L/ε) − h1(0)) ≤ ε(h1(dL/(εT )e T ) − h1(0)) L L L ≤ ε α ≤ ε + 1 α ≤ + ε α. εT εT T 0

3.3 Ordinary diﬀerential equations with impulses

We now use the theorem from the previous section to obtain a periodic averaging theorem for ordinary diﬀerential equations with impulses. Given a set B ⊂ Rn, a function f : B × [0, ∞) → Rn, an increasing n sequence of numbers 0 ≤ t1 < t2 < ··· , and a sequence of mappings Ii : B → R , i ∈ N, consider the impulsive diﬀerential equation

0 2 x (t) = εf(x(t), t) + ε g(x(t), t, ε), t ∈ [0, ∞)\{t1, t2,...},

+ ∆ x(ti) = εIi(x(ti)), i ∈ N, + where ∆ x(ti) = x(ti+) − x(ti). Since we are interested in deriving a periodic averaging theorem, we will assume that f is T -periodic in the second argument and that the impulses are periodic in the following sense: There exists a k ∈ N such that 0 ≤ t1 < t2 < ··· < tk < T and for every integer i > k, we have ti = ti−k + T , Ii = Ii−k. It is known (see Chapter 5 in [67]) that if f is a bounded function which is Lipschitz-continuous in the ﬁrst argument and continuous in the second argument, and if the impulse operators Ii are bounded and Lipschitz-continuous, then the impulsive diﬀerential equation

0 x (t) = f(x(t), t), t ∈ [0, ∞)\{t1, t2,...},

+ ∆ x(ti) = Ii(x(ti)), i ∈ N, is equivalent to a generalized ordinary diﬀerential equation with the right-hand side

∞ Z t X Z t X F (x, t) = f(x, s) ds + Ii(x) = f(x, s) ds + Ii(x)Hti (t), 0 0 i; 0≤ti v. n Theorem 3.3.1. Assume that B ⊂ R , Ω = B × [0, ∞), T > 0, ε0 > 0, L > 0. Consider functions n n f :Ω → R and g :Ω × (0, ε0] → R which are bounded, Lipschitz-continuous in the ﬁrst argument and continuous in the second argument. Moreover, let f be T -periodic in the second argument. Assume that n k ∈ N, 0 ≤ t1 < t2 < ··· < tk < T , and that Ii : B → R , i = 1, 2, . . . , k are bounded and Lipschitz- continuous functions. For every integer i > k, deﬁne ti and Ii by the recursive formulas ti = ti−k + T and Ii = Ii−k. Denote k 1 Z T 1 X f (x) = f(x, s) ds and I (x) = I (x) 0 T 0 T i 0 i=1 for every x ∈ B. Suppose that for every ε ∈ (0, ε0], the impulsive equation

0 2 x (t) = εf(x(t), t) + ε g(x(t), t, ε) t ∈ [0, ∞)\{t1, t2,...},

28 + ∆ x(ti) = εIi(x(ti)), i ∈ N, x(0) = x0(ε) and the ordinary diﬀerential equation

0 y (t) = ε(f0(y(t)) + I0(y(t))), y(0) = y0(ε)

L have solutions xε, yε : [0, ε ] → B. If there is a constant J > 0 such that kx0(ε) − y0(ε)k ≤ Jε for every ε ∈ (0, ε0], then there exists a constant K > 0 such that

kxε(t) − yε(t)k ≤ Kε

L for every ε ∈ (0, ε0] and t ∈ [0, ε ]. Proof. Let ∞ Z t X F (x, t) = f(x, s) ds + Ii(x)Hti (t), 0 i=1 Z t G(x, t, ε) = g(x, s, ε) ds. 0

Given an ε ∈ (0, ε0], the function xε satisﬁes dx ε = D[εF (x , t) + ε2G(x , t, ε)]. dτ ε ε According to the assumptions, there exists a constant C > 0 such that

kf(x, t)k ≤ C, kf(x, t) − f(y, t)k ≤ Ckx − yk for every x, y ∈ B, t ∈ [0, ∞), a constant D > 0 such that

kIi(x)k ≤ D, kIi(x) − Ii(y)k ≤ Dkx − yk for every x, y ∈ B and i ∈ N, and a constant N > 0 such that kg(x, t, ε)k ≤ N, kg(x, t, ε) − g(y, t, ε)k ≤ Nkx − yk for every x, y ∈ B, t ∈ [0, ∞), ε ∈ (0, ε0]. The function h1 : [0, ∞) → R given by

∞ X h1(t) = Ct + D Hti (t) i=1 is left-continuous and nondecreasing. If 0 ≤ u ≤ t, then

∞ Z t X kF (x, t) − F (x, u)k = f(x, s) ds + I (x)(H (t) − H (u)) ≤ i ti ti u i=1

∞ ∞ Z t X X ≤ kf(x, s)k ds + kIi(x)k(Hti (t) − Hti (u)) ≤ C(t − u) + D (Hti (t) − Hti (u)) = h1(t) − h1(u) u i=1 i=1 and

∞ Z t X kF (x, t) − F (x, u) − F (y, t) + F (y, u)k = (f(x, s) − f(y, s)) ds + (I (x) − I (y)) (H (t) − H (u)) i i ti ti u i=1

∞ Z t X ≤ kf(x, s) − f(y, s)k ds + kIi(x) − Ii(y)k (Hti (t) − Hti (u)) ≤ u i=1

29 ∞ ! X ≤ kx − yk C(t − u) + D (Hti (t) − Hti (u)) = kx − yk(h1(t) − h1(u)). i=1

It follows that F belongs to the class F(Ω, h1). Deﬁne h2 : [0, ∞) → R by h2(t) = Nt. If 0 ≤ u ≤ t, then

Z t

kG(x, t, ε) − G(x, u, ε)k = g(x, s, ε) ds ≤ N(t − u) = h2(t) − h2(u). u

Also, if 0 ≤ u ≤ t and x, y ∈ B, we have

Z t

kG(x, t, ε) − G(x, u, ε) − G(y, t, ε) + G(y, u, ε)k = (g(x, s, ε) − g(y, s, ε)) ds ≤ u

≤ Nkx − yk(t − u) = kx − yk(h2(t) − h2(u)).

Therefore, for every ﬁxed ε ∈ (0, ε0], the function (x, t) 7→ G(x, t, ε) belongs to the class F(Ω, h2). It is clear that F (x, 0) = 0 and G(x, 0, ε) = 0 for every x ∈ B. Moreover, for every t ≥ 0, the diﬀerence

Z t+T X Z T X F (x, t + T ) − F (x, t) = f(x, s) ds + Ii(x) = f(x, s) ds + Ii(x) t 0 i; t≤ti

Z T k X kM(x)k = kF (x, T ) − F (x, 0)k = f(x, s) ds + Ii(x) ≤ CT + kD 0 i=1

kM(x) − M(y)k = kF (x, T ) − F (y, T ) − F (x, 0) + F (y, 0)k =

Z T k Z T k X X = (f(x, s) − f(y, s)) ds + (Ii(x) − Ii(y)) ≤ kf(x, s) − f(y, s)k ds + kIi(x) − Ii(y)k 0 i=1 0 i=1

≤ CT kx − yk + kDkx − yk = kx − yk(CT + kD)

For every j ∈ N, we have

∞ ∞ X X h1(jT ) − h1((j − 1)T ) = CjT + D Hti (jT ) − C(j − 1)T − D Hti ((j − 1)T ) = i=1 i=1

X = CT + D 1 = CT + Dk.

i;(j−1)T ≤ti

Finally, note that |h2(t)/t| = N for every t > 0. We see that the assumptions of Theorem 3.2.6 are satisfied. To conclude the proof, it is now sufficient to define

k F (x, T ) 1 Z T 1 X F (x) = = f(x, s) ds + I (x) = f (x) + I (x) 0 T T T i 0 0 0 i=1 and apply Theorem 3.2.6.

30 3.4 Dynamic equations on time scales

In this section, we use Theorem 3.2.6 to derive a periodic averaging theorem for dynamic equations on time scales. We assume that the reader is familiar with the basic notions of time scales calculus as described in [8], and with integration on time scales as presented in [9]. According to Chapter 2, dynamic equations on time scales can be converted to generalized ordinary diﬀerential equations; let us recall the main ideas. Let T be a time scale. If t is a real number such that t ≤ sup T, let

∗ t = inf{s ∈ T; s ≥ t}.

Since T is a closed set, we have t∗ ∈ T. Further, let (−∞, sup ] if sup < ∞, ∗ = T T T (−∞, ∞) otherwise.

Given a function f : T → Rn, we deﬁne a function f ∗ : T∗ → Rn by

∗ ∗ ∗ f (t) = f(t ), t ∈ T . The following theorem, which is a special case of Theorem 2.3.7, describes a one-to-one correspondence between the solutions of a dynamic equation and the solutions of a certain generalized ordinary diﬀerential equation.

Theorem 3.4.1. Consider a bounded set B ⊂ Rn and a bounded Lipschitz-continuous function f : B×T → Rn. Moreover, assume that f is rd-continuous, i.e. the function t 7→ f(x(t), t) is rd-continuous whenever x : T → B is a continuous function. If x : T → B is a solution of

∆ x (t) = f(x(t), t), t ∈ T, (3.4.1) then x∗ : T∗ → B is a solution of generalized ordinary diﬀerential equation dx = DF (x, t), t ∈ ∗, (3.4.2) dτ T where Z t ∗ ∗ F (x, t) = f(x, s ) du(s), x ∈ B, t ∈ T , t0 ∗ ∗ ∗ t0 ∈ T is an arbitrary ﬁxed number, and u(s) = s for every s ∈ T . Conversely, every solution y : T → B of (3.4.2) has the form y = x∗ , where x : T → B is a solution of (3.4.1). We now proceed to the periodic averaging theorem for dynamic equations on time scales.

Deﬁnition 3.4.2. Let T > 0 be a real number. A time scale T is called T -periodic if t ∈ T implies t + T ∈ T and µ(t) = µ(t + T ). n Theorem 3.4.3. Let T be a T -periodic time scale, t0 ∈ T, ε0 > 0, L > 0, B ⊂ R bounded. Consider a pair n n of bounded Lipschitz-continuous functions f : B × [t0, ∞)T → R and g : B × [t0, ∞)T × (0, ε0] → R . Assume that f is T -periodic in the second variable, and that both f and g are rd-continuous. Deﬁne n f0 : B → R by 1 Z t0+T f0(x) = f(x, s)∆s, x ∈ B. T t0

Suppose that for every ε ∈ (0, ε0], the dynamic equation

∆ 2 x (t) = εf(x(t), t) + ε g(x(t), t, ε), x(t0) = x0(ε)

L has a solution xε :[t0, t0 + ε ]T → B, and the ordinary diﬀerential equation 0 y (t) = εf0(y(t)), y(t0) = y0(ε)

31 L has a solution yε :[t0, t0 + ε ] → B. If there is a constant J > 0 such that kx0(ε) − y0(ε)k ≤ Jε for every ε ∈ (0, ε0], then there exists a constant K > 0 such that

kxε(t) − yε(t)k ≤ Kε,

L for every ε ∈ (0, ε0] and t ∈ [t0, t0 + ε ]T.

Proof. Without loss of generality, we can assume that t0 = 0; otherwise, consider a shifted problem with ˜ ˜ the time scale T = {t − t0; t ∈ T} and right-hand side f(x, t) = f(x, t0 + t). According to the assumptions, there exist constants m, l > 0 such that

kf(x, t)k ≤ m, kg(x, t, ε)k ≤ m,

kf(x, t) − f(y, t)k ≤ lkx − yk, kg(x, t, ε) − g(y, t, ε)k ≤ lkx − yk

∗ for every x, y ∈ B, t ∈ [0, ∞)T, ε ∈ (0, ε0]. Let u(t) = t , h1(t) = h2(t) = (m + l)u(t), Z t F (x, t) = f(x, s∗) du(s), x ∈ B, t ∈ [0, ∞), 0

Z t G(x, t, ε) = g(x, s∗, ε) du(s), x ∈ B, t ∈ [0, ∞). 0

If 0 ≤ t1 ≤ t2 and x, y ∈ B, then

Z t2 ∗ kF (x, t2) − F (x, t1)k = f(x, s ) du(s) ≤ m(u(t2) − u(t1)) ≤ h1(t2) − h1(t1), t1

Z t2 ∗ ∗ kF (x, t2) − F (x, t1) − F (y, t2) + F (y, t1)k = (f(x, s ) − f(y, s )) du(s) ≤ t1

≤ lkx − yk(u(t2) − u(t1)) ≤ kx − yk(h1(t2) − h1(t1)).

It follows that F belongs to the class F(Ω, h1). Similarly, if 0 ≤ t1 ≤ t2 and x, y ∈ B, then

Z t2 ∗ kG(x, t2, ε) − G(x, t1, ε)k = g(x, s , ε) du(s) ≤ m(u(t2) − u(t1)) ≤ h2(t2) − h2(t1), t1

Z t2 ∗ ∗ kG(x, t2, ε) − G(x, t1, ε) − G(y, t2, ε) + G(y, t1, ε)k = (g(x, s , ε) − g(y, s , ε)) du(s) ≤ t1

≤ lkx − yk(u(t2) − u(t1)) ≤ kx − yk(h2(t2) − h2(t1)).

Therefore, for every ﬁxed ε ∈ (0, ε0], the function (x, t) 7→ G(x, t, ε) belongs to the class F(Ω, h2). It is clear that F (x, 0) = 0 and G(x, 0, ε) = 0. Since T is a T -periodic time scale, the function u is also T -periodic. The function f is T -periodic in the second argument and it follows that the diﬀerence

Z t+T Z T F (x, t + T ) − F (x, t) = f(x, s∗) du(s) = f(x, s∗) du(s) t 0 does not depend on t. The function M(x) = F (x, t + T ) − F (x, t) satisﬁes

Z T ∗ kM(x)k = f(x, s ) du(s) ≤ m(u(T ) − u(0)) = mT, 0

Z T ∗ ∗ kM(x) − M(y)k = (f(x, s ) − f(y, s )) du(s) ≤ lkx − yk(u(T ) − u(0)) = lkx − ykT, 0

32 i.e. M is a bounded Lipschitz-continuous function. For every i ∈ N, we have

h1(iT ) − h1((i − 1)T ) = (m + l)(u(iT ) − u((i − 1)T )) = (m + l)(iT − (i − 1)T ) = (m + l)T.

If t ≥ L/ε0, then

∗ h2(t) t t + T T T ε0 = (m + l) ≤ (m + l) = (m + l) 1 + ≤ (m + l) 1 + . t t t t L

Thus we have checked that all assumptions of Theorem 3.2.6 are satisﬁed. Moreover,

Z T F (x, T ) 1 ∗ F0(x) = = f(x, s ) du(s) = f0(x), T T 0 where the last equality follows from Theorem 2.2.5. By Theorem 3.4.1, for every ε ∈ (0, ε0], the function ∗ L xε :[t0, t0 + ε ] → B satisﬁes dx∗ ε = D[εF (x∗, t) + ε2G(x∗, t, ε)], x∗(0) = x (ε). dτ ε ε ε 0 According to Theorem 3.2.6, there exists a constant K > 0 such that

∗ kxε(t) − yε(t)k ≤ Kε

L for every t ∈ [0, ε ], which proves the theorem.

Note that in our Theorem 3.4.3, the averaged equation is an ordinary diﬀerential equation, while in a similar Theorem 9 obtained in [79], the averaged equation is a dynamic equation on the same time scale as the original equation.

3.5 Functional diﬀerential equations

Let r > 0 be a given number. The theory of functional diﬀerential equations is usually concerned with the initial-value problem 0 x (t) = f(xt, t), xt0 = φ, where xt is given by the formula xt(θ) = x(t + θ), θ ∈ [−r, 0]. The equivalent integral form is

Z t

x(t) = x(t0) + f (xs, s) ds, xt0 = φ. t0 We will focus on slightly more general problems of the form

Z t

x(t) = x(t0) + f (xs, s) dh(s), xt0 = φ, t0 where the Kurzweil-Stieltjes integral on the right-hand side is taken with respect to a nondecreasing function h. More precisely, we are interested in deriving a periodic averaging theorem for the equation

Z t Z t 2 x(t) = x(0) + ε f(xs, s) dh(s) + ε g(xs, s, ε) dh(s), x0 = φ. 0 0 Before proceeding to the averaging theorem, we need the following auxiliary lemma.

n Lemma 3.5.1. If y :[a − r, b] → R is a regulated function, then s 7→ kysk∞ is regulated on [a, b].

33 Proof. We will show that lims→s0− kysk∞ exists for every s0 ∈ (a, b]. The function y is regulated, and therefore satisﬁes the Cauchy condition at s0 − r and s0: Given an arbitrary ε > 0, there exists a δ ∈ (0, s0 − a) such that ky(u) − y(v)k < ε, u, v ∈ (s0 − r − δ, s0 − r), (3.5.1) and

ky(u) − y(v)k < ε, u, v ∈ (s0 − δ, s0). (3.5.2)

Now, consider a pair of numbers s1, s2 such that s0 − δ < s1 < s2 < s0. For every s ∈ [s1 − r, s2 − r], it follows from (3.5.1) that

ky(s)k ≤ ky(s2 − r)k + ε ≤ kys2 k∞ + ε.

It is also clear that ky(s)k ≤ kys2 k∞ for every s ∈ [s2 − r, s1]. Consequently, kys1 k∞ ≤ kys2 k∞ + ε.

Using (3.5.2) in a similar way, we obtain kys2 k∞ ≤ kys1 k∞ + ε. It follows that

kys1 k∞ − kys2 k∞ ≤ ε, s1, s2 ∈ (s0 − δ, s0),

i.e. the Cauchy condition for the existence of lims→s0− kysk∞ is satisﬁed. The existence of lims→s0+ kysk∞ for s0 ∈ [a, b) can be proved similarly.

The next proof of the periodic averaging theorem follows the same basic idea as the proof of Theorem 3.2.6. Certain details are inspired by the paper [22], which is devoted to nonperiodic averaging. Given a set B ⊂ Rn, we use the symbol G([a, b],B) to denote the set of all regulated functions f :[a, b] → B.

n Theorem 3.5.2. Let ε0 > 0, L > 0, B ⊂ R , X = G([−r, 0],B). Consider a pair of bounded functions n n f : X × [0, ∞) → R , g : X × [0, ∞) × (0, ε0] → R and a nondecreasing left-continuous function h : [0, ∞) → R such that the following conditions are satisﬁed: R b 1. The integral 0 f(yt, t) dh(t) exists for every b > 0 and y ∈ G([−r, b],B). 2. f is T -periodic in the second variable.

3. There is a constant α > 0 such that h(t + T ) − h(t) = α for every t ≥ 0.

4. There is a constant C > 0 such that for x, y ∈ X and t ∈ [0, ∞),

kf(x, t) − f(y, t)k ≤ C kx − yk∞ .

5. The integral 1 Z T f0(x) = f(x, s) dh(s) T 0 exists for every x ∈ X.

Let φ ∈ X. Suppose that for every ε ∈ (0, ε0], the initial-value problems

Z t Z t 2 x(t) = x(0) + ε f(xs, s) dh(s) + ε g(xs, s, ε) dh(s), x0 = φ, 0 0 Z t y(t) = y(0) + ε f0(ys) ds, y0 = φ 0

ε ε L have solutions x , y : −r, ε → B. Then there exists a constant J > 0 such that

kxε(t) − yε(t)k ≤ Jε

L for every ε ∈ (0, ε0] and t ∈ [0, ε ].

34 Proof. There is a constant M > 0 such that kf(x, t)k ≤ M and kg(x, t, ε)k ≤ M for every x ∈ X, t ∈ [0, ∞) and ε ∈ (0, ε0]. It follows that

1 Z T M Mα

kf0(x)k = f(x, s) dh(s) ≤ (h(T ) − h(0)) = T 0 T T

ε for every x ∈ X. Thus if ε ∈ (0, ε0], s, t ∈ [0, ∞), s ≥ t, the solution y satisﬁes

Z s+θ εM(s − t)α ε ε ε ky (s + θ) − y (t + θ)k = ε f0(yσ) dσ ≤ , θ ∈ [−r, 0], t+θ T

ε ε ε ε εM(s − t)α kys − yt k∞ = sup ky (s + θ) − y (t + θ)k ≤ . (3.5.3) θ∈[−r,0] T For every t ∈ [0, L/ε], we have

Z t Z t Z t ε ε ε 2 ε ε kx (t) − y (t)k = ε f(xs, s) dh(s) + ε g(xs, s, ε) dh(s) − ε f0(ys) ds ≤ 0 0 0 Z t Z t Z t Z t ε ε ε ε 2 ε ≤ ε (f(xs, s) − f(ys, s)) dh(s) + ε f(ys, s) dh(s) − f0(ys) ds + ε g(xs, s, ε) dh(s) ≤ 0 0 0 0 Z t Z t Z t ε ε ε ε 2 ≤ ε Ckxs − ysk∞ dh(s) + ε f (ys, s) dh(s) − f0 (ys) ds + ε M(h(t) − h(0)). (3.5.4) 0 0 0 R t ε ε (Note that the integral 0 Ckxs − ysk∞ dh(s) is guaranteed to exist by Lemma 3.5.1, while the existence R t ε of 0 f (ys, s) dh(s) follows from assumption 1.) First, we estimate the second term. Let p be the largest integer such that pT ≤ t. Then Z t Z t ε ε f (ys, s) dh(s) − f0 (ys) ds ≤ 0 0

p p Z iT Z iT Z iT X ε ε X ε ε ≤ (f(ys, s) − f(y(i−1)T , s)) dh(s) + f(y(i−1)T , s) dh(s) − f0(y(i−1)T ) ds i=1 (i−1)T i=1 (i−1)T (i−1)T p Z iT Z t Z t X ε ε ε ε + (f0(y ) − f0(y )) ds + f(y , s) dh(s) − f0(y ) ds . (i−1)T s s s i=1 (i−1)T pT pT For every i ∈ {1, 2, . . . , p} and every s ∈ [(i − 1)T, iT ], inequality (3.5.3) gives

Mεα(s − (i − 1)T ) kyε − yε k ≤ ≤ Mεα. s (i−1)T ∞ T

L Using this estimate together with the fact that pT ≤ ε , we obtain

p p X Z iT X CMLα2 (f(yε, s) − f(yε , s)) dh(s) ≤ CMεα(h(iT ) − h((i − 1)T )) = CMεα2p ≤ . s (i−1)T T i=1 (i−1)T i=1 When s ≥ t ≥ 0 and y ∈ G([−r, s],B), then

1 Z T C C

kf0(ys) − f0(yt)k = (f(ys, σ) − f(yt, σ)) dh(σ) ≤ kys − ytk∞(h(T ) − h(0)) = kys − ytk∞α. T 0 T T Thus p p Z iT Z iT X ε ε X ε ε (f0(ys) − f0(y(i−1)T )) ds ≤ f0(ys) − f0(y(i−1)T ) ds ≤ i=1 (i−1)T i=1 (i−1)T

35 p p C X Z iT C X MCLα2 ≤ α kyε − yε k ds ≤ α εMαT = εMCα2p ≤ . T s (i−1)T ∞ T T i=1 (i−1)T i=1 The fact that f is T -periodic in the second variable and the deﬁnition of f 0 imply

p Z iT Z iT X ε ε f(y(i−1)T , s) dh(s) − f0(y(i−1)T ) ds = i=1 (i−1)T (i−1)T

p Z T X ε ε = f(y(i−1)T , s) dh(s) − f0(y(i−1)T )T = 0. i=1 0 Finally, Z t Z t Z t Z t ε ε ε ε f(ys, s) dh(s) − f0(ys) ds ≤ f(ys, s) dh(s) + kf0(ys)k ds ≤ pT pT pT pT Mα Mα ≤ M(h(t) − h(pT )) + (t − pT ) ≤ M(h((p + 1)T ) − h(pT )) + T = Mα + Mα = 2Mα. T T By combination of the previous results, we obtain

Z t Z t 2 ε ε 2MCLα f(ys, s) dh(s) − f0(ys) ds ≤ + 2Mα. 0 0 T Denote the constant on the right-hand side by K. Returning back to inequality (3.5.4), we see that

Z t ε ε ε ε 2 kx (t) − y (t)k ≤ ε Ckxs − ysk∞ dh(s) + εK + ε M(h(t) − h(0)). 0

ε ε ε ε Let ψ(s) = supτ∈[0,s] kx (τ) − y (τ)k. Since x and y are regulated, it is not diﬃcult to see that ψ is also regulated and therefore Kurzweil-Stieltjes integrable with respect to the function h. For every u ∈ [0, t], we have Z u Z t kxε(u)−yε(u)k ≤ ε Cψ(s) dh(s)+εK +ε2M(h(u)−h(0)) ≤ ε Cψ(s) dh(s)+εK +ε2M(h(t)−h(0)). 0 0 Consequently, Z t ψ(t) ≤ ε Cψ(s) dh(s) + εK + ε2M(h(t) − h(0)). 0 Next, note that

L L L ε(h(t) − h(0)) ≤ ε(h(L/ε) − h(0)) ≤ ε(h(dL/(εT )e T ) − h(0)) ≤ ε α ≤ ε + 1 α ≤ + ε α. εT εT T 0

Thus Z t L ψ(t) ≤ ε Cψ(s) dh(s) + εK + εM + ε0 α. 0 T Gronwall’s inequality from Theorem 3.2.5 gives εC(h(t)−h(0)) L C L +ε α L ψ(t) ≤ e K + M + ε α ε ≤ e ( T 0) K + M + ε α ε. T 0 T 0

L C( T +ε0)α L It follows that if we let J = e K + M T + ε0 α , then kxε(t) − yε(t)k ≤ ψ(t) ≤ Jε

L for every ε ∈ (0, ε0] and t ∈ [0, ε ].

36 In the special case h(t) = t, we obtain a periodic averaging theorem for the usual type of functional diﬀerential equations. However, our theorem is much more general. The following example shows that it is applicable even to diﬀerence equations.

Example 3.5.3. Consider the function h(t) = dte and assume that x :[−r, ∞) → Rn satisﬁes Z t x(t) = x(0) + f(xs, s) dh(s), t ∈ [0, ∞). (3.5.5) 0 It follows from the properties of the Kurzweil-Stieltjes integral that for every integer k ≥ 0, the function x is constant on (k, k + 1], and x(k+) = x(k) + f(xk, k)(h(k+) − h(k)) = x(k) + f(xk, k). Using this observation, we see that a diﬀerence equation of the form

a(k + 1) − a(k) = F (k, a(k), a(k − 1), . . . , a(k − r)), k ∈ N0, is equivalent to the integral equation (3.5.5), where h(t) = dte and f(y, t) = F (dte, y(0), y(−1), . . . , y(−r)) for every t ≥ 0 and y ∈ G([−r, 0], Rn). Indeed, every solution x of this integral equation must be constant on (k, k + 1] and satisfy

x(k + 1) = x(k+) = x(k) + f(xk, k) = x(k) + F (k, x(k), x(k − 1), . . . , x(k − r)) for every integer k ≥ 0. Thus our averaging theorem is applicable to diﬀerence equations of the form

a(k + 1) − a(k) = εF (k, a(k), a(k − 1), . . . , a(k − r)), k ∈ N0, where ε ∈ (0, ε0] is a small parameter. Assuming that F is T -periodic in the ﬁrst argument (where T is a positive integer), the corresponding averaged equation has the form

Z t y(t) = y(0) + ε f0(ys) ds, 0 where the function f0 is given by

T −1 1 Z T 1 X Z i+1 f (y) = f(y, s) dh(s) = f(y, s) dh(s) = 0 T T 0 i=0 i

T −1 T −1 1 X 1 X = f(y, i) = F (i, y(0), y(−1), . . . , y(−r)) T T i=0 i=0 for every y ∈ G([−r, 0], Rn).

37 Chapter 4

Measure functional diﬀerential equations and functional dynamic equations

4.1 Introduction

Let r, σ > 0 be given numbers and t0 ∈ R. The classical theory of functional diﬀerential equations (see e.g. [36]) is concerned with problems of the form

0 x (t) = f(xt, t), t ∈ [t0, t0 + σ],

n n where f :Ω × [t0, t0 + σ] → R ,Ω ⊂ C([−r, 0], R ), and xt is given by xt(θ) = x(t + θ), θ ∈ [−r, 0], for every t ∈ [t0, t0 + σ]. The equivalent integral form is

Z t x(t) = x(t0) + f(xs, s) ds, t ∈ [t0, t0 + σ], t0 where the integral can be considered, for instance, in the sense of Riemann, Lebesgue or Henstock-Kurzweil. In this chapter, we focus our attention on more general problems of the form

Z t x(t) = x(t0) + f(xs, s) dg(s), t ∈ [t0, t0 + σ], (4.1.1) t0 where the integral on the right-hand side is the Kurzweil-Stieltjes integral with respect to a nondecreasing function g. We call these equations measure functional differential equations. While the theories of measure differential equations (including the more general abstract measure differential equations) and measure delay differential equations are well-developed (see e.g. [13, 16, 17, 46, 75]), the literature concerning measure functional differential equations is scarce. We start our exposition by describing the relation between measure functional differential equations and generalized ordinary differential equations. The idea of converting functional differential equations to generalized ordinary differential equations first appeared in the papers [45, 61] written by C. Imaz, F. Oliva, and Z. Vorel. It was later generalized by M. Federson and S.ˇ Schwabik in the paper [24], which is devoted to impulsive retarded functional differential equations. Using the existing theory of generalized ordinary differential equations, we obtain results on the existence and uniqueness of a solution and on the continuous dependence of a solution on parameters for measure functional differential equations. Both differential equations and difference equations are important special cases of measure differential equations. Another unification of continuous-time and discrete-time equations is provided by the theory of dynamic equations on time scales, which has its roots in the work of S. Hilger (see [38]), and became increasingly popular during the past two decades (see [8, 9] and the references there). In Chapter 2,

38 we established the relation between dynamic equations on time scales, measure differential equations, and generalized ordinary differential equations. In the present chapter, we extend this correspondence to functional dynamic equations on time scales and measure functional differential equations. Using this relation, we obtain theorems on the existence and uniqueness of a solution, continuous dependence of a solution on a parameter, and a periodic averaging theorem for functional dynamic equations on time scales. This chapter is organized as follows. In the second section, we recall two basic results concerning the Kurzweil-Stieltjes integral. In the third section, using some ideas from [24], we present the correspondence between measure functional differential equations and generalized ordinary differential equations. We also explain that impulsive functional differential equations represent a special case of measure functional differential equations. The fourth section is devoted to functional dynamic equations on time scales. In this section, we establish the relation between functional dynamic equations on time scales and measure functional differential equations. In the fifth section, we present theorems concerning existence and uniqueness of a solution. In the sixth section, we prove continuous dependence results. Finally, a periodic averaging theorem for functional dynamic equations on time scales is presented in the last section.

4.2 Preliminaries

A function f :[a, b] → X is called regulated, if the limits lim f(s) = f(t−), t ∈ (a, b], and lim f(s) = f(t+), t ∈ [a, b) s→t− s→t+ exist. The space of all regulated functions f :[a, b] → X will be denoted by G([a, b],X), and is a Banach space under the usual supremum norm kfk∞ = supa≤t≤b kf(t)k. The subspace of all continuous functions f :[a, b] → X will be denoted by C([a, b],X). A proof of the next result can be found in [67], Corollary 1.34; the inequalities follow directly from the deﬁnition of the Kurzweil-Stieltjes integral.

Theorem 4.2.1. If f :[a, b] → Rn is a regulated function and g :[a, b] → R is a nondecreasing function, R b then the integral a f dg exists and

Z b Z b

f(s) dg(s) ≤ kf(s)k dg(s) ≤ kfk∞(g(b) − g(a)). a a The following result, which describes the properties of the indeﬁnite Kurzweil-Stieltjes integral, is a special case of Theorem 1.16 in [67].

Theorem 4.2.2. Let f :[a, b] → Rn and g :[a, b] → R be a pair of functions such that g is regulated and R b a f dg exists. Then the function Z t h(t) = f(s) dg(s), t ∈ [a, b], a is regulated and satisﬁes h(t+) = h(t) + f(t)∆+g(t), t ∈ [a, b), h(t−) = h(t) − f(t)∆−g(t), t ∈ (a, b], where ∆+g(t) = g(t+) − g(t) and ∆−g(t) = g(t) − g(t−).

4.3 Measure functional diﬀerential equations and generalized ordinary diﬀerential equations

In the present section, our goal is to establish a correspondence between measure functional diﬀerential equations and generalized ordinary diﬀerential equations. We start by recalling the following simple property of solutions to generalized ODEs.

39 Lemma 4.3.1. Let X be a Banach space. Consider a set O ⊂ X, an interval [a, b] ⊂ R and a function F : O × [a, b] → X. If x :[a, b] → O is a solution of the generalized ordinary diﬀerential equation

dx = DF (x, t) dτ and F ∈ F(O × [a, b], h), then x is a regulated function.

n n Let O ⊂ G([t0 − r, t0 + σ], R ) and P = {yt; y ∈ O, t ∈ [t0, t0 + σ]} ⊂ G([−r, 0], R ). Consider n a nondecreasing function g :[t0, t0 + σ] → R and a function f : P × [t0, t0 + σ] → R . We will show that under certain assumptions, a measure functional diﬀerential equation of the form

Z t y(t) = y(t0) + f(ys, s) dg(s), t ∈ [t0, t0 + σ], (4.3.1) t0

n where the solution y :[t0 − r, t0 + σ] → R is supposed to be a regulated function, can be converted to a generalized ordinary differential equation of the form dx = DF (x, t), (4.3.2) dτ where x takes values in O, i.e. we transform the original measure functional differential equation, whose solution takes values in Rn, into a generalized ordinary equation, whose solution takes values in an infinite- dimensional Banach space. The right-hand side F of this generalized equation will be given by  0, t − r ≤ ϑ ≤ t ,  0 0 R ϑ f(xs, s) dg(s), t0 ≤ ϑ ≤ t ≤ t0 + σ, F (x, t)(ϑ) = t0 (4.3.3) R t  f(xs, s) dg(s), t ≤ ϑ ≤ t0 + σ t0 for every x ∈ O and t ∈ [t0, t0 + σ]. As we will show, the relation between the solution x of (4.3.2) and the solution y of (4.3.1) is described by ( y(ϑ), ϑ ∈ [t − r, t], x(t)(ϑ) = 0 y(t), ϑ ∈ [t, t0 + σ], where t ∈ [t0, t0 + σ]. The following property will be important for us because it ensures that if y ∈ O, then x(t) ∈ O for every t ∈ [t0, t0 + σ].

n Deﬁnition 4.3.2. Let O be a subset of G([t0 − r, t0 + σ], R ). We will say that O has the prolongation property, if for every y ∈ O and every t¯∈ [t0 − r, t0 + σ], the functiony ¯ given by ( y(t), t − r ≤ t ≤ t,¯ y¯(t) = 0 y(t¯), t¯ < t ≤ t0 + σ is also an element of O.

For example, let B be an arbitrary subset of Rn. Then both the set of all regulated functions f : [t0 − r, t0 + σ] → B and the set of all continuous functions f :[t0 − r, t0 + σ] → B have the prolongation property. n n Recall that O ⊂ G([t0 − r, t0 + σ], R ), P = {yt; y ∈ O, t ∈ [t0, t0 + σ]}, f : P × [t0, t0 + σ] → R , and g :[t0, t0 + σ] → R is nondecreasing. We introduce the following three conditions, which will be used throughout the rest of this chapter:

R t0+σ (A) The integral f(yt, t) dg(t) exists for every y ∈ O. t0

40 (B) There exists a constant M > 0 such that

kf(y, t)k ≤ M

for every y ∈ P and every t ∈ [t0, t0 + σ]. (C) There exists a constant L > 0 such that

kf(y, t) − f(z, t)k ≤ Lky − zk∞

for every y, z ∈ P and every t ∈ [t0, t0 + σ].

Before proceeding further, we need the following property of regulated functions.

n Lemma 4.3.3. If y :[t0 −r, t0 +σ] → R is a regulated function, then s 7→ kysk∞ is regulated on [t0, t0 +σ].

Proof. We will show that lims→s0− kysk∞ exists for every s0 ∈ (t0, t0 + σ]. The function y is regulated, and therefore satisﬁes the Cauchy condition at s0 − r and s0: Given an arbitrary ε > 0, there exists a δ ∈ (0, s0 − t0) such that

ky(u) − y(v)k < ε, u, v ∈ (s0 − r − δ, s0 − r), (4.3.4) and ky(u) − y(v)k < ε, u, v ∈ (s0 − δ, s0). (4.3.5)

Now, consider a pair of numbers s1, s2 such that s0 − δ < s1 < s2 < s0. For every s ∈ [s1 − r, s2 − r], it follows from (4.3.4) that

ky(s)k ≤ ky(s2 − r)k + ε ≤ kys2 k∞ + ε.

It is also clear that ky(s)k ≤ kys2 k∞ for every s ∈ [s2 − r, s1]. Consequently, kys1 k∞ ≤ kys2 k∞ + ε. Using

(4.3.5) in a similar way, we obtain kys2 k∞ ≤ kys1 k∞ + ε. It follows that

kys1 k∞ − kys2 k∞ ≤ ε, s1, s2 ∈ (s0 − δ, s0),

i.e. the Cauchy condition for the existence of lims→s0− kysk∞ is satisﬁed. The existence of lims→s0+ kysk∞ for s0 ∈ [t0, t0 + σ) can be proved similarly.

n Lemma 4.3.4. Let O ⊂ G([t0 − r, t0 + σ], R ) and P = {yt; y ∈ O, t ∈ [t0, t0 + σ]}. Assume that n g :[t0, t0 + σ] → R is a nondecreasing function and that f : P × [t0, t0 + σ] → R satisﬁes conditions (A), n (B), (C). Then the function F : O × [t0, t0 + σ] → G([t0 − r, t0 + σ], R ) given by (4.3.3) belongs to the class F(O × [t0, t0 + σ], h), where

h(t) = (L + M)(g(t) − g(t0)), t ∈ [t0, t0 + σ].

Proof. Condition (A) implies that the integrals in the deﬁnition of F exist. Given y ∈ O and t0 ≤ s1 < s2 ≤ t0 + σ, we see that

 0, t − r ≤ τ ≤ s ,  0 1  R τ F (y, s )(τ) − F (y, s )(τ) = f(ys, s) dg(s), s1 ≤ τ ≤ s2, 2 1 s1  R s2  f(ys, s) dg(s), s2 ≤ τ ≤ t0 + σ. s1

Hence for an arbitrary y ∈ O and for t0 ≤ s1 < s2 ≤ t0 + σ, we have by (B)

kF (y, s2) − F (y, s1)k∞ = sup kF (y, s2)(τ) − F (y, s1)(τ)k = t0−r≤τ≤t0+σ

Z τ

= sup kF (y, s2)(τ) − F (y, s1)(τ)k = sup f(ys, s) dg(s) ≤ s1≤τ≤s2 s1≤τ≤s2 s1

41 Z s2 ≤ M dg(s) ≤ h(s2) − h(s1). s1

Similarly, using (C), if y, z ∈ O and t0 ≤ s1 ≤ s2 ≤ t0 + σ, then

kF (y, s2) − F (y, s1) − F (z, s2) + F (z, s1)k∞ = Z τ Z τ

= sup [f(ys, s) − f(zs, s)] dg(s) ≤ sup Lkys − zsk∞ dg(s) ≤ s1≤τ≤s2 s1 s1≤τ≤s2 s1

Z s2 ≤ ky − zk∞ L dg(s) ≤ ky − zk∞(h(s2) − h(s1)) s1

(note that the function s 7→ kys − zsk∞ is regulated according to Lemma 4.3.3, and therefore the integral R τ Lkys − zsk∞ dg(s) exists). s1 The following statement is a slightly modiﬁed version of Lemma 3.3 from [24] (which is concerned with the special case g(t) = t). The proof from [24] can be carried over without any changes and we repeat it here for reader’s convenience.

n Lemma 4.3.5. Let O be a subset of G([t0 − r, t0 + σ], R ) with the prolongation property and P = {yt; y ∈ O, t ∈ [t0, t0 + σ]}. Assume that φ ∈ P , g :[t0, t0 + σ] → R is a nondecreasing function, and n R t0+σ f : P × [t0, t0 + σ] → is such that the integral f(yt, t) dg(t) exists for every y ∈ O. Consider F R t0 given by (4.3.3) and assume that x :[t0, t0 + σ] → O is a solution of dx = DF (x, t) dτ with initial condition x(t0)(ϑ) = φ(ϑ) for ϑ ∈ [t0 − r, t0], and x(t0)(ϑ) = x(t0)(t0) for ϑ ∈ [t0, t0 + σ]. If v ∈ [t0, t0 + σ] and ϑ ∈ [t0 − r, t0 + σ], then

x(v)(ϑ) = x(v)(v), ϑ ≥ v, (4.3.6) and x(v)(ϑ) = x(ϑ)(ϑ), v ≥ ϑ. (4.3.7)

Proof. Assume that ϑ ≥ v. Since x is a solution of

dx = DF (x, t), dτ we have Z v x(v)(v) = x(t0)(v) + DF (x(τ), t)(v) t0 and similarly Z v x(v)(ϑ) = x(t0)(ϑ) + DF (x(τ), t)(ϑ). t0

Since x(t0)(ϑ) = x(t0)(v) by the properties of the initial condition, we have Z v Z v x(v)(ϑ) − x(v)(v) = DF (x(τ), t)(ϑ) − DF (x(τ), t)(v). t0 t0

It follows from the existence of the integral R v DF (x(τ), t) that for every ε > 0, there is a gauge δ on t0 [t0, t0 + σ] such that if {(τi, [si−1, si]), i = 1, . . . , k} is a δ-ﬁne tagged partition of [t0, v], then

k Z v X (F (x(τi), si) − F (x(τi), si−1)) − DF (x(τ), t) < ε. t i=1 0 ∞

42 Therefore we have

k k X X kx(v)(ϑ) − x(v)(v)k < 2ε + (F (x(τi), si) − F (x(τi), si−1))(ϑ) − (F (x(τi), si) − F (x(τi), si−1))(v) . i=1 i=1

By the deﬁnition of F in (4.3.3), it is a matter of routine to check that, for every i ∈ {1, . . . , k}, we have

F (x(τi), si)(ϑ) − F (x(τi), si−1)(ϑ) = F (x(τi), si)(v) − F (x(τi), si−1)(v), and consequently kx(v)(ϑ) − x(v)(v)k < 2ε. Since this holds for an arbitrary ε > 0, the relation (4.3.6) is satisﬁed. To prove the second statement, assume that ϑ ≤ v. Similarly to the ﬁrst part of the proof, we have

Z v x(v)(ϑ) = x(t0)(ϑ) + DF (x(τ), t)(ϑ) t0 and Z ϑ x(ϑ)(ϑ) = x(t0)(ϑ) + DF (x(τ), t)(ϑ). t0 Hence Z v x(v)(ϑ) − x(ϑ)(ϑ) = DF (x(τ), t)(ϑ). ϑ

Now, if {(τi, [si−1, si]), i = 1, . . . , k} is an arbitrary tagged partition of [ϑ, v], it is straightforward to check by (4.3.3) that, for every i ∈ {1, . . . , k}, we have

F (x(τi), si)(ϑ) − F (x(τi), si−1)(ϑ) = 0. R v This means that ϑ DF (x(τ), t)(ϑ) = 0 and x(v)(ϑ) = x(ϑ)(ϑ).

The proofs of the following two theorems are inspired by similar proofs from the paper [24], which describes the special case g(t) = t, i.e. the usual type of functional diﬀerential equations.

n Theorem 4.3.6. Assume that X is a closed subspace of G([t0 − r, t0 + σ], R ), O is a subset of X with the prolongation property, P = {xt; x ∈ O, t ∈ [t0, t0 + σ]}, φ ∈ P , g :[t0, t0 + σ] → R is a n nondecreasing function, f : P ×[t0, t0 +σ] → R satisﬁes conditions (A), (B), (C). Let F : O×[t0, t0 +σ] → n G([t0 − r, t0 + σ], R ) be given by (4.3.3) and assume that F (x, t) ∈ X for every x ∈ O, t ∈ [t0, t0 + σ]. Let y ∈ O be a solution of the measure functional diﬀerential equation

Z t y(t) = y(t0) + f(ys, s) dg(s), t ∈ [t0, t0 + σ], t0

yt0 = φ.

For every t ∈ [t0 − r, t0 + σ], let ( y(ϑ), ϑ ∈ [t − r, t], x(t)(ϑ) = 0 y(t), ϑ ∈ [t, t0 + σ].

Then the function x :[t0, t0 + σ] → O is a solution of the generalized ordinary diﬀerential equation

dx = DF (x, t). dτ

43 R v Proof. We will show that, for every v ∈ [t0, t0 + σ], the integral DF (x(τ), t) exists and t0 Z v x(v) − x(t0) = DF (x(τ), t). t0

Let an arbitrary ε > 0 be given. Since g is nondecreasing, it can have only a ﬁnite number of points t ∈ [t0, v] + + such that ∆ g(t) ≥ ε/M; denote these points by t1, . . . , tm. Consider a gauge δ :[t0, t0 + σ] → R such that t − t δ(τ) < min k k−1 , k = 2, . . . , m , τ ∈ [t , t + σ] 2 0 0 and δ(τ) < min {|τ − tk|; k = 1, . . . , m} , τ ∈ [t0, t0 + σ]. These conditions assure that if a point-interval pair (τ, [c, d]) is δ-ﬁne, then [c, d] contains at most one of the points t1, . . . , tm, and, moreover, τ = tk whenever tk ∈ [c, d].

Since ytk = x(tk)tk , it follows from Theorem 4.2.2 that Z s + lim Lkys − x(tk)sk∞ dg(s) = Lkytk − x(tk)tk k∞∆ g(tk) = 0 s→tk+ tk for every k ∈ {1, . . . , m}. Thus the gauge δ might be chosen in such a way that

Z tk+δ(tk) ε Lkys − x(tk)sk∞ dg(s) < , k ∈ {1, . . . , m}. tk 2m + 1 Using Theorem 4.2.2 again, we see that ε ky(τ+) − y(τ)k = kf(y , τ)∆+g(τ)k < M = ε, τ ∈ [t , t + σ]\{t , . . . , t }. τ M 0 0 1 m Thus we can assume that the gauge δ is such that

ky(ρ) − y(τ)k ≤ ε for every τ ∈ [t0, t0 + σ]\{t1, . . . , tm} and ρ ∈ [τ, τ + δ(τ)). Assume now that {(τi, [si−1, si]), i = 1, . . . , l} is a δ-fine tagged partition of the interval [t0, v]. Using the definition of x, it can be easily shown that  0, ϑ ∈ [t0 − r, si−1],  R ϑ x(s ) − x(s ) (ϑ) = f(ys, s) dg(s), ϑ ∈ [si−1, si, ] i i−1 si−1 R si  f(ys, s) dg(s), ϑ ∈ [si, t0 + σ]. si−1 Similarly, it follows from the definition of F that  0, ϑ ∈ [t0 − r, si−1],  R ϑ F (x(τ ), s ) − F (x(τ ), s ) (ϑ) = f(x(τi)s, s) dg(s), ϑ ∈ [si−1, si], i i i i−1 si−1 R si  f(x(τi)s, s) dg(s), ϑ ∈ [si, t0 + σ]. si−1 By combination of the previous equalities, we obtain x(si) − x(si−1) (ϑ) − F (x(τi), si) − F (x(τi), si−1) (ϑ) =  0, ϑ ∈ [t0 − r, si−1],  R ϑ = f(ys, s) − f(x(τi)s, s) dg(s), ϑ ∈ [si−1, si], si−1 R si  f(ys, s) − f(x(τi)s, s) dg(s), ϑ ∈ [si, t0 + σ]. si−1

44 Consequently, kx(si) − x(si−1) − F (x(τi), si) − F (x(τi), si−1) k∞ = = sup k x(si) − x(si−1) (ϑ) − F (x(τi), si) − F (x(τi), si−1) (ϑ)k = ϑ∈[t0−r,t0+σ]

Z ϑ = sup f(ys, s) − f(x(τi)s, s) dg(s) . ϑ∈[si−1,si] si−1

By the deﬁnition of x, we see that x(τi)s = ys whenever s ≤ τi. Thus

ϑ ( Z 0, ϑ ∈ [si−1, τi], f(y , s) − f(x(τ ) , s) dg(s) = s i s R ϑ si−1 f(ys, s) − f(x(τi)s, s) dg(s), ϑ ∈ [τi, si]. τi Then condition (C) implies

Z ϑ Z ϑ Z si f(ys, s) − f(x(τi)s, s) dg(s) ≤ Lkys − x(τi)sk∞ dg(s) ≤ Lkys − x(τi)sk∞ dg(s). τi τi τi

Given a particular point-interval pair (τi, [si−1, si]), there are two possibilities:

(i) The intersection of [si−1, si] and {t1, . . . , tm} contains a single point tk = τi.

(ii) The intersection of [si−1, si] and {t1, . . . , tm} is empty. In case (i), it follows from the deﬁnition of the gauge δ that

Z si ε Lkys − x(τi)sk∞ dg(s) ≤ , τi 2m + 1 i.e. ε kx(s ) − x(s ) − F (x(τ ), s ) − F (x(τ ), s )k ≤ . i i−1 i i i i−1 ∞ 2m + 1 In case (ii), we have kys − x(τi)sk∞ = sup ky(ρ) − y(τi)k ≤ ε, s ∈ [τi, si] ρ∈[τi,s] by the deﬁnition of the gauge δ. Thus

Z si kx(si) − x(si−1) − F (x(τi), si) − F (x(τi), si−1) k∞ ≤ ε L dg(s). τi Combining cases (i) and (ii) and using the fact that case (i) occurs at most 2m times, we obtain

l Z t0+σ 2mε X x(v) − x(t0) − F (x(τi), si) − F (x(τi), si−1) ≤ ε L dg(s) + . t 2m + 1 i=1 ∞ 0 Since ε is arbitrary, it follows that Z v x(v) − x(t0) = DF (x(τ), t). t0

n Theorem 4.3.7. Assume that X is a closed subspace of G([t0 − r, t0 + σ], R ), O is a subset of X with the prolongation property, P = {xt; x ∈ O, t ∈ [t0, t0 + σ]}, φ ∈ P , g :[t0, t0 + σ] → R is a n nondecreasing function, f : P ×[t0, t0 +σ] → R satisﬁes conditions (A), (B), (C). Let F : O×[t0, t0 +σ] → n G([t0 − r, t0 + σ], R ) be given by (4.3.3) and assume that F (x, t) ∈ X for every x ∈ O, t ∈ [t0, t0 + σ]. Let x :[t0, t0 + σ] → O be a solution of the generalized ordinary diﬀerential equation dx = DF (x, t), dτ

45 with the initial condition ( φ(ϑ − t0), t0 − r ≤ ϑ ≤ t0, x(t0)(ϑ) = x(t0)(t0), t0 ≤ ϑ ≤ t0 + σ. Then the function y ∈ O deﬁned by ( x(t )(ϑ), t − r ≤ ϑ ≤ t , y(ϑ) = 0 0 0 x(ϑ)(ϑ), t0 ≤ ϑ ≤ t0 + σ. is a solution of the measure functional diﬀerential equation Z t y(t) = y(t0) + f(ys, s) dg(s), t ∈ [t0, t0 + σ], t0

yt0 = φ.

Proof. The equality yt0 = φ follows easily from the definitions of y and x(t0). It remains to prove that if v ∈ [t0, t0 + σ], then Z v y(v) − y(t0) = f(ys, s) dg(s). t0 Using Lemma 4.3.5, we obtain Z v y(v) − y(t0) = x(v)(v) − x(t0)(t0) = x(v)(v) − x(t0)(v) = DF (x(τ), t) (v). t0 Thus Z v Z v Z v y(v) − y(t0) − f(ys, s) dg(s) = DF (x(τ), t) (v) − f(ys, s) dg(s). (4.3.8) t0 t0 t0 Let an arbitrary ε > 0 be given. Since g is nondecreasing, it can have only a finite number of points + t ∈ [t0, v] such that ∆ g(t) ≥ ε/(L + M); denote these points by t1, . . . , tm. Consider a gauge δ : + [t0, t0 + σ] → R such that t − t δ(τ) < min k k−1 , k = 2, . . . , m , τ ∈ [t , t + σ] 2 0 0 and δ(τ) < min {|τ − tk|; k = 1, . . . , m} , τ ∈ [t0, t0 + σ]. As in the proof of Theorem 4.3.6, these conditions assure that if a point-interval pair (τ, [c, d]) is δ-fine, then [c, d] contains at most one of the points t1, . . . , tm, and, moreover, τ = tk whenever tk ∈ [c, d]. Again, the gauge δ might be chosen in such a way that

Z tk+δ(tk) ε Lkys − x(tk)sk∞ dg(s) < , k ∈ {1, . . . , m}. tk 2m + 1

According to Lemma 4.3.4, the function F given by (4.3.3) belongs to the class F(O ×[t0, t0 +σ], h), where

h(t) = (L + M)(g(t) − g(t0)). Since + kh(τ+) − h(τ)k = k(L + M)∆ g(τ)k < ε, τ ∈ [t0, t0 + σ]\{t1, . . . , tm}, we can assume that the gauge δ satisﬁes kh(ρ) − h(τ)k ≤ ε for every ρ ∈ [τ, τ + δ(τ)). Finally, the gauge δ should be such that

Z v l X DF (x(τ), t) − F (x(τi), si) − F (x(τi), si−1) < ε (4.3.9) t 0 i=1 ∞

46 for every δ-fine partition {(τi, [si−1, si]), i = 1, . . . , l} of [t0, v]. The existence of such a gauge follows from the definition of the Kurzweil integral. Choose a particular δ-fine partition {(τi, [si−1, si]), i = 1, . . . , l} of [t0, v]. By (4.3.8) and (4.3.9), we have Z v Z v Z v

y(v) − y(t0) − f(ys, s) dg(s) = DF (x(τ), t) (v) − f(ys, s) dg(s) < t0 t0 t0

l Z v X < ε + F (x(τi), si) − F (x(τi), si−1) (v) − f(ys, s) dg(s) ≤ i=1 t0 l Z si X ≤ ε + F (x(τi), si) − F (x(τi), si−1) (v) − f(ys, s) dg(s) . i=1 si−1 The deﬁnition of F yields

Z si F (x(τi), si) − F (x(τi), si−1) (v) = f(x(τi)s, s) dg(s), si−1 which implies

Z si F (x(τi), si) − F (x(τi), si−1) (v) − f(ys, s) dg(s) = si−1

Z si Z si Z si

= f(x(τi)s, s) dg(s) − f(ys, s) dg(s) = [f(x(τi)s, s) − f(ys, s)] dg(s) . si−1 si−1 si−1

By Lemma 4.3.5, for every i ∈ {1, . . . , l}, we have x(τi)s = x(s)s = ys for s ∈ [si−1, τi] and ys = x(s)s = x(si)s for s ∈ [τi, si]. Therefore

Z si Z si

[f(x(τi)s, s) − f(ys, s)] dg(s) = [f(x(τi)s, s) − f(ys, s)] dg(s) = si−1 τi

Z si Z si

= [f(x(τi)s, s) − f(x(si)s, s)] dg(s) ≤ Lkx(τi)s − x(si)sk∞ dg(s), τi τi where the last inequality follows from condition (C). Again, we distinguish two cases:

(i) The intersection of [si−1, si] and {t1, . . . , tm} contains a single point tk = τi.

(ii) The intersection of [si−1, si] and {t1, . . . , tm} is empty. In case (i), it follows from the deﬁnition of the gauge δ that

Z si ε Lkys − x(τi)sk∞ dg(s) ≤ , τi 2m + 1 i.e.

Z si ε F (x(τ ), s ) − F (x(τ ), s )(v) − f(y , s) dg(s) ≤ . i i i i−1 s 2m + 1 si−1 In case (ii), we use Lemma 4.3.1 to obtain the estimate

kx(si)s − x(τi)sk∞ ≤ kx(si) − x(τi)k∞ ≤ h(si) − h(τi) ≤ ε for every s ∈ [τi, si], and thus

Z si Z si F (x(τi), si) − F (x(τi), si−1) (v) − f(ys, s) dg(s) ≤ ε L dg(s). si−1 τi

47 Combining cases (i) and (ii) and using the fact that case (i) occurs at most 2m times, we obtain

l X Z si Z t0+σ 2mε F (x(τ ), s ) − F (x(τ ), s )(v) − f(y , s) dg(s) ≤ ε L dg(s) + < i i i i−1 s 2m + 1 i=1 si−1 t0

Z t0+σ < ε 1 + L dg(s) . t0 Consequently, Z v Z t0+σ

y(v) − y(t0) − f(ys, s) dg(s) < ε 2 + L dg(s) , t0 t0 which completes the proof.

Remark 4.3.8. It follows from Lemma 4.3.5 that the relation ( x(t )(ϑ), t − r ≤ ϑ ≤ t , y(ϑ) = 0 0 0 x(ϑ)(ϑ), t0 ≤ ϑ ≤ t0 + σ. from the previous theorem can be replaced by a single equality

y(ϑ) = x(t0 + σ)(ϑ), t0 − r ≤ ϑ ≤ t0 + σ.

Remark 4.3.9. Before proceeding further, we stop for a moment to discuss conditions (A), (B), (C), which appear in the statements of Theorem 4.3.6 and Theorem 4.3.7. R t0+σ Condition (A) requires the existence of the Kurzweil-Stieltjes integral f(yt, t) dg(t) for every t0 y ∈ O. The class of integrable functions is quite large; for example, if g(t) = t, the Kurzweil-Stieltjes integral reduces to the well-known Henstock-Kurzweil integral, which generalizes both Lebesgue and Newton integrals (see e.g. [31]). For a general nondecreasing function g, Theorem 4.2.1 provides a useful suﬃcient condition: the Kurzweil-Stieltjes integral exists whenever t 7→ f(yt, t) is a regulated function. Conditions (B) and (C) can be replaced by slightly weaker statements: An inspection of the proofs of Theorem 4.3.6 and Theorem 4.3.7 shows that it is enough to assume the existence of constants M, L > 0 such that

Z b

f(yt, t) dg(t) ≤ M(g(b) − g(a)) a and

Z b Z b

(f(yt, t) − f(zt, t)) dg(t) ≤ L kyt − ztk dg(t) a a for every a, b ∈ [t0, t0 + σ], y, z ∈ O. Remark 4.3.10. The paper [24] deals with impulsive functional diﬀerential equations of the form

0 y (t) = f(yt, t), t ∈ [t0, t0 + σ]\{t1, . . . , tm}, + ∆ y(tk) = Ik(y(tk)), k ∈ {1, . . . , m},

yt0 = φ,

n n n n where f : P × [t0, t0 + σ] → R , P ⊂ G([−r, 0], R ), t0 ≤ t1 < . . . < tm < t0 + σ and Ik : R → R for every k ∈ {1, . . . , m}. The solution y is assumed to be diﬀerentiable on [t0, t0 + σ]\{t1, . . . , tm} and left-continuous at the points t1, . . . , tm. The integral form of this impulsive problem is

m Z t X Z t X y(t) = y(t0) + f(ys, s) ds + Ik(y(tk)) = y(t0) + f(ys, s) ds + Ik(y(tk))Htk (t), t0 t0 k; t0≤tk

yt0 = φ,

48 where Hv denotes the characteristic function of (v, ∞), i.e. Hv(t) = 0 for t ≤ v and Hv(t) = 1 for t > v. We claim that this problem is equivalent to Z t ˜ y(t) = y(t0) + f(ys, s) dg(s), t ∈ [t0, t0 + σ], (4.3.10) t0

yt0 = φ, Pm where g(t) = t + k=1 Htk (t) and ( f(y, t) if t ∈ [t , t + σ]\{t , . . . , t }, f˜(y, t) = 0 0 1 m Ik(y(0)) if t = tk for some k ∈ {1, . . . , m} for every t ∈ [t0, t0 + σ] and y ∈ P . Indeed, assume that y satisﬁes (4.3.10). Then we have Z t Z t m Z t ˜ ˜ X ˜ y(t) = y(t0) + f(ys, s) dg(s) = y(t0) + f(ys, s) ds + f(ys, s) dHtk (s) = t0 t0 k=1 t0 Z t Z t X ˜ + X = y(t0) + f(ys, s) ds + f(ytk , tk)∆ Htk (tk) = y(t0) + f(ys, s) ds + Ik(y(tk)). t0 t0 k; t0≤tk

Note that if f is bounded and Lipschitz-continuous and if the impulse operators Ik are bounded and Lipschitz-continuous, then f˜ has the same properties. Thus we see that our measure functional diﬀerential equations are general enough to encompass impulsive behavior and there is no need to consider impulses separately like in [24].

4.4 Functional dynamic equations on time scales

This section starts with a short overview of some basic concepts in the theory of time scales. Then we suggest a new approach to functional dynamic equations on time scales and explain their relation to measure functional differential equations. Let T be a time scale, i.e. a closed nonempty subset of R. For every t ∈ T, we define the forward jump operator by σ(t) = inf{s ∈ T, s > t} (where we make the convention that inf ∅ = sup T) and the graininess function by µ(t) = σ(t) − t. If σ(t) > t, we say that t is right-scattered; otherwise, t is right-dense. A function f : T → R is called rd-continuous if it is regulated on T and continuous at right-dense points of T. Given a pair of numbers a, b ∈ T, the symbol [a, b]T will be used to denote a closed interval in T, i.e. [a, b]T = {t ∈ T; a ≤ t ≤ b}. On the other hand, [a, b] is the usual closed interval on the real line, i.e. [a, b] = {t ∈ R; a ≤ t ≤ b}. This notational convention should help the reader to distinguish between ordinary and time scale intervals. In the time scale calculus, the usual derivative f 0(t) is replaced by the ∆-derivative f ∆(t). Similarly, R b R b the usual integral a f(t) dt is replaced by the ∆-integral a f(t) ∆t, where f :[a, b]T → R. The definitions and properties of the ∆-derivative and ∆-integral can be found in [8] and [9]. In the rest of this chapter, we use the same notation as in Chapter 2: Given a real number t ≤ sup T, let ∗ t = inf{s ∈ T; s ≥ t}. (Note that t∗ might be different from σ(t).) Since T is a closed set, we have t∗ ∈ T. Further, let (−∞, sup ] if sup < ∞, ∗ = T T T (−∞, ∞) otherwise.

Given a function f : T → Rn, we consider its extension f ∗ : T∗ → Rn given by ∗ ∗ ∗ f (t) = f(t ), t ∈ T . According to the following theorem (which was proved in Chapter 2), the ∆-integral of a time scale function f is in fact equivalent to the Kurzweil-Stieltjes integral of the extended function f ∗.

49 Theorem 4.4.1. Let f : T → Rn be a rd-continuous function. Choose an arbitrary a ∈ T and deﬁne Z t F1(t) = f(s) ∆s, t ∈ T, a Z t ∗ ∗ F2(t) = f (s) dg(s), t ∈ T , a

∗ ∗ ∗ where g(s) = s for every s ∈ T . Then F2 = F1 ; in particular, F2(t) = F1(t) for every t ∈ T.

R b ∗ However, it is useful to note that the Kurzweil-Stieltjes integral a f dg does not change if we replace f ∗ by a diﬀerent function which coincides with f on [a, b] ∩ T. This is the content of the next theorem.

Theorem 4.4.2. Let T be a time scale, g(s) = s∗ for every s ∈ T∗, [a, b] ⊂ T∗. Consider a pair of n R b R b functions f1, f2 :[a, b] → R such that f1(t) = f2(t) for every t ∈ [a, b] ∩ T. If a f1 dg exists, then a f2 dg exists as well and both integrals have the same value.

R b + Proof. Denote I = a f1 dg. Given an arbitrary ε > 0, there is a gauge δ1 :[a, b] → R such that

k X f1(τi)(g(si) − g(si−1)) − I < ε i=1 for every δ1-ﬁne partition with division points a = s0 ≤ s1 ≤ · · · ≤ sk = b and tags τi ∈ [si−1, si], i = 1, . . . , k. Now, let ( δ1(t) if t ∈ [a, b] ∩ T, δ2(t) = 1 min (δ1(t), 2 inf {|t − s|, s ∈ T}) if t ∈ [a, b]\T.

Note that each δ2-fine partition is also δ1-fine. Consider an arbitrary δ2-fine partition with division points a = s0 ≤ s1 ≤ · · · ≤ sk = b and tags τi ∈ [si−1, si], i ∈ {1, . . . , k}. For every i ∈ {1, . . . , k}, there are two possibilities: Either [si−1, si] ∩ T = ∅, or τi ∈ T. In the first case, g(si−1) = g(si), and therefore

f2(τi)(g(si) − g(si−1)) = 0 = f1(τi)(g(si) − g(si−1)).

In the second case, f1(τi) = f2(τi) and

f2(τi)(g(si) − g(si−1)) = f1(τi)(g(si) − g(si−1)).

Thus we have

k k X X f2(τi)(g(si) − g(si−1)) − I = f1(τi)(g(si) − g(si−1)) − I < ε. i=1 i=1

R b Since ε can be arbitrarily small, we conclude that a f2 dg = I. We would like to study dynamic equations on time scales such that the ∆-derivative of the unknown function x : T → Rn at t ∈ T depends on the values of x(s), where s ∈ [t − r, t] ∩ T. But, unlike the classical case, there is a difficulty: The function xt is now defined on a subset of [−r, 0], and this subset can ∗ depend on t. We overcome this problem by considering the function xt instead; throughout this and the ∗ ∗ ∗ following sections, xt stands for (x )t. Clearly, xt contains the same information as xt, but it is defined on the whole interval [−r, 0]. Thus, it seems reasonable to consider functional dynamic equations of the form ∆ ∗ x (t) = f(xt , t). We now show that this equation is equivalent to a certain measure functional differential equation. The n n symbol C([a, b]T, R ) will be used to denote the set of all continuous functions f :[a, b]T → R .

50 n Theorem 4.4.3. Let [t0 − r, t0 + σ]T be a time scale interval, t0 ∈ T, B ⊂ R , C = C([t0 − r, t0 + σ]T,B), ∗ n P = {xt ; x ∈ C, t ∈ [t0, t0 + σ]}, f : P × [t0, t0 + σ]T → R , φ ∈ C([t0 − r, t0]T,B). Assume that for every ∗ ∗ x ∈ C, the function t 7→ f(xt , t) is rd-continuous on [t0, t0 +σ]T. Deﬁne g(s) = s for every s ∈ [t0, t0 +σ]. If x :[t0 − r, t0 + σ]T → B is a solution of the functional dynamic equation

∆ ∗ x (t) = f(xt , t), t ∈ [t0, t0 + σ]T, (4.4.1)

x(t) = φ(t), t ∈ [t0 − r, t0]T, (4.4.2)

∗ then x :[t0 − r, t0 + σ] → B satisﬁes

Z t ∗ ∗ ∗ ∗ x (t) = x (t0) + f(xs, s ) dg(s), t ∈ [t0, t0 + σ], t0 x∗ = φ∗. t0

Conversely, if y :[t0 − r, t0 + σ] → B is a solution of the measure functional diﬀerential equation

Z t ∗ y(t) = y(t0) + f(ys, s ) dg(s), t ∈ [t0, t0 + σ], t0 ∗ yt0 = φ ,

∗ then y = x , where x :[t0 − r, t0 + σ]T → B satisﬁes (4.4.1) and (4.4.2). Proof. Assume that ∆ ∗ x (t) = f(xt , t), t ∈ [t0, t0 + σ]T. Then Z t ∗ x(t) = x(t0) + f(xs, s) ∆s, t ∈ [t0, t0 + σ]T, t0 and, by Theorem 4.4.1,

Z t ∗ ∗ ∗ ∗ x (t) = x (t0) + f(xs∗ , s ) dg(s), t ∈ [t0, t0 + σ]. t0

∗ ∗ ∗ ∗ Since f(xs∗ , s ) = f(xs, s ) for every s ∈ T, we can use Theorem 4.4.2 to conclude that Z t ∗ ∗ ∗ ∗ x (t) = x (t0) + f(xs, s ) dg(s), t ∈ [t0, t0 + σ]. t0 Conversely, assume that y satisﬁes

Z t ∗ y(t) = y(t0) + f(ys, s ) dg(s), t ∈ [t0, t0 + σ]. t0

Note that g is constant on every interval (α, β], where β ∈ T and α = sup{τ ∈ T; τ < β}. Thus y has the ∗ same property and it follows that y = x for some x :[t0 − r, t0 + σ]T → B. Using Theorem 4.2.2, it is easy to see that x is continuous on [t0 − r, t0 + σ]T. By reversing our previous reasoning, we conclude that x satisﬁes (4.4.1) and (4.4.2).

Example 4.4.4. There is a fairly large number of papers devoted to delay dynamic equations of the form

∆ x (t) = h(t, x(t), x(τ1(t)), . . . , x(τk(t))), (4.4.3) where τi : T → T are functions corresponding to the delays, i.e. τi(t) ≤ t for every t ∈ T and every i = 1, . . . , k, and where the function t 7→ h(t, x(t), x(τ1(t)), . . . , x(τk(t))) is rd-continuous whenever x is a continuous function.

51 Suppose that the delays are bounded, i.e. there exists a constant r > 0 such that t − r ≤ τi(t) ≤ t, or equivalently −r ≤ τi(t) − t ≤ 0. Then it is possible to write (4.4.3) in the form

∆ ∗ x (t) = f(xt , t) by taking f(y, t) = h(t, y(0), y(τ1(t) − t), . . . , y(τk(t) − t)) n ∗ for every y :[−r, 0] → R . Also, if x is continuous on [t0 − r, t0 + σ]T, then the function t 7→ f(xt , t) is rd-continuous on [t0, t0 + σ]T. An important special case is represented by linear delay dynamic equations of the form

k ∆ X x (t) = pi(t)x(τi(t)) + q(t), i=1 where q, p1, . . . , pk and τ1, . . . , τk are rd-continuous functions on [t0, t0 +σ]T. The corresponding functional dynamic equation is ∆ ∗ x (t) = f(xt , t), where k X f(y, t) = pi(t)y(τi(t) − t) + q(t) i=1 n ∗ for every y :[−r, 0] → R . Again, we see that t 7→ f(xt , t) is rd-continuous on [t0, t0 + σ]T whenever x is n continuous on [t0 − r, t0 + σ]T. Moreover, for each pair of functions y, z :[−r, 0] → R , we have

k k ! X X kf(y, t) − f(z, t)k ≤ kpi(t)(y(τi(t) − t) − z(τi(t) − t))k ≤ kpi(t)k ky − zk∞ ≤ Lky − zk∞ i=1 i=1 with k X L = sup kpi(t)k, t∈[t ,t +σ] 0 0 T i=1 i.e. f is Lipschitz-continuous in the ﬁrst variable.

4.5 Existence-uniqueness theorems

The following existence-uniqueness theorem for generalized ordinary diﬀerential equations was proved in [24], Theorem 2.15.

Theorem 4.5.1. Assume that X is a Banach space, O ⊂ X an open set, and F : O × [t0, t0 + σ] → X belongs to the class F(O × [t0, t0 + σ], h), where h :[t0, t0 + σ] → R is a left-continuous nondecreasing function. If x0 ∈ O is such that x0 + F (x0, t0+) − F (x0, t0) ∈ O, then there exists a δ > 0 and a function x :[t0, t0 + δ] → X which is a unique solution of the generalized ordinary differential equation dx = DF (x, t), x(t ) = x . dτ 0 0 Remark 4.5.2. An estimate for the value δ which corresponds to the length of interval where the solution exists can be obtained by inspection of the proof of Theorem 2.15 in [24]. Let B(x0, r) denote the closed ball {x ∈ X, kx−x0k ≤ r}. Then we find that δ ∈ (0, σ] can be any number such that B(x0, h(t0+δ)−h(t0+)) ⊂ O and h(t0 + δ) − h(t0+) < 1. (Note that the proof in [24] assumes that h(t0 + δ) − h(t0+) < 1/2, but a careful examination reveals that h(t0 + δ) − h(t0+) < 1 is sufficient). We now use the previous result to obtain an existence-uniqueness theorem for measure functional differential equations.

52 n Theorem 4.5.3. Assume that X is a closed subspace of G([t0 − r, t0 + σ], R ), O is an open subset of X with the prolongation property, P = {xt; x ∈ O, t ∈ [t0, t0 + σ]}, g :[t0, t0 + σ] → R is a n left-continuous nondecreasing function, f : P × [t0, t0 + σ] → R satisﬁes conditions (A), (B), (C). Let n F : O × [t0, t0 + σ] → G([t0 − r, t0 + σ], R ) be given by (4.3.3) and assume that F (x, t) ∈ X for every x ∈ O, t ∈ [t0, t0 + σ]. If φ ∈ P is such that the function ( φ(t − t0), t ∈ [t0 − r, t0], z(t) = + φ(0) + f(φ, t0)∆ g(t0), t ∈ (t0, t0 + σ]

n belongs to O, then there exists a δ > 0 and a function y :[t0 − r, t0 + δ] → R which is a unique solution of the measure functional diﬀerential equation

Z t y(t) = y(t0) + f(ys, s) dg(s), t0

yt0 = φ.

Proof. According to Lemma 4.3.4, the function F belongs to the class F(O × [a, b], h), where

h(t) = (M + L)(g(t) − g(t0)).

Further, let ( φ(ϑ − t0), ϑ ∈ [t0 − r, t0], x0(ϑ) = φ(0), ϑ ∈ [t0, t0 + σ].

It is clear that x0 ∈ O. We also claim that x0 + F (x0, t0+) − F (x0, t0) ∈ O. First, note that F (x0, t0) = 0. The limit F (x0, t0+) is taken with respect to the supremum norm and we know it must exist since F is regulated with respect to the second variable (this follows from the fact that F ∈ F(O × [a, b], h)). Thus it is suﬃcient to calculate the pointwise limit F (x0, t0+)(ϑ) for every ϑ ∈ [t0 −r, t0 +σ]. Using Theorem 4.2.2, we obtain ( 0, t ∈ [t0 − r, t0], F (x0, t0+)(ϑ) = + f(φ, t0)∆ g(t0), t ∈ (t0, t0 + σ].

It follows that x0 + F (x0, t0+) − F (x0, t0) = z ∈ O. Since all the assumptions of Theorem 4.5.1 are satisﬁed, there exists a δ > 0 and a unique solution x :[t0, t0 + δ] → X of the generalized ordinary diﬀerential equation dx = DF (x, t), x(t ) = x . (4.5.1) dτ 0 0

According to Theorem 4.3.7, the function y :[t0 − r, t0 + δ] given by ( x(t )(ϑ), t − r ≤ ϑ ≤ t , y(ϑ) = 0 0 0 x(ϑ)(ϑ), t0 ≤ ϑ ≤ t0 + δ is a solution of the measure functional diﬀerential equation

Z t y(t) = y(t0) + f(ys, s) dg(s), t0

yt0 = φ.

This solution must be unique; otherwise, Theorem 4.3.6 would imply that x is not the only solution of the generalized ordinary diﬀerential equation (4.5.1).

Remark 4.5.4. Since the assumptions of Theorem 4.5.3 might look complicated, we mention two typical choices for the sets X, O and P :

53 n n • g(t) = t for every t ∈ [t0, t0 + σ], X = C([t0 − r, t0 + σ], R ), B ⊂ R is an open set, O = C([t0 − r, t0 + σ],B), P = C([−r, 0],B). Both conditions F (x, t) ∈ X and z ∈ O from Theorem 4.5.3 are always satisﬁed (by Theorem 4.2.2, F (x, t) is a continuous function and therefore F (x, t) ∈ X).

n n • X = G([t0 − r, t0 + σ], R ), B ⊂ R is an open set, O = G([t0 − r, t0 + σ],B), P = G([−r, 0],B). The condition F (x, t) ∈ X from Theorem 4.5.3 is always satisﬁed (by Theorem 4.2.2, F (x, t) is a regulated + function and therefore F (x, t) ∈ X). The condition z ∈ O reduces to φ(0) + f(φ, t0)∆ g(t0) ∈ B. Note that if Z t y(t) = y(t0) + f(ys, s) dg(s), t0 + then y(t0+) = φ(0) + f(φ, t0)∆ g(t0). In other words, the condition ensures that the solution does not leave the set B immediately after time t0. In both cases, we can use Remark 4.5.2 to obtain an estimate for the value δ which corresponds to the length of interval where the solution exists. Assume there exists a ρ > 0 such that ky − φ(t)k < ρ implies y ∈ B for every t ∈ [−r, 0] (in other words, a ρ-neighborhood of φ is contained in B). Since we have

h(t) = (M + L)(g(t) − g(t0)), we see that δ ∈ (0, σ] can be any number such that min(1, ρ) g(t + δ) − g(t +) < . 0 0 M + L We now prove an existence-uniqueness theorem for functional dynamic equations (cf. [47], [56]). n Theorem 4.5.5. Let [t0 − r, t0 + σ]T be a time scale interval, t0 ∈ T, B ⊂ R open, C = C([t0 − r, t0 + ∗ n σ]T,B), P = {yt ; y ∈ C, t ∈ [t0, t0 + σ]}, f : P × [t0, t0 + σ]T → R a bounded function, which is Lipschitz- ∗ continuous in the ﬁrst argument and such that t 7→ f(yt , t) is rd-continuous on [t0, t0 +σ]T for every y ∈ C. If φ :[t − r, t ] → B is a continuous function such that φ(t ) + f(φ∗ , t )µ(t ) ∈ B, then there exists a 0 0 T 0 t0 0 0 δ > 0 such that δ ≥ µ(t0) and t0 + δ ∈ T, and a function y :[t0 − r, t0 + δ]T → B which is a unique solution of the functional dynamic equation

∆ ∗ y (t) = f(yt , t), t ∈ [t0, t0 + δ],

y(t) = φ(t), t ∈ [t0 − r, t0]T. ∗ n ∗ ∗ Proof. Let X = {y ; y ∈ C([t0 − r, t0 + σ]T, R )}, O = {y ; y ∈ C}, and g(t) = t for every t ∈ [t0, t0 + σ]. n n Note that C([t0 −r, t0 +σ]T, R ) is a closed (Banach) space and the operator T : C([t0 −r, t0 +σ]T, R ) → X ∗ given by T (y) = y is an isometric isomorphism; it follows that X is a closed subspace of G([t0 − r, t0 + n n σ], R ). The function F : O × [t0, t0 + σ] → G([t0 − r, t0 + σ], R ) given by (4.3.3) satisﬁes F (x, t) ∈ X for every x ∈ O, t ∈ [t0, t0 + σ]. Indeed, by deﬁnition (4.3.3), we see that F (x, t) is constant on [t0 − r, t0] and on every interval (α, β) ⊂ [t0, t0 + r] which contains no time scale points (because g is constant on such intervals). The function g is left-continuous, and it follows from Theorem 4.2.2 that F (x, t) is left- continuous on [t0, t0 + σ] and right-continuous at all points of [t0, t0 + σ] where g is right-continuous, i.e. at ∗ all right-dense points of [t0, t0 + r]T. Thus F (x, t) must have the form y for some rd-continuous function n y :[t0 − r, t0 + σ]T → R and F (x, t) ∈ X. It is also clear that O is an open subset of X and has the prolongation property. ∗ ∗ Let f (y, t) = f(y, t ) for every y ∈ P and t ∈ [t0, t0 + σ]. Consider an arbitrary y ∈ O. Since R t0+σ t 7→ f(yt, t) is rd-continuous on [t0, t0 + σ] , the integral f(yt, t)∆t exists. Using Theorem 4.4.1 and T t0 Theorem 4.4.2, we have

Z t0+σ Z t0+σ Z t0+σ Z t0+σ ∗ ∗ ∗ f(yt, t)∆t = f(yt∗ , t ) dg(t) = f(yt, t ) dg(t) = f (yt, t) dg(t), t0 t0 t0 t0 i.e. the last integral exists. Since ∆+g(t ) = µ(t ) and φ(t ) + f(φ∗ , t )µ(t ) ∈ B, it follows that the 0 0 0 t0 0 0 function ( ∗ φ (t − t0), t ∈ [t0 − r, t0], z(t) = t0 φ∗ (0) + f(φ∗ , t )∆+g(t ), t ∈ (t , t + σ] t0 t0 0 0 0 0

54 belongs to O. Therefore the functions f ∗, g and φ∗ satisfy all assumptions of Theorem 4.5.3, and there t0 exists a δ > 0 and a function u :[t0 − r, t0 + δ] → B which is the unique solution of

Z t ∗ u(t) = u(t0) + f (us, s) dg(s), t ∈ [t0, t0 + δ] t0 u = φ∗ . t0 t0

∗ By Theorem 4.4.3, u = y , where y :[t0 − r, t0 + δ]T → B is a solution of

∆ ∗ y (t) = f(yt , t), t ∈ [t0, t0 + δ]

y(t) = φ(t), t ∈ [t0 − r, t0]T. Without loss of generality, we can assume that δ ≥ µ(t ); otherwise, let y(σ(t )) = φ(t ) + f(φ∗ , t )µ(t ) 0 0 0 t0 0 0 to obtain a solution deﬁned on [t0 − r, t0 + µ(t0)]T. Again by Theorem 4.4.3, it follows that the solution y is unique.

Remark 4.5.6. Similarly to the previous existence-uniqueness results, we can estimate the value of δ which corresponds to length of interval where the solution exists. Assume there exists a ρ > 0 such that ky − φ(t)k < ρ implies y ∈ B for every t ∈ [t0 − r, t0]T (in other words, a ρ-neighborhood of φ is contained in B). By Remark 4.5.4, we know that δ ∈ (0, σ] can be any number such that

min(1, ρ) g(t + δ) − g(t +) < , 0 0 M + L where M is the bound and L is the Lipschitz constant for the function f on P × [t0, t0 + σ]T. In our ∗ particular case, we have g(t) = t for every t ∈ [t0, t0 + σ]. Since g(t0 + δ) = t0 + δ and g(t0+) = σ(t0), we obtain min(1, ρ) δ < µ(t ) + . 0 M + L 4.6 Continuous dependence results

In this section, we use an existing continuous dependence theorem for generalized ordinary diﬀerential equations to derive continuous dependence theorems for measure functional diﬀerential equations and for functional dynamic equations on time scales. We need the following proposition from [30], Theorem 2.18.

Theorem 4.6.1. The following conditions are equivalent:

1. A set A ⊂ G([α, β], Rn) is relatively compact. 2. The set {x(α); x ∈ A} is bounded and there is an increasing continuous function η : [0, ∞) → [0, ∞), η(0) = 0 and an increasing function K :[α, β] → R such that

kx(t2) − x(t1)k ≤ η(K(t2) − K(t1))

for every x ∈ A, α ≤ t1 ≤ t2 ≤ β. The following continuous dependence result for generalized ordinary diﬀerential equations is a Banach space version of Theorem 2.4 from [29]; the proof for the case X = Rn from [29] is still valid in this more general setting.

Theorem 4.6.2. Let X be a Banach space, O ⊂ X an open set, and hk :[a, b] → R, k ∈ N0, a sequence of nondecreasing left-continuous functions such that hk(b) − hk(a) ≤ c for some c > 0 and every k ∈ N0. Assume that for every k ∈ N0, Fk : O × [a, b] → X belongs to the class F(O × [a, b], hk), and that

lim Fk(x, t) = F0(x, t), x ∈ O, t ∈ [a, b], k→∞

55 lim Fk(x, t+) = F0(x, t+) x ∈ O, t ∈ [a, b). k→∞

For every k ∈ N, let xk :[a, b] → O be a solution of the generalized ordinary diﬀerential equation dx = DF (x, t). dτ k

If there exists a function x0 :[a, b] → O such that limk→∞ xk(t) = x0(t) uniformly for t ∈ [a, b], then x0 is a solution of dx = DF (x, t), t ∈ [a, b]. dτ 0

It should be remarked that Theorem 2.4 in [29] assumes that the functions Fk are defined on O×(−T,T ), where [a, b] ⊂ (−T,T ), and similarly the functions hk are defined in the open interval (−T,T ). However, it is easy to extend the functions defined on [a, b] to (−T,T ) by letting Fk(x, t) = Fk(x, a) for t ∈ (−T, a), Fk(x, t) = Fk(x, b) for t ∈ (b, T ), and similarly for hk. Note that the extended functions Fk now belong to the class F(O × (−T,T ), hk), as assumed in [29]. We are now ready to prove a continuous dependence theorem for measure functional differential equations.

n Theorem 4.6.3. Assume that X is a closed subspace of G([t0 − r, t0 + σ], R ), O is an open subset of X with the prolongation property, P = {yt; y ∈ O, t ∈ [t0, t0 + σ]}, g :[t0, t0 + σ] → R is a nondecreasing n left-continuous function, and fk : P × [t0, t0 + σ] → R , k ∈ N0, is a sequence of functions which satisfy the following conditions:

R t0+σ 1. The integral fk(yt, t) dg(t) exists for every k ∈ 0, y ∈ O. t0 N 2. There exists a constant M > 0 such that

kfk(y, t)k ≤ M

for every k ∈ N, y ∈ P and t ∈ [t0, t0 + σ]. 3. There exists a constant L > 0 such that

kfk(y, t) − fk(z, t)k ≤ Lky − zk∞

for every k ∈ N, y, z ∈ P and t ∈ [t0, t0 + σ]. 4. For every y ∈ O, Z t Z t lim fk(ys, s) dg(s) = f0(ys, s) dg(s) k→∞ t0 t0

uniformly with respect to t ∈ [t0, t0 + σ].

n 5. For every k ∈ N, x ∈ O, t ∈ [t0, t0 + σ], the function Fk(x, t):[t0 − r, t0 + σ] → R given by  0, t − r ≤ ϑ ≤ t ,  0 0 R ϑ fk(xs, s) dg(s), t0 ≤ ϑ ≤ t ≤ t0 + σ, Fk(x, t)(ϑ) = t0 R t  fk(xs, s) dg(s), t ≤ ϑ ≤ t0 + σ. t0 is an element of X.

Consider a sequence of functions φk ∈ P , k ∈ N0, such that limk→∞ φk = φ0 uniformly on [−r, 0]. Let yk ∈ O, k ∈ N, be solutions of Z t yk(t) = yk(t0) + fk((yk)s, s) dg(s), t ∈ [t0, t0 + σ], t0

(yk)t0 = φk.

56 If there exists a function y0 ∈ O such that limk→∞ yk = y0 on [t0, t0 + σ], then y0 is a solution of

Z t y0(t) = y0(t0) + f0((y0)s, s) dg(s), t ∈ [t0, t0 + σ], t0

(y0)t0 = φ0.

Proof. The assumptions imply that for every x ∈ O, limk→∞ Fk(x, t) = F0(x, t) uniformly with respect to t ∈ [t0, t0 + σ]. By the Moore-Osgood theorem, we have limk→∞ Fk(x, t+) = F0(x, t+) for every x ∈ O and t ∈ [t0, t0 + σ). Also, since X is a closed subspace, we have F0(x, t) ∈ X. It follows from Lemma 4.3.4 that Fk ∈ F(O × [t0, t0 + σ], h) for every k ∈ N, where

h(t) = (L + M)(g(t) − g(t0)), t ∈ [t0, t0 + σ].

Since limk→∞ Fk(x, t) = F0(x, t), we have F0 ∈ F(O × [t0, t0 + σ], h). For every k ∈ N0, t ∈ [t0, t0 + σ], let ( yk(ϑ), ϑ ∈ [t0 − r, t], xk(t)(ϑ) = yk(t), ϑ ∈ [t, t0 + σ].

According to Theorem 4.3.6, the function xk, where k ∈ N, is a solution of the generalized ordinary diﬀerential equation dx = DF (x, t). dτ k

When k ∈ N and t0 ≤ t1 ≤ t2 ≤ t0 + σ, we have

Z t2

kyk(t2) − yk(t1)k = fk((yk)s, s) dg(s) ≤ M(g(t2) − g(t1)) ≤ η(K(t2) − K(t1)), t1 where η(t) = Mt for every t ∈ [0, ∞) and K(t) = g(t) + t for every t ∈ [t0, t0 + σ]; note that K is ∞ an increasing function. Moreover, the sequence {yk(t0)}k=1 is bounded. Thus we see that condition 2 ∞ from Theorem 4.6.1 is satisfied and it follows that {yk}k=1 contains a subsequence which is uniformly ∞ convergent in [t0, t0 + σ]. Without loss of generality, we can denote this subsequence again by {yk}k=1. ∞ Since (yk)t0 = φk, we see that {yk}k=1 is in fact uniformly convergent in [t0 − r, t0 + σ]. By the definition of xk, we have lim xk(t) = x0(t) k→∞ uniformly with respect to t ∈ [t0, t0 + σ]. It follows from Theorem 4.6.2 that x0 is a solution of dx = DF (x, t) dτ 0 on [t0, t0 + σ]. The proof is finished by applying Theorem 4.3.7, which guarantees that y0 satisfies

Z t y0(t) = y0(t0) + f0((y0)s, s) dg(s), t ∈ [t0, t0 + σ], t0

(y0)t0 = φ0.

Remark 4.6.4. We remind the reader that although assumption 5 in the previous theorem looks compli- n cated, it is automatically satisﬁed if either g(t) = t for every t ∈ [t0, t0 + σ] and X = C([t0 − r, t0 + σ], R ), n or if X = G([t0 − r, t0 + σ], R ); see Remark 4.5.4. Using the previous result, we prove a continuous dependence theorem for functional dynamic equations on time scales.

57 n Theorem 4.6.5. Let [t0 − r, t0 + σ]T be a time scale interval, t0 ∈ T, B ⊂ R open, C = C([t0 − r, t0 + ∗ n σ]T,B), P = {yt ; y ∈ C, t ∈ [t0, t0 + σ]}. Consider a sequence of functions fk : P × [t0, t0 + σ]T → R , k ∈ N0, such that the following conditions are satisﬁed:

∗ 1. For every y ∈ C and k ∈ N0, the function t 7→ fk(yt , t) is rd-continuous on [t0, t0 + σ]T. 2. There exists a constant M > 0 such that

kfk(y, t)k ≤ M

for every k ∈ N0, y ∈ P and t ∈ [t0 − r, t0 + σ]T. 3. There exists a constant L > 0 such that

kfk(y, t) − fk(z, t)k ≤ Lky − zk∞

for every k ∈ N0, y, z ∈ P and t ∈ [t0 − r, t0 + σ]T. 4. For every y ∈ C, Z t Z t ∗ ∗ lim fk(ys , s)∆s = f0(ys , s)∆s k→∞ t0 t0

uniformly with respect to t ∈ [t0, t0 + σ]T.

Assume that φk ∈ C([t0−r, t0]T,B), k ∈ N0, is a sequence of functions such that limk→∞ φk = φ0 uniformly on [t0 − r, t0]T. Let yk ∈ C, k ∈ N be solutions of

∆ ∗ yk (t) = fk((yk)t, t), t ∈ [t0, t0 + σ]T,

yk(t) = φk(t), t ∈ [t0 − r, t0]T.

If there exists a function y0 ∈ C such that limk→∞ yk = y0 on [t0, t0 + σ]T, then y0 is a solution of

∆ ∗ y0 (t) = f0((y0 )s, s), t ∈ [t0, t0 + σ]T,

y0(t) = φ0(t), t ∈ [t0 − r, t0]T.

∗ n ∗ ∗ Proof. Let X = {y ; y ∈ C([t0 − r, t0 + σ]T, R )}, O = {y ; y ∈ C}, and g(t) = t for every t ∈ [t0, t0 + σ]. ∗ ∗ Note that O is an open subset of X and has the prolongation property. Further, let fk (y, t) = fk(y, t ) ∗ for every k ∈ N0, y ∈ P and t ∈ [t0, t0 + σ]. Consider an arbitrary y ∈ O and k ∈ N0. Since t 7→ fk(yt , t) R t0+σ is rd-continuous on [t0, t0 + σ] , the integral fk(yt, t)∆t exists. Using Theorem 4.4.1 and Theorem T t0 4.4.2, we have

Z t0+σ Z t0+σ Z t0+σ Z t0+σ ∗ ∗ ∗ fk(yt, t)∆t = fk(yt∗ , t ) dg(t) = fk(yt, t ) dg(t) = fk (yt, t) dg(t), t0 t0 t0 t0 i.e. the last integral exists. Using Theorem 4.4.1 again, we obtain

Z t Z t∗ Z t∗ Z t ∗ ∗ lim fk (ys, s) dg(s) = lim fk(ys, s)∆s = f0(ys, s)∆s = f0 (ys, s) dg(s) k→∞ k→∞ t0 t0 t0 t0

∗ ∗ uniformly with respect to t ∈ [t0, t0 + σ]. Further, it is clear that limk→∞ yk = y0 on [t0, t0 + σ], and ∗ ∗ limk→∞ φk = φ0 uniformly on [t0 − r, t0]. By Theorem 4.4.3, we have

Z t ∗ ∗ ∗ ∗ yk(t) = yk(t0) + fk((yk)s, s ) dg(s), t ∈ [t0, t0 + σ], t0 (y∗) = φ∗. k t0 k

58 ∗ ∗ ∗ for every k ∈ N0. The functions fk , yk and φk, k ∈ N0, satisfy the assumptions of Theorem 4.6.3, and we conclude that Z t ∗ ∗ ∗ ∗ y0 (t) = y0 (t0) + f0((y0 )s, s ) dg(s), t ∈ [t0, t0 + σ], t0 (y∗) = φ∗. 0 t0 0

n By Theorem 4.4.3, it follows that y0 :[t0 − r, t0 + σ]T → R satisﬁes

∆ ∗ y0 (t) = f0((y0 )s, s), t ∈ [t0, t0 + σ]T,

y0(t) = φ0(t), t ∈ [t0 − r, t0]T.

Remark 4.6.6. An inspection of the proof of Theorem 4.6.3 reveals that the hypothesis limk→∞ yk = y0 ∞ is not necessary to conclude that {yk}k=1 has a uniformly convergent subsequence. However, we need a condition guaranteeing that the limit function belongs to O. Thus, instead of requiring that limk→∞ yk = 0 0 y0 ∈ O, it is possible to assume the existence of a closed set O ⊂ O such that yk ∈ O for every k ∈ N. ∞ Then it follows that {yk}k=1 has a subsequence which is uniformly convergent to a function y0 ∈ O such that Z t y0(t) = y0(t0) + f0((y0)s, s) dg(s), t ∈ [t0, t0 + σ], t0

(y0)t0 = φ0.

0 Similarly, in Theorem 4.6.5, we may assume the existence of a closed set B ⊂ B such that yk takes values 0 in B for every k ∈ N, and omit the condition limk→∞ yk = y0.

4.7 Periodic averaging theorems

Averaging theorems provide a useful tool for approximating solutions of a non-autonomous equation by solutions of an autonomous equation whose right-hand side is obtained by averaging the original right-hand side with respect to t (see e.g. [66]). The approximation is especially good in the case when the original right-hand side is periodic with respect to t. In this section, we use the following periodic averaging theorem for measure functional diﬀerential equations from Chapter 3 to obtain a new theorem on periodic averaging for functional dynamic equations.

n Theorem 4.7.1. Let ε0 > 0, L > 0, T > 0, B ⊂ R , X = G([−r, 0],B). Consider a pair of bounded n n functions f : X × [0, ∞) → R , g : X × [0, ∞) × (0, ε0] → R and a nondecreasing left-continuous function h : [0, ∞) → R such that the following conditions are satisﬁed: R b 1. The integral 0 f(yt, t) dh(t) exists for every b > 0 and y ∈ G([−r, b],B). 2. f is Lipschitz-continuous in the ﬁrst variable.

3. f is T -periodic in the second variable.

4. There is a constant α > 0 such that h(t + T ) − h(t) = α for every t ≥ 0.

5. The integral 1 Z T f0(x) = f(x, s) dh(s) T 0 exists for every x ∈ X.

59 Let φ ∈ X. Suppose that for every ε ∈ (0, ε0], the initial-value problems

Z t Z t 2 x(t) = x(0) + ε f(xs, s) dh(s) + ε g(xs, s, ε) dh(s), x0 = φ, 0 0 Z t y(t) = y(0) + ε f0(ys) ds, y0 = φ 0 have solutions xε, yε :[−r, L/ε] → B. Then there exists a constant J > 0 such that

kxε(t) − yε(t)k ≤ Jε for every ε ∈ (0, ε0] and t ∈ [0, L/ε]. To be able to speak about periodic functions on time scales, we need the following concept of a periodic time scale.

Deﬁnition 4.7.2. Let T > 0 be a real number. A time scale T is called T -periodic if t ∈ T implies t + T ∈ T and µ(t) = µ(t + T ). We now proceed to the periodic averaging theorem for functional dynamic equations on time scales.

n Theorem 4.7.3. Assume that T > 0, T is a T -periodic time scale, t0 ∈ T, ε0 > 0, L > 0, B ⊂ R . n Consider a pair of bounded functions f : G([−r, 0],B) × [t0, ∞)T → R , g : G([−r, 0],B) × [t0, ∞)T × n (0, ε0] → R such that the following conditions are satisﬁed:

1. For every b > t0 and y ∈ G([t0 − r, b],B), the function t 7→ f(yt, t) is regulated on [t0, b]T. ∗ 2. For every b > t0 and y ∈ C([t0 − r, b]T,B), the function t 7→ f(yt , t) is rd-continuous on [t0, b]T. 3. f is Lipschitz-continuous in the ﬁrst variable. 4. f is T -periodic and rd-continuous in the second variable. Denote 1 Z t0+T f0(y) = f(y, s)∆s, y ∈ G([−r, 0],B). T t0

Let φ ∈ C([t0 − r, t0]T,B). Suppose that for every ε ∈ (0, ε0], the functional dynamic equation

∆ ∗ 2 ∗ x (t) = εf(xt , t) + ε g(xt , t, ε),

x(t) = φ(t), t ∈ [t0 − r, t0]T

n has a solution xε :[t0 − r, t0 + L/ε]T → R , and that the functional diﬀerential equation Z t y(t) = y(t0) + ε f0(ys) ds, t0 ∗ yt0 = φ

n has a solution yε :[t0 − r, t0 + L/ε] → R . Then there exists a constant J > 0 such that

kxε(t) − yε(t)k ≤ Jε, for every ε ∈ (0, ε0] and t ∈ [t0, t0 + L/ε]T.

Proof. Without loss of generality, we can assume that t0 = 0; otherwise, consider a shifted problem with the time scale Te = {t − t0; t ∈ T} and the right-hand side fe(x, t) = f(x, t + t0). For every t ∈ [0, ∞), y ∈ G([−r, 0],B) and ε ∈ (0, ε0], let

f ∗(y, t) = f(y, t∗) and g∗(y, t, ε) = g(y, t∗, ε).

60 Also, let h(t) = t∗ for every t ∈ [0, ∞). It follows directly from the deﬁnition of h and the fact that T is T -periodic that h(t + T ) − h(t) = T, t ≥ 0. ∗ By Theorem 4.4.3, xε satisﬁes Z t Z t ∗ ∗ ∗ ∗ 2 ∗ ∗ xε(t) = xε(0) + ε f ((xε)s, s) dh(s) + ε g ((xε)s, s, ε) dh(s), t ∈ [0, L/ε] 0 0 ∗ ∗ (xε)0 = φ . for every ε ∈ (0, ε0]. From Theorem 4.4.1, we have

Z T Z T 1 1 ∗ f0(y) = f(y, s)∆s = f (y, s) dh(s), y ∈ G([−r, 0],B). T 0 T 0

For every b ∈ [0, ∞)T and y ∈ G([−r, b],B), the function u(t) = f(yt, t) is regulated on [0, b]T. Conse- n quently, there is a sequence of continuous functions un : [0, b]T → R , n ∈ N, which is uniformly convergent ∗ ∞ ∗ to u. It follows that {un}n=1 is uniformly convergent to u on [0, b]. Using Theorem 4.4.1 and uniform convergence theorems for the Kurzweil-Stieltjes and ∆-integrals, we obtain

Z b Z b Z b Z b Z b ∗ ∗ ∗ u(t)∆t = lim un(t)∆t = lim un(t) dh(t) = u (t) dh(t) = f(yt∗ , t ) dh(t). 0 n→∞ 0 n→∞ 0 0 0

R b ∗ Theorem 4.4.2 implies the existence of 0 f (yt, t) dh(t). Since f ∗ and g∗ satisfy all assumptions of Theorem 4.7.1, there exists a constant J > 0 such that

∗ kxε(t) − yε(t)k ≤ Jε,

∗ for every ε ∈ (0, ε0] and t ∈ [0, L/ε]. The proof is ﬁnished by observing that xε(t) = xε(t) for t ∈ [0, L/ε]T.

61 Chapter 5

Basic results for functional diﬀerential and dynamic equations involving impulses

5.1 Introduction

Let r, σ > 0 be given numbers and t0 ∈ R. In Chapter 4, we have introduced equations of the form Z t x(t) = x(t0) + f(xs, s) dg(s), t ∈ [t0, t0 + σ], t0

xt0 = φ, where xt denotes the function xt(θ) = x(t + θ), θ ∈ [−r, 0], for every t ∈ [t0, t0 + σ]. The integral on the right-hand side should be understood as the Kurzweil-Stieltjes integral taken with respect to a nondecreasing function g :[t0, t0 + σ] → R. These equations are called measure functional differential equations; they generalize the usual type of functional differential equation which corresponds to the case g(t) = t. We have shown that functional dynamic equations on time scales represent a special case of measure functional differential equations, and obtained various results concerning the existence and uniqueness of solutions, continuous dependence, and periodic averaging for both types of equations. Our aim in this chapter is to demonstrate that measure functional differential equations represent an adequate tool for dealing with differential and dynamic equations involving impulses. Section 2 presents some results about Kurzweil-Stieltjes integrals. In Section 3, we introduce impulsive measure functional differential equations and show how to transform them into measure functional differential equations without impulses. In Section 4, we explain the relation between time scale integrals and Kurzweil-Stieltjes integrals (this part is independent of the previous sections and might be useful for readers interested in integration theory on time scales). Section 5 discusses impulsive functional dynamic equations on time scales and demonstrates how to convert them to impulsive measure functional differential equations. Using the results from the last two sections, we are able to relate impulsive functional dynamic equations and measure functional differential equations. In the final three sections, we employ this correspondence to obtain theorems on the existence and uniqueness of solutions, continuous dependence of solutions on parameters, and periodic averaging for impulsive equations. It is worth mentioning here that in the classical theory of functional differential equations of the form

0 x (t) = f(xt, t), t ∈ [t0, t0 + σ], (5.1.1) it is common to use the following assumptions on the right-hand side of the equation: • There exists a constant M > 0 such that kf(x, t)k ≤ M for each x in a certain subset of the phase space and every t ∈ [t0, t0 + σ].

62 • There exists a constant L > 0 such that kf(x, t) − f(y, t)k ≤ Lkx − yk for each x, y in a certain subset of the phase space and every t ∈ [t0, t0 + σ]. However, it became clear (see e.g. [24, 27]) that it is suﬃcient to impose certain conditions on the indeﬁnite integral of the function on the right-hand side of (5.1.1) rather than on the right-hand side itself. In the present chapter, we consider the following weaker conditions: • There exists a constant M > 0 such that

Z u2

f(xt, t) dg(t) ≤ M(g(u2) − g(u1)) u1

for all u1, u2 ∈ [t0, t0 + σ] and all x in a certain subset of the phase space. • There exists a constant L > 0 such that

Z u2 Z u2

(f(xt, t) − f(yt, t)) dg(t) ≤ L kxt − ytk∞ dg(t) u1 u1

for all u1, u2 ∈ [t0, t0 + σ] and all x, y in a certain subset of the phase space.

5.2 Preliminaries

A function f :[a, b] → Rn is called regulated, if the limits lim f(s) = f(t−) ∈ n, t ∈ (a, b] and lim f(s) = f(t+) ∈ n, t ∈ [a, b) s→t− R s→t+ R exist. The set of all regulated functions f :[a, b] → B, where B ⊂ Rn, will be denoted by G([a, b],B). n Note that G([a, b], R ) is a Banach space under the usual supremum norm kfk∞ = supa≤t≤b kf(t)k. Given a regulated function f, the symbols ∆+f(t) and ∆−f(t) will be used throughout this chapter to denote ∆+f(t) = f(t+) − f(t) and ∆−f(t) = f(t) − f(t−). In the following sections, we often assume the existence of certain Kurzweil-Stieltjes integrals. The next result from [67, Corollary 1.34] is not really necessary for us, but we mention it here as it provides a useful suﬃcient condition for the existence of the Kurzweil-Stieltjes integral.

Theorem 5.2.1. If f :[a, b] → Rn is a regulated function and g :[a, b] → R is a nondecreasing function, R b then the integral a f dg exists. The following Hake-type theorem for the Kurzweil-Stieltjes integral is a special case of Theorem 1.14 in [67] (see also Remark 1.15 in the same book).

Theorem 5.2.2. Consider a pair of functions f :[a, b] → Rn and g :[a, b] → R. R t 1. Assume that the integral a f dg exists for every t ∈ [a, b) and Z t lim f dg + f(b)(g(b) − g(t)) = I. t→b− a R b Then a f dg = I. R b 2. Assume that the integral t f dg exists for every t ∈ (a, b] and

Z b ! lim f dg + f(a)(g(t) − g(a)) = I. t→a+ t

R b Then a f dg = I.

63 We also need the following related result, which is a special case of Theorem 1.16 in [67].

Theorem 5.2.3. Let f :[a, b] → Rn and g :[a, b] → R be a pair of functions such that g is regulated and R b a f dg exists. Then the functions Z t Z b h(t) = f dg and k(t) = f dg a t are regulated on [a, b] and satisfy

h(t+) = h(t) + f(t)∆+g(t), t ∈ [a, b), h(t−) = h(t) − f(t)∆−g(t), t ∈ (a, b], k(t+) = k(t) − f(t)∆+g(t), t ∈ [a, b), k(t−) = k(t) + f(t)∆−g(t), t ∈ (a, b].

We remark here that, according to the previous theorem, solutions of measure functional diﬀerential equations must be regulated functions.

Lemma 5.2.4. Let m ∈ N, a ≤ t1 < t2 < ··· < tm ≤ b. Consider a pair of functions f :[a, b] → R and ˜ g :[a, b] → R, where g is regulated, left-continuous on [a, b], and continuous at t1, . . . , tm. Let f :[a, b] → R ˜ and g˜ :[a, b] → R be such that f(t) = f(t) for every t ∈ [a, b]\{t1, . . . , tm} and g˜ − g is constant on each R b ˜ of the intervals [a, t1], (t1, t2],..., (tm−1, tm], (tm, b]. Then the integral a f d˜g exists if and only if the R b integral a f dg exists; in that case, we have Z b Z b ˜ X ˜ + f d˜g = f dg + f(tk)∆ g˜(tk). a a k∈{1,...,m}, tk

Z t1 f˜d(˜g − g) = 0. a It follows from Theorem 5.2.2 and the deﬁnition of the Kurzweil-Stieltjes integral that

Z tk+1 Z tk+1 ˜ ˜ ˜ + ˜ + f d(˜g − g) = lim f d(˜g − g) + f(tk)∆ (˜g − g)(tk) = f(tk)∆ g˜(tk) τ→tk+ tk τ R b ˜ for every k ∈ {1, . . . , m − 1}. If tm = b, then f d(˜g − g) = 0; otherwise, tm Z b Z b ˜ ˜ ˜ + ˜ + f d(˜g − g) = lim f d(˜g − g) + f(tm)∆ (˜g − g)(tm) = f(tm)∆ g˜(tm). τ→tm+ tm τ R b ˜ Consequently, a f d(˜g − g) exists and Z b ˜ X ˜ + f d(˜g − g) = f(tk)∆ g˜(tk). a k∈{1,...,m}, tk

Z t1 Z τ Z τ Z t1 f˜dg = lim f˜dg = lim f dg = f dg, a τ→t1− a τ→t1− a a

Z tk+1 Z τ Z τ Z tk+1 f˜dg = lim f˜dg = lim f dg = f dg, k ∈ {1, . . . , m − 1}, σ→tk+, σ→tk+, tk σ σ tk τ→tk+1− τ→tk+1−

64 Z b Z b Z b Z b f˜dg = lim f˜dg = lim f dg = f dg. τ→tm+ τ→tm+ tm τ τ tm These three relations might be read not only from left to right, but also from right to left; in other words, the integrals on the left-hand sides exist if and only if the integrals on the right-hand sides exist. Combining R b ˜ R b the three relations, we see that a f dg exists if and only if a f dg exists; in this case, their values are equal. To conclude the proof, it is suﬃcent to observe that Z b Z b Z b Z b ˜ ˜ ˜ X ˜ + f d˜g = f dg + f d(˜g − g) = f dg + f(tk)∆ g˜(tk). a a a a k∈{1,...,m}, tk

A measure functional diﬀerential equation has the form Z t x(t) = x(t0) + f(xs, s) dg(s), t ∈ [t0, t0 + σ], t0

xt0 = φ, where the Kurzweil-Stieltjes integral on the right-hand side is taken with respect to a nondecreasing function g :[t0, t0 + σ] → R; these equations have been studied in Chapter 4. We assume that g is a left-continuous function and consider the possibility of adding impulses at preassigned times t1, . . . , tm, where t0 ≤ t1 < ··· < tm < t0 + σ. For every k ∈ {1, . . . , m}, the impulse n n + at tk is described by the operator Ik : R → R . In other words, the solution x should satisfy ∆ x(tk) = Ik(x(tk)). This leads us to the following problem: R v x(v) − x(u) = u f(xs, s) dg(s), whenever u, v ∈ Jk for some k ∈ {0, . . . , m}, + ∆ x(tk) = Ik(x(tk)), k ∈ {1, . . . , m},

xt0 = φ, where J0 = [t0, t1], Jk = (tk, tk+1] for k ∈ {1, . . . , m − 1}, and Jm = (tm, t0 + σ]. R v The value of the integral u f(xs, s) dg(s), where u, v ∈ Jk, does not change if we replace g by a function g˜ such that g − g˜ is a constant function on Jk (this follows easily from the deﬁnition of the Kurzweil- + Stieltjes integral). Thus, without loss of generality, we can assume that g is such that ∆ g(tk) = 0 for every k ∈ {1, . . . , m}. Since g is a left-continuous function, it follows that g is continuous at t1, . . . , tm. Under this assumption, our problem can be rewritten as

Z t X x(t) = x(t0) + f(xs, s) dg(s) + Ik(x(tk)), t ∈ [t0, t0 + σ], (5.3.1) t0 k∈{1,...,m}, tk

xt0 = φ. R t Indeed, the function t 7→ f(xs, s) dg(s) is continuous at t1, . . . , tm (see Theorem 5.2.3), and therefore t0 + ∆ x(tk) = Ik(x(tk)) for every k ∈ {1, . . . , m}. Pm Alternatively, the sum on the right-hand side of (5.3.1) might be written as k=1 Ik(x(tk))Htk (t), where Hv denotes the characteristic function of (v, ∞), i.e. Hv(t) = 0 for t ≤ v and Hv(t) = 1 for t > v. The following theorem shows that impulsive measure functional diﬀerential equations of the form (5.3.1) can always be transformed to measure functional diﬀerential equations without impulses.

n n Theorem 5.3.1. Let m ∈ N, t0 ≤ t1 < ··· < tm < t0 + σ, B ⊂ R , I1,...,Im : B → R , P = n G([−r, 0],B), f : P × [t0, t0 + σ] → R . Assume that g :[t0, t0 + σ] → R is a regulated left-continuous function which is continuous at t1, . . . , tm. For every y ∈ P , deﬁne ( f(y, t), t ∈ [t , t + σ]\{t , . . . , t }, f˜(y, t) = 0 0 1 m Ik(y(0)), t = tk for some k ∈ {1, . . . , m}.

65 Moreover, let the function g˜ :[t0, t0 + σ] → R be given by  g(t), t ∈ [t , t ],  0 1 g˜(t) = g(t) + k, t ∈ (tk, tk+1] for some k ∈ {1, . . . , m − 1},  g(t) + m, t ∈ (tm, t0 + σ].

Then x ∈ G([t0 − r, t0 + σ],B) is a solution of Z t X x(t) = x(t0) + f(xs, s) dg(s) + Ik(x(tk)), t ∈ [t0, t0 + σ], (5.3.2) t0 k∈{1,...,m}, tk

xt0 = φ if and only if Z t ˜ x(t) = x(t0) + f(xs, s) d˜g(s), t ∈ [t0, t0 + σ], (5.3.3) t0

xt0 = φ. + Proof. By the deﬁnition ofg ˜, we have ∆ g˜(tk) = 1 for every k ∈ {1, . . . , m}. According to Lemma 5.2.4, we obtain Z t Z t ˜ X + f(xs, s) d˜g(s) = f(xs, s) dg(s) + f(xtk , tk)∆ g˜(tk) t0 t0 k∈{1,...,m}, tk

Remark 5.3.2. When g(t) = t for every t ∈ [t0, t0 + σ], Eq. (5.3.2) reduces to the usual type of impulsive functional diﬀerential equation Z t X x(t) = x(t0) + f(xs, s) ds + Ik(x(tk)), t ∈ [t0, t0 + σ]. t0 k∈{1,...,m}, tk

Moreover, let the function g˜ :[t0, t0 + σ] → R be given by  g(t), t ∈ [t , t ],  0 1 g˜(t) = g(t) + k, t ∈ (tk, tk+1] for some k ∈ {1, . . . , m − 1},  g(t) + m, t ∈ (tm, t0 + σ]. Then the following statements are true:

66 1. The function g˜ is nondecreasing.

+ 2. Assume there exist constants M1,M2 ∈ R such that

Z u2

f(yt, t) dg(t) ≤ M1(g(u2) − g(u1)) u1

whenever t0 ≤ u1 ≤ u2 ≤ t0 + σ, y ∈ O, and

kIk(x)k ≤ M2

for every k ∈ {1, . . . , m} and x ∈ B. Then

Z u2 ˜ f(yt, t) d˜g(t) ≤ (M1 + M2)(˜g(u2) − g˜(u1)) u1

whenever t0 ≤ u1 ≤ u2 ≤ t0 + σ and y ∈ O.

+ 3. Assume there exist constants L1,L2 ∈ R , such that

Z u2 Z u2

(f(yt, t) − f(zt, t)) dg(t) ≤ L1 kyt − ztk∞ dg(t) u1 u1

whenever t0 ≤ u1 ≤ u2 ≤ t0 + σ, y, z ∈ O, and

kIk(x) − Ik(y)k ≤ L2kx − yk

for every k ∈ {1, . . . , m} and x, y ∈ B. Then

Z u2 Z u2 ˜ ˜ f(yt, t) − f(zt, t) d˜g(t) ≤ (L1 + L2) kyt − ztk∞ d˜g(t) u1 u1

whenever t0 ≤ u1 ≤ u2 ≤ t0 + σ and y, z ∈ O. Proof. It is clear from the deﬁnition ofg ˜ that it is nondecreasing if g is nondecreasing. Moreover,

g˜(v) − g˜(u) ≥ g(v) − g(u) (5.3.4) whenever t0 ≤ u ≤ v ≤ t0 + σ. To prove the second statement, let t0 ≤ u1 ≤ u2 ≤ t0 + σ, y ∈ O. From Lemma 5.2.4, we obtain

Z u2 Z u2 ˜ X + f(yt, t) d˜g(t) = f(yt, t) dg(t) + Ik(y(tk))∆ g˜(tk), u1 u1 k∈{1,...,m}, u1≤tk

u1 k∈{1,...,m}, u1≤tk

≤ M1(˜g(u2) − g˜(u1)) + M2(˜g(u2) − g˜(u1)) = (M1 + M2)(˜g(u2) − g˜(u1)).

To prove the third statement, let t0 ≤ u1 ≤ u2 ≤ t0 + σ and y, z ∈ O. Using Lemma 5.2.4 again, we obtain

Z u2 ˜ ˜ f(yt, t) − f(zt, t) d˜g(t) u1

Z u2 X + = (f(yt, t) − f(zt, t)) dg(t) + (Ik(y(tk)) − Ik(z(tk)))∆ g˜(tk). u1 k∈{1,...,m}, u1≤tk

67 Consequently,

Z u2 Z u2 ˜ ˜ X + f(yt, t) − f(zt, t) d˜g(t) ≤ L1 kyt − ztk∞ dg(t) + L2 ky(tk) − z(tk)k∆ g˜(tk).

u1 u1 k∈{1,...,m}, u1≤tk

Using Eq. (5.3.4) and the deﬁnition of the Kurzweil-Stieltjes integral, we see that

Z u2 Z u2 kyt − ztk∞ dg(t) ≤ kyt − ztk∞ d˜g(t). u1 u1 Next, we observe that the function Z s h(s) = kyt − ztk∞ d˜g(t), s ∈ [t0, t0 + σ], t0

+ + is nondecreasing and ∆ h(tk) = kytk − ztk k∞∆ g˜(tk) for k ∈ {1, . . . , m}. Therefore

X + X + L2 ky(tk) − z(tk)k∆ g˜(tk) ≤ L2 kytk − ztk k∞∆ g˜(tk) k∈{1,...,m}, k∈{1,...,m}, u1≤tk

Z u2 X + = L2 ∆ h(tk) ≤ L2(h(u2) − h(u1)) = L2 kyt − ztk∞ d˜g(t), k∈{1,...,m}, u1 u1≤tk

Z u2 Z u2 ˜ ˜ f(yt, t) − f(zt, t) d˜g(t) ≤ (L1 + L2) kyt − ztk∞ d˜g(t). u1 u1 5.4 Integration on time scales

In this section, we clarify the relation between time scale integrals and Kurzweil-Stieltjes integrals. A function f : T → R is called rd-continuous, if it is regulated on T and continuous at right-dense n points of T. For each pair of numbers a, b ∈ T, a ≤ b, let [a, b]T = [a, b] ∩ T. Given a set B ⊂ R , the symbol G([a, b]T,B) will be used to denote the set of all regulated functions f :[a, b]T → B. 0 R b In the time scale calculus, the usual derivative f (t) and integral a f(t) dt of a function f :[a, b] → R ∆ R b are replaced by the ∆-derivative f (t) and ∆-integral a f(t) ∆t, where f :[a, b]T → R. Similarly to the R b classical case, there exist various definitions of the ∆-integral a f(t) ∆t, such as the Riemann ∆-integral or Lebesgue ∆-integral; these definitions as well as the definition of the ∆-derivative can be found in [8, 9]. The more general Kurzweil-Henstock ∆-integral was introduced in [63] (see below). Let us recall the notation from Chapter 2: Given a real number t ≤ sup T, let

∗ t = inf{s ∈ T; s ≥ t}.

Since T is a closed set, we have t∗ ∈ T. Further, let (−∞, sup ] if sup < ∞, ∗ = T T T (−∞, ∞) otherwise.

Finally, given a function f : T → Rn, we consider its extension f ∗ : T∗ → Rn given by

∗ ∗ ∗ f (t) = f(t ), t ∈ T . The following theorem from Chapter 2 describes the relation between the ∆-integral and the Kurzweil- Stieltjes integral.

68 Theorem 5.4.1. Let f : T → Rn be an rd-continuous function. Choose an arbitrary a ∈ T and define Z t F1(t) = f(s)∆s, t ∈ T, a Z t ∗ ∗ F2(t) = f (s) dg(s), t ∈ T , a ∗ ∗ ∗ where g(s) = s for every s ∈ T . Then F2 = F1 . n In particular, if f :[a, b]T → R is an rd-continuous function, we obtain Z b Z b f(s)∆s = f ∗(s) dg(s). (5.4.1) a a Since the solutions of impulsive equations are discontinuous, we need to relax the assumption of rd- continuity. It is not difficult to show that Eq. (5.4.1) remains true in the more general case where f is a regulated function; it is sufficient to use uniform convergence theorems for both types of integrals and the fact that every regulated function is a uniform limit of continuous functions. Although regulated functions are general enough for our purposes, we take this opportunity to prove a much stronger result: All we need to require for Eq. (5.4.1) to hold is that f is ∆-integrable in Kurzweil-Henstock’s sense. At first, we recall the definition of the Kurzweil-Henstock ∆-integral as introduced by A. Peterson and B. Thompson in [63].

Let δ = (δL, δR) be a pair of nonnegative functions deﬁned on [a, b]T. We say that δ is a ∆-gauge for [a, b]T provided δL(t) > 0 on (a, b] ∩ T, δR(t) > 0 on [a, b) ∩ T, and δR(t) ≥ µ(t) for all t ∈ [a, b) ∩ T. A tagged partition of [a, b]T consists of division points s0, . . . , sm ∈ [a, b]T such that a = s0 < s1 < ··· < sm = b, and tags τ1, . . . , τm ∈ [a, b]T such that τi ∈ [si−1, si] for every i ∈ {1, . . . , m}. Such a partition is called δ-ﬁne if τi − δL(τi) ≤ si−1 < si ≤ τi + δR(τi), i ∈ {1, . . . , m}. n n A function f :[a, b]T → R is called Kurzweil-Henstock ∆-integrable, if there exists a vector I ∈ R such that for every ε > 0, there is a ∆-gauge δ on [a, b]T such that

m X f(τi)(si − si−1) − I < ε i=1 for every δ-ﬁne tagged partition of [a, b]T. In this case, I is called the Kurzweil-Henstock ∆-integral of f R b over [a, b]T and will be denoted by a f(t)∆t. Here is the promised result which shows that ∆-integrals are in fact special cases of Kurzweil-Stieltjes integrals.

n ∗ Theorem 5.4.2. Let f :[a, b]T → R be an arbitrary function. Deﬁne g(t) = t for every t ∈ [a, b]. Then R b R b ∗ the Kurzweil-Henstock ∆-integral a f(t)∆t exists if and only if the Kurzweil-Stieltjes integral a f (t) dg(t) exists; in this case, both integrals have the same value.

Proof. For an arbitrary tagged partition P of [a, b] consisting of division points a = s0 < s1 < ··· < sm = b and tags τ1, . . . , τm, let

m m X ∗ X ∗ ∗ ∗ S(P ) = f (τi)(g(si) − g(si−1)) = f(τi )(si − si−1). (5.4.2) i=1 i=1

R b At ﬁrst, assume that a f(t)∆t exists. Then, given an arbitrary ε > 0, there is a ∆-gauge δ = (δL, δR) on [a, b] such that T m Z b X f(τi)(si − si−1) − f(t)∆t < ε i=1 a

69 ˜ + for every δ-fine tagged partition of [a, b]T. We construct a gauge δ :[a, b] → R in the following way:  min(δL(t), sup{d; t + d ∈ [a, b] , d ≤ δR(t)}) if t ∈ (a, b) ∩ T,  T sup{d; a + d ∈ [a, b] , d ≤ δ (a)} if t = a, δ˜(t) = T R δ (b) if t = b,  L  1 2 inf {|t − s|, s ∈ T} if t ∈ [a, b]\T. ˜ Let P be an arbitrary δ-fine tagged partition of [a, b] with division points a = s0 < s1 < ··· < sm = b and tags τi ∈ [si−1, si], i ∈ {1, . . . , m}. For every i ∈ {1, . . . , m}, there are two possibilities: either τi ∈ T, or [si−1, si] ∩ T = ∅. 0 The division points s0, . . . , sm and tags τ1, . . . , τm need not belong to T, but we can find a partition P whose division points and tags belong to T, S(P ) = S(P 0), and P 0 is δ-fine. We proceed by induction: Clearly, s0 = a ∈ T. Now, consider an interval [si−1, si] with si−1 ∈ T. Since [si−1, si] ∩ T 6= ∅, we must ∗ have τi ∈ T. If si ∈/ T, we replace the division point si by si , delete all division points sj belonging ∗ ∗ to (si, si ), and also all tags τj belonging to (si, si ). This operation keeps the value of the integral sum ∗ (5.4.2) unchanged: The contributions of the intervals [si−1, si] and [si−1, si ] to the value of the sum are ∗ ∗ ∗ ∗ the same, and the contributions of intervals [sj−1, sj] contained in (si, si ) are zero because sj−1 = sj = si . It remains to check that the modified partition is δ-fine. Let M = sup([a, τi + δR(τi)] ∩ T). Obviously, ˜ M ∈ [a, b]T. Since our original partition was δ-fine, it follows that ˜ si ≤ τi + δ(τi) ≤ τi + sup{d; τi + d ∈ [a, b]T, d ≤ δR(τi)} = M. ∗ ∗ But si ∈/ T and M ∈ T implies si ≤ M, because si is the smallest time scale point larger than si. ∗ Consequently, si ≤ M ≤ τi + δR(τi). 0 Now, P is a δ-fine tagged partition of [a, b]T, and therefore

Z b Z b 0 S(P ) − f(t)∆t = S(P ) − f(t)∆t < ε, a a

R b ∗ R b which proves that a f (t) dg(t) exists and equals a f(t)∆t. R b ∗ ˜ Conversely, assume that a f (t) dg(t) exists. Then, given an arbitrary ε > 0, there is a gauge δ : [a, b] → R+ such that

m Z b X ∗ ∗ ∗ ∗ f(τi )(si − si−1) − f (t) dg(t) < ε i=1 a ˜ for every δ-fine tagged partition of [a, b]. We construct a ∆-gauge δ = (δL, δR) on [a, b] by letting ˜ ˜ T δL(t) = δ(t) and δR(t) = max(δ(t), µ(t)) for every t ∈ [a, b]T. Consider an arbitrary δ-fine tagged partition P of [a, b]T with division points a = s0 < s1 < ··· < sm = b and tags τi ∈ [si−1, si], i ∈ {1, . . . , m}; by definition, all these points belong to T. Our δ-fine partition need not be δ˜-fine: for certain values of i ∈ {1, . . . , m}, it can happen that ˜ δR(τi) + τi ≥ si > δ(τi) + τi. In this case, we have δR(τi) = µ(τi), the point τi is right-scattered, and 0 ˜ si = σ(τi). We claim that it is possible to find a modified tagged partition P of [a, b] which is δ-fine and 0 ˜ S(P ) = S(P ). To this end, replace the division point si by τi + δ(τi) while keeping τi as the tag for the ˜ ˜ ˜ interval [si−1, τi + δ(τi)], and cover the interval [τi + δ(τi), si] by an arbitrary δ-fine partition. The equality 0 ∗ S(P ) = S(P ) follows from the fact that t = si for every t ∈ (τi, si]. The proof is concluded by observing that

m Z b X ∗ f(τi)(si − si−1) − f (t) dg(t) i=1 a

Z b Z b ∗ 0 ∗ = S(P ) − f (t) dg(t) = S(P ) − f (t) dg(t) < ε, a a R b R b ∗ which implies that a f(t)∆t exists and equals a f (t) dg(t).

70 Remark 5.4.3. Several authors have been interested in Stieltjes-type integrals on time scales (see e.g. [42, R b n 60]). For example, the Riemann-Stieltjes ∆-integral a f(t)∆g(t) of a function f :[a, b]T → R with respect to a function g :[a, b]T → R can be deﬁned in a straightforward way by taking the deﬁnition of the Riemann ∆-integral and replacing the usual integral sums by

m X f(τi)(g(si) − g(si−1)). i=1 Alternatively, we can start with the deﬁnition of the Kurzweil-Henstock ∆-integral and modify the integral sums in the same way. Using exactly the same reasoning as in the proof of Theorem 5.4.2, one can show that the resulting Stieltjes-type ∆-integral satisﬁes

Z b Z b f(t)∆g(t) = f ∗(t) dg∗(t). a a Consequently, many properties of the ∆-integrals can be simply derived from the known properties of the Kurzweil-Stieltjes integrals.

Lemma 5.4.4. Let a, b ∈ T, a < b, g(t) = t∗ for every t ∈ [a, b]. If f :[a, b] → Rn is such that the integral R b a f(t) dg(t) exists, then Z d Z d∗ f(t) dg(t) = f(t) dg(t) c c∗ for every c, d ∈ [a, b]. Proof. Using the deﬁnition of the Kurzweil-Stieltjes integral and the fact that g is constant on [c, c∗] and ∗ ∗ ∗ R c R d on [d, d ], we see that c f(t) dg(t) = 0 and d f(t) dg(t) = 0. Therefore Z d Z c∗ Z d Z d∗ Z d∗ f(t) dg(t) = f(t) dg(t) + f(t) dg(t) + f(t) dg(t) = f(t) dg(t). c c c∗ d c∗

n R b Theorem 5.4.5. Let f : T → R be a function such that the Kurzweil-Henstock integral a f(s)∆s exists for every a, b ∈ T, a < b. Choose an arbitrary a ∈ T and deﬁne Z t F1(t) = f(s)∆s, t ∈ T, a Z t ∗ ∗ F2(t) = f (s) dg(s), t ∈ T , a ∗ ∗ ∗ where g(s) = s for every s ∈ T . Then F2 = F1 . Proof. The statement is a simple consequence of Lemma 5.4.4 and Theorem 5.4.2:

Z t Z t∗ Z t∗ ∗ ∗ ∗ ∗ F2(t) = f (s) dg(s) = f (s) dg(s) = f(s)∆s = F1(t ) = F1 (t) a a a 5.5 Impulsive functional dynamic equations on time scales

In this section, we focus our attention on functional dynamic equations with impulses. In particular, we explain the relation between this type of equations and impulsive measure functional diﬀerential equations, which were discussed in Section 3. In Chapter 4, we dealt with functional dynamic equations of the form

∆ ∗ x (t) = f(xt , t), t ∈ [t0, t0 + σ]T,

x(t) = φ(t), t ∈ [t0 − r, t0]T.

71 ∗ ∗ ∗ The symbol xt should be understood as (x )t; as explained in Chapter 4, the advantage of using xt rather ∗ than xt stems from the fact that xt is always deﬁned on the whole interval [−r, 0], whereas xt is deﬁned only on a subset of [−r, 0]; moreover, this subset depends on t. Our aim here is to study functional dynamic equations with impulses. Several authors have already considered impulsive dynamic equations on time scales (see for example [5, 6, 11, 32]); to this end, let n n t1, . . . , tm ∈ T, t0 ≤ t1 < t2 < ··· < tm < t0 + σ and I1,...,Im : R → R . The usual condition which can be found in the existing literature is that the solution should satisfy

x(tk+) − x(tk−) = Ik(x(tk−)), k ∈ {1, . . . , m}. (5.5.1) The convention is that x(t+) = x(t) when t ∈ T is a right-scattered point and x(t−) = x(t) when t ∈ T is left-scattered. Moreover, it is usually assumed that the solution x should be left-continuous. In this case, Eq. (5.5.1) reduces to x(tk+) − x(tk) = Ik(x(tk)), k ∈ {1, . . . , m}. (5.5.2)

Note that if tk is right-scattered, then the left-hand side of Eq. (5.5.2) is zero. In other words, it makes sense to consider impulses at right-dense points only (the same assumption is made in [6], [11]). This motivates us to consider impulsive functional dynamic equations of the form ∆ ∗ x (t) = f(xt , t), t ∈ [t0, t0 + σ]T\{t1, . . . , tm}, + ∆ x(tk) = Ik(x(tk)), k ∈ {1, . . . , m},

x(t) = φ(t), t ∈ [t0 − r, t0]T, n n where t1, . . . , tm ∈ T are right-dense points, t0 ≤ t1 < t2 < ··· < tm < t0 + σ, and I1,...,Im : R → R . The solution is assumed to be left-continuous. It is not diﬃcult to see that the above problem can be written more compactly in the form Z t ∗ X x(t) = x(t0) + f(xs, s)∆s + Ik(x(tk)), t ∈ [t0, t0 + σ]T, t0 k∈{1,...,m}, tk

x(t) = φ(t), t ∈ [t0 − r, t0]T. Our immediate goal is to rewrite this equation as an impulsive measure functional differential equation. We need the following proposition from Chapter 4. Theorem 5.5.1. Let T be a time scale, g(s) = s∗ for every s ∈ T∗, [a, b] ⊂ T∗. Consider a pair of n R b functions f1, f2 :[a, b] → R such that f1(t) = f2(t) for every t ∈ [a, b] ∩ T. If a f1(s) dg(s) exists, then R b a f2(s) dg(s) exists as well and both integrals have the same value. The following theorem describes the relation between impulsive functional dynamic equations and impulsive measure functional differential equations. n Theorem 5.5.2. Let [t0 − r, t0 + σ]T be a time scale interval, t0 ∈ T, B ⊂ R , f : G([−r, 0],B) × [t0, t0 + n ∗ σ]T → R , φ ∈ G([t0 − r, t0]T,B). Define g(s) = s for every s ∈ [t0, t0 + σ]. If x :[t0 − r, t0 + σ]T → B is a solution of the impulsive functional dynamic equation Z t ∗ X x(t) = x(t0) + f(xs, s) ∆s + Ik(x(tk)), t ∈ [t0, t0 + σ]T, (5.5.3) t0 k∈{1,...,m}, tk

x(t) = φ(t), t ∈ [t0 − r, t0]T, (5.5.4) ∗ then x :[t0 − r, t0 + σ] → B is a solution of the impulsive measure functional diﬀerential equation Z t ∗ X y(t) = y(t0) + f(ys, s ) dg(s) + Ik(y(tk)), t ∈ [t0, t0 + σ], (5.5.5) t0 k∈{1,...,m}, tk

72 Proof. Assume that x satisﬁes (5.5.3) and (5.5.4). Clearly, x∗ = φ∗ . By Theorem 5.4.5, t0 t0 Z t ∗ ∗ ∗ ∗ X x (t) = x (t0) + f(xs∗ , s ) dg(s) + Ik(x(tk)), t ∈ [t0, t0 + σ]. t0 k∈{1,...,m}, ∗ tk

∗ ∗ We have tk ∈ T for every k ∈ {1, . . . , m}. It follows that x(tk) = x (tk), and tk < t if and only if tk < t. ∗ ∗ ∗ ∗ Moreover, since f(xs∗ , s ) = f(xs, s ) for every s ∈ T, we can use Theorem 5.5.1 to conclude that Z t ∗ ∗ ∗ ∗ X ∗ x (t) = x (t0) + f(xs, s ) dg(s) + Ik(x (tk)), t ∈ [t0, t0 + σ], t0 k∈{1,...,m}, tk

Lemma 5.5.3. Let [t0 − r, t0 + σ]T be a time scale interval, t0 ∈ T, O = G([t0 − r, t0 + σ],B), P = n ∗ ∗ ∗ G([−r, 0],B), f : P × [t0, t0 + σ]T → R an arbitrary function. Deﬁne g(t) = t and f (y, t) = f(y, t ) for every y ∈ P and t ∈ [t0, t0 + σ].

R t0+σ R t0+σ ∗ 1. If the integral f(yt, t)∆t exists for every y ∈ O, then the integral f (yt, t) dg(t) exists for t0 t0 every y ∈ O. 2. Assume there exists a constant M > 0 such that

Z u2

f(yt, t)∆t ≤ M(u2 − u1) u1

for every y ∈ O and u1, u2 ∈ [t0, t0 + σ]T, u1 ≤ u2. Then

Z u2 ∗ f (yt, t) dg(t) ≤ M(g(u2) − g(u1)) u1

whenever t0 ≤ u1 ≤ u2 ≤ t0 + σ and y ∈ O. 3. Assume there exists a constant L > 0 such that

Z u2 Z u2

(f(yt, t) − f(zt, t)) ∆t ≤ L kyt − ztk∞∆t u1 u1

for every y, z ∈ O and u1, u2 ∈ [t0, t0 + σ]T, u1 ≤ u2. Then

Z u2 Z u2 ∗ ∗ (f (yt, t) − f (zt, t)) dg(t) ≤ L kyt − ztk∞ dg(t) u1 u1

whenever t0 ≤ u1 ≤ u2 ≤ t0 + σ and y, z ∈ O.

R t0+σ Proof. Consider an arbitrary y ∈ O. If the integral f(yt, t)∆t exists, then, using Theorems 5.4.2 t0 and 5.5.1, we have

Z t0+σ Z t0+σ Z t0+σ Z t0+σ ∗ ∗ ∗ f(yt, t)∆t = f(yt∗ , t ) dg(t) = f(yt, t ) dg(t) = f (yt, t) dg(t), t0 t0 t0 t0 i.e. the last integral exists as well. This proves the ﬁrst part. The remaining two statements follow from Theorem 5.5.1, Lemma 5.4.4, and Theorem 5.4.2. In the ﬁrst case, we have

∗ ∗ Z u2 Z u2 Z u2 Z u2 ∗ ∗ ∗ f (yt, t) dg(t) = f(yt∗ , t ) dg(t) = f(yt∗ , t ) dg(t) = f(yt, t)∆t ∗ ∗ u1 u1 u1 u1

73 ∗ ∗ ≤ M(u2 − u1) = M(g(u2) − g(u1)). In the second case, we obtain

Z u2 Z u2 ∗ ∗ ∗ ∗ (f (yt, t) − f (zt, t)) dg(t) = (f(yt∗ , t ) − f(zt∗ , t )) dg(t) u1 u1

∗ ∗ Z u2 Z u2 ∗ ∗ = (f(yt∗ , t ) − f(zt∗ , t )) dg(t) = (f(yt, t) − f(zt, t)) ∆t ∗ ∗ u1 u1

∗ ∗ Z u2 Z u2 Z u2 ≤ L kyt − ztk∞∆t = L kyt − ztk∞ dg(t) = L kyt − ztk∞ dg(t). ∗ ∗ u1 u1 u1

5.6 Existence-uniqueness theorems

In this section, we present results on local existence and uniqueness of solutions for impulsive measure functional differential equations and impulsive functional dynamic equations on time scales. Our main tools in the proofs of these results are the correspondence between measure functional differential equations and impulsive measure functional differential equations presented in Section 3 (see Theorem 5.3.1), and the relation between this last type of equations and impulsive functional dynamic equations on time scales (see Theorem 5.5.2). We also make use of the existence-uniqueness theorem for measure functional differential equations, which was proved in Chapter 4.

n n Theorem 5.6.1. Assume that X = G([t0 −r, t0 +σ], R ), B ⊂ R is an open set, O = G([t0 −r, t0 +σ],B), P = G([−r, 0],B), m ∈ N, t0 ≤ t1 < t2 < . . . < tm < t0 + σ, g :[t0, t0 + σ] → R is a left-continuous n nondecreasing function which is continuous at t1, . . . , tm. Also, suppose that I1,...,Im : B → R and n f : P × [t0, t0 + σ] → R satisfy the following conditions:

R t0+σ 1. The integral f(yt, t) dg(t) exists for every y ∈ O. t0

2. There exists a constant M1 > 0 such that

Z u2

f(yt, t) dg(t) ≤ M1(g(u2) − g(u1)) u1

whenever t0 ≤ u1 ≤ u2 ≤ t0 + σ and y ∈ O.

3. There exists a constant L1 > 0 such that

Z u2 Z u2

(f(yt, t) − f(zt, t)) dg(t) ≤ L1 kyt − ztk∞ dg(t) u1 u1

whenever t0 ≤ u1 ≤ u2 ≤ t0 + σ and y, z ∈ O.

4. There exists a constant M2 > 0 such that

kIk(x)k ≤ M2

for every k ∈ {1, . . . , m} and x ∈ B.

5. There exists a constant L2 > 0 such that

kIk(x) − Ik(y)k ≤ L2kx − yk

for every k ∈ {1, . . . , m} and x, y ∈ B.

74 + Let φ ∈ P and assume that either t0 < t1 and φ(0)+f(φ, t0)∆ g(t0) ∈ B, or t0 = t1 and φ(0)+I1(φ(0)) ∈ n B. Then there exists δ > 0 and a function y :[t0 − r, t0 + δ] → R which is a unique solution of the impulsive measure functional diﬀerential equation

Z t X y(t) = y(t0) + f(ys, s) dg(s) + Ik(y(tk)), t ∈ [t0, t0 + δ], t0 k∈{1,...,m}, (5.6.1) tk

yt0 = φ. Proof. For every y ∈ P , deﬁne ( f(y, t), t ∈ [t , t + σ]\{t , . . . , t }, f˜(y, t) = 0 0 1 m Ik(y(0)), t = tk for some k ∈ {1, . . . , m}.

Moreover, let the functiong ˜ :[t0, t0 + σ] → R be given by  g(t), t ∈ [t , t ],  0 1 g˜(t) = g(t) + k, t ∈ (tk, tk+1] for some k ∈ {1, . . . , m − 1},  g(t) + m, t ∈ (tm, t0 + σ].

Since g is nondecreasing and left-continuous,g ˜ has the same properties. ˜ + + We have either t0 < t1 and φ(0) + f(φ, t0)∆ g˜(t0) = φ(0) + f(φ, t0)∆ g(t0) ∈ B, or t0 = t1 and ˜ + φ(0) + f(φ, t0)∆ g˜(t0) = φ(0) + I1(φ(0)) ∈ B. Using these facts and Lemma 5.3.3, we see that the functions f˜,g ˜, and φ satisfy all hypotheses of the existence and uniqueness theorem for measure functional diﬀerential equations (see Theorem 4.5.3 and n Remark 4.3.9). Consequently, there exist δ > 0 and a function y :[t0 − r, t0 + δ] → R which is a unique solution of the measure functional diﬀerential equation

Z t ˜ y(t) = y(t0) + f(ys, s) d˜g(s), t0

yt0 = φ.

Finally, by Theorem 5.3.1, the function y is also a unique solution of (5.6.1) on [t0 − r, t0 + δ]. In the sequel, we prove a result on local existence and uniqueness of solutions of impulsive functional dynamic equations on time scales.

n Theorem 5.6.2. Assume that [t0 − r, t0 + σ]T is a time scale interval, t0 ∈ T, B ⊂ R is an open set, O = G([t0 − r, t0 + σ],B), P = G([−r, 0],B), m ∈ N, t1, . . . , tm ∈ [t0, t0 + σ]T are right-dense points such n n that t0 ≤ t1 < ··· < tm < t0 + σ. Let f : P × [t0, t0 + σ]T → R and I1,...,Im : B → R be functions which satisfy the following conditions:

R t0+σ 1. The integral f(yt, t)∆t exists for every y ∈ O. t0

2. There exists a constant M1 > 0 such that

Z u2

f(yt, t)∆t ≤ M1(u2 − u1) u1

for every y ∈ O and u1, u2 ∈ [t0, t0 + σ]T, u1 ≤ u2.

3. There exists a constant L1 > 0 such that

Z u2 Z u2

(f(yt, t) − f(zt, t)) ∆t ≤ L1 kyt − ztk∞∆t u1 u1

for every y, z ∈ O and u1, u2 ∈ [t0, t0 + σ]T, u1 ≤ u2.

75 4. There exists a constant M2 > 0 such that

kIk(y)k ≤ M2

for every k ∈ {1, . . . , m} and y ∈ B.

5. There exists a constant L2 > 0 such that

kIk(x) − Ik(y)k ≤ L2kx − yk

for every k ∈ {1, . . . , m} and x, y ∈ B.

Let φ :[t − r, t ] → B be a regulated function such that either t < t and φ(t ) + f(φ∗ , t )µ(t) ∈ B, 0 0 T 0 1 0 t0 0 or t0 = t1 and φ(t0) + I1(φ(t0)) ∈ B. Then there exist a δ > 0 such that δ ≥ µ(t0) and t0 + δ ∈ T, and a function y :[t0 − r, t0 + δ]T → B which is a unique solution of the impulsive functional dynamic equation Z t ∗ X y(t) = y(t0) + f(ys , s) ∆s + Ik(y(tk)), t ∈ [t0, t0 + δ]T, t0 k∈{1,...,m}, tk

y(t) = φ(t), t ∈ [t0 − r, t0]T.

∗ ∗ ∗ + Proof. Let g(t) = t and f (y, t) = f(y, t ) for every t ∈ [t0, t0 + σ] and y ∈ P . Note that ∆ g(t0) = µ(t0). Using the hypotheses and Lemma 5.5.3, we see that the functions f ∗, g and φ∗ satisfy all assumptions of t0 Theorem 5.6.1. Consequently, there exist δ > 0 and a function u :[t0 − r, t0 + δ] → B which is a unique solution of Z t ∗ X u(t) = u(t0) + f (us, s) dg(s) + Ik(u(tk)), t ∈ [t0, t0 + δ], t0 k∈{1,...,m}, tk

∗ Then, by Theorem 5.5.2, u = y , where y :[t0 − r, t0 + δ]T → B is a solution of Z t ∗ X y(t) = y(t0) + f(ys , s) ∆s + Ik(y(tk)), t ∈ [t0, t0 + δ]T, t0 k∈{1,...,m}, tk

y(t) = φ(t), t ∈ [t0 − r, t0]T.

Without loss of generality, we can assume that δ ≥ µ(t0); otherwise, t0 is right-scattered, t0 < t1, and we can let y(σ(t )) = φ(t ) + f(φ∗ , t )µ(t ) 0 0 t0 0 0 to obtain a solution deﬁned on [t0 − r, t0 + µ(t0)]T. Again, by Theorem 5.5.2, the solution y is unique.

5.7 Continuous dependence results

In Chapter 4, we have obtained a continuous dependence theorem for measure functional differential equations. Since we already know that impulsive functional differential and dynamic equations are in fact special cases of measure functional different equations, we can use the existing result from Chapter 4 to derive continuous dependence theorems for both types of impulsive equations; this is the content of the present section.

n n Theorem 5.7.1. Assume that X = G([t0 −r, t0 +σ], R ), B ⊂ R is an open set, O = G([t0 −r, t0 +σ],B), P = G([−r, 0],B), m ∈ N, t0 ≤ t1 < t2 < . . . < tm < t0 + σ, g :[t0, t0 + σ] → R is a nondecreasing n left-continuous function which is continuous at t1, . . . , tm. Finally, let fp : P × [t0, t0 + σ] → R , p ∈ N0, p p n and I1 ,...,Im : B → R , p ∈ N0, be functions which satisfy the following conditions:

76 R t0+σ 1. The integral fp(yt, t) dg(t) exists for every p ∈ 0, y ∈ O. t0 N

2. There exists a constant M1 > 0 such that

Z u2

fp(yt, t) dg(t) ≤ M1(g(u2) − g(u1)) u1

whenever p ∈ N, t0 ≤ u1 ≤ u2 ≤ t0 + σ and y ∈ O.

3. There exists a constant L1 > 0 such that

Z u2 Z u2

(fp(yt, t) − fp(zt, t)) dg(t) ≤ L1 kyt − ztk∞dg(t) u1 u1

whenever p ∈ N, t0 ≤ u1 ≤ u2 ≤ t0 + σ and y, z ∈ O. 4. For every y ∈ O, Z t Z t lim fp(ys, s) dg(s) = f0(ys, s) dg(s) p→∞ t0 t0

uniformly with respect to t ∈ [t0, t0 + σ].

5. There exists a constant M2 > 0 such that

p kIk (x)k ≤ M2

for every k ∈ {1, . . . , m}, p ∈ N0 and x ∈ B.

6. There exists a constant L2 > 0 such that

p p kIk (x) − Ik (y)k ≤ L2kx − yk

for every k ∈ {1, . . . , m}, p ∈ N0 and x, y ∈ B.

p 0 7. For every y ∈ B and k ∈ {1, . . . , m}, limp→∞ Ik (y) = Ik (y).

Consider functions φp ∈ P , p ∈ N0, such that limp→∞ φp = φ0 uniformly on [−r, 0]. Let yp ∈ O, p ∈ N, be solutions of

Z t X p yp(t) = yp(t0) + fp((yp)s, s) dg(s) + Ik (yp(tk)), t ∈ [t0, t0 + σ], (5.7.1) t0 k∈{1,...,m}, tk

(yp)t0 = φp, (5.7.2) such that limp→∞ yp = y0 ∈ O. Then y0 satisﬁes

Z t X 0 y0(t) = y0(t0) + f0((y0)s, s) dg(s) + Ik (y0(tk)), t ∈ [t0, t0 + σ], (5.7.3) t0 k∈{1,...,m}, tk

(y0)t0 = φ0. (5.7.4)

Proof. We already know that (5.7.1) and (5.7.2) imply

Z t ˜ yp(t) = yp(t0) + fp((yp)s, s) d˜g(s), t ∈ [t0, t0 + σ], t0

(yp)t0 = φp,

77 ˜ where the construction of fp andg ˜ is described in Theorem 5.3.1. Since g is nondecreasing and left- continuous,g ˜ posseses the same properties. For every t ∈ [t0, t0 + σ], we have Z t Z t ˜ X p lim fp(ys, s) d˜g(s) = lim fp(ys, s) dg(s) + lim I (y(tk)) = p→∞ p→∞ p→∞ k t0 t0 k∈{1,...,m}, tk

(y0)t0 = φ0.

The proof is ﬁnished by applying Theorem 5.3.1, which implies that y0 satisﬁes (5.7.3) and (5.7.4). The second result in this section is a continuous dependence theorem for impulsive functional dynamic equations on time scales.

n Theorem 5.7.2. Assume that [t0 − r, t0 + σ]T is a time scale interval, t0 ∈ T, B ⊂ R is an open set, O = G([t0 − r, t0 + σ],B), P = G([−r, 0],B), m ∈ N, t1, . . . , tm ∈ [t0, t0 + σ]T are right-dense points such n p p n that t0 ≤ t1 < ··· < tm < t0 + σ. Let fp : P × [t0, t0 + σ]T → R , p ∈ N0, and I1 ,...,Im : B → R , p ∈ N0, be functions which satisfy the following conditions: R t0+σ 1. The integral fp(yt, t)∆t exists for every y ∈ O and p ∈ 0. t0 N

2. There exists a constant M1 > 0 such that

Z u2

fp(yt, t)∆t ≤ M1(u2 − u1) u1

for every p ∈ N, y ∈ O and u1, u2 ∈ [t0, t0 + σ]T, u1 ≤ u2.

3. There exists a constant L1 > 0 such that

Z u2 Z u2

(fp(yt, t) − fp(zt, t)) ∆t ≤ L1 kyt − ztk∞∆t u1 u1

for every p ∈ N, y, z ∈ O and u1, u2 ∈ [t0, t0 + σ]T, u1 ≤ u2. 4. For every y ∈ O, Z t Z t lim fp(ys, s)∆s = f0(ys, s)∆s p→∞ t0 t0

uniformly with respect to t ∈ [t0, t0 + σ]T.

5. There exists a constant M2 > 0 such that p kIk (x)k ≤ M2

for every k ∈ {1, . . . , m}, p ∈ N0 and x ∈ B.

6. There exists a constant L2 > 0 such that p p kIk (x) − Ik (y)k ≤ L2kx − yk

for every k ∈ {1, . . . , m}, p ∈ N0 and x, y ∈ B.

78 p 0 7. For every x ∈ B and k ∈ {1, . . . , m}, limp→∞ Ik (x) = Ik (x).

Assume that φp ∈ G([t0 −r, t0]T,B), p ∈ N0, is a sequence of functions such that limp→∞ φp = φ0 uniformly on [t0 − r, t0]T. Let yp :[t0 − r, t0 + σ]T → B, p ∈ N be solutions of Z t ∗ X p yp(t) = yp(t0) + fp((yp)s, s) ∆s + Ik (yp(tk)), t ∈ [t0, t0 + σ]T, t0 k∈{1,...,m}, tk

If there exists a function y0 :[t0 − r, t0 + σ]T → B such that limp→∞ yp = y0, then y0 satisﬁes Z t ∗ X 0 y0(t) = y0(t0) + f0((y0 )s, s) ∆s + Ik (y0(tk)), t ∈ [t0, t0 + σ]T, (5.7.5) t0 k∈{1,...,m}, tk

y0(t) = φ0(t), t ∈ [t0 − r, t0]T. (5.7.6) ∗ Proof. Let g(t) = t for every t ∈ [t0, t0 + σ]; then g is a left-continuous nondecreasing function which is ∗ ∗ continuous at t1, . . . , tm. Further, let fp (y, t) = fp(y, t ) for every p ∈ N0, y ∈ P and t ∈ [t0, t0 + σ]. By R t0+σ ∗ Lemma 5.5.3, the integral f (yt, t) dg(t) exists for every y ∈ O and p ∈ 0. By Theorems 5.4.5 and t0 p N 5.5.1, we obtain Z t Z t∗ Z t∗ ∗ lim fp (ys, s) dg(s) = lim fp(ys, s)∆s = f0(ys, s)∆s p→∞ p→∞ t0 t0 t0 Z t Z t ∗ ∗ = f0(ys∗ , s ) dg(s) = f0 (ys, s) dg(s), t0 t0 where the convergence is uniform with respect to t ∈ [t0, t0 + σ]. ∗ ∗ ∗ ∗ Further, it is clear that limp→∞ yp = y0 on [t0, t0 + σ], and limp→∞ φp = φ0 uniformly on [t0 − r, t0]. By Theorem 5.5.2, we have Z t ∗ ∗ ∗ ∗ X p ∗ yp(t) = yp(t0) + fp ((yp)s, s) dg(s) + Ik (yp(tk)), t ∈ [t0, t0 + σ], t0 k∈{1,...,m}, tk

By Theorem 5.5.2, it follows that y0 satisﬁes (5.7.5) and (5.7.6). Remark 5.7.3. According to Remark 4.6.6, the assumptions of Theorem 5.7.1 might be modiﬁed in the following way: Instead of requiring the existence of a function y0 ∈ O such that limk→∞ yk = y0, it is 0 0 enough to assume the existence of a closed set B ⊂ B such that the functions yk, k ∈ N, take values in B . ∞ Under this hypothesis, the conclusion is that {yk}k=1 has a subsequence which is uniformly convergent to a function y0 ∈ O such that Z t y0(t) = y0(t0) + f0((y0)s, s) dg(s), t ∈ [t0, t0 + σ], t0

(y0)t0 = φ0. Theorem 5.7.2 can be modiﬁed in a similar way.

79 5.8 Periodic averaging theorems

The basic idea behind averaging theorems is that one can approximate solutions of a non-autonomous equation by solutions of an autonomous equation whose right-hand side corresponds to the average of the original right-hand side. The method is quite general and can be applied to many types of equations (see Chapter 3); it is especially powerful in the case when the original right-hand side is periodic in t. In this section, we use an existing periodic averaging theorem for measure functional diﬀerential equations to obtain periodic averaging theorems for functional diﬀerential and dynamic equations involving impulses.

n Theorem 5.8.1. Assume that ε0 > 0, L > 0, B ⊂ R , X = G([−r, 0],B), m ∈ N and 0 ≤ t1 < t2 < ··· < n n tm < T . Consider a pair of bounded functions f : X × [0, ∞) → R , g : X × [0, ∞) × (0, ε0] → R and n a nondecreasing left-continuous function h : [0, ∞) → R which is continuous at t1, . . . , tm. Let Ik : B → R , k ∈ {1, 2, . . . , m} be bounded and Lipschitz-continuous functions. For every integer k > m, define tk and Ik by the recursive formulas tk = tk−m + T and Ik = Ik−m. Suppose that the following conditions are satisfied: R b 1. The integral 0 f(yt, t) dh(t) exists for every b > 0 and y ∈ G([−r, b],B). 2. f is Lipschitz-continuous with respect to the first variable. 3. f is T -periodic in the second variable. 4. There is a constant α > 0 such that h(t + T ) − h(t) = α for every t ≥ 0. 5. The integral 1 Z T f0(x) = f(x, s) dh(s) T 0 exists for every x ∈ X. Denote m 1 X I (y) = I (y), y ∈ B. 0 T k k=1

Let φ ∈ X and suppose for every ε ∈ (0, ε0], the initial value problems Z t Z t 2 X x(t) = x(0) + ε f(xs, s) dh(s) + ε g(xs, s, ε) dh(s) + ε Ik(x(tk)), 0 0 k∈N, tk 0 such that

kxε(t) − yε(t)k ≤ Jε for every ε ∈ (0, ε0] and t ∈ [0, L/ε].

Proof. Deﬁne the function h˜(t) : [0, ∞) → R by ( h(t), t ∈ [0, t ], h˜(t) = 1 h(t) + k, t ∈ (tk, tk+1] for some k ∈ N.

+˜ ˜ Note that ∆ h(tk) = 1 for every k ∈ N, h is nondecreasing, left-continuous, and for every t ≥ 0, we have h˜(t + T ) − h˜(t) =α ˜, whereα ˜ = h(t + T ) − h(t) + m = α + m.

80 By the assumptions, it follows that Z t ε ε ε 2 ε X ε x (t) = x (0) + εf(xs, s) + ε g(xs, s, ε) dh(s) + εIk(x (tk)) 0 k∈N, tk

F ε(y, t) = εf˜(y, t) + ε2g˜(y, t, ε), (5.8.2) where ( f(y, t), t∈ / {t , t ,...}, f˜(y, t) = 1 2 Ik(y(0)), t = tk for some k ∈ N and ( g(y, t, ε), t∈ / {t , t ,...}, g˜(y, t, ε) = 1 2 0, t = tk for some k ∈ N. ε It follows from (5.8.1) and (5.8.2) that for every ε ∈ (0, ε0], the function x :[−r, L/ε] → B is a solution of the initial value problem Z t Z t ˜ ˜ 2 ˜ x(t) = x(0) + ε f(xs, s) dh(s) + ε g˜(xs, s, ε) dh(s), 0 0 x0 = φ.

The function f˜ is Lipschitz-continuous with respect to the ﬁrst variable and T -periodic in the second variable. Using Lemma 5.2.4, we have

Z T Z T m Z T m ˜ ˜ X ˜ +˜ X f(x, s) dh(s) = f(x, s) dh(s) + f(x, tk)∆ h(tk) = f(x, s) dh(s) + Ik(x(0)) 0 0 k=1 0 k=1 for every x ∈ X. Consequently, the function

Z T ˜ 1 ˜ ˜ f0(x) = f(x, s) dh(s), x ∈ X, T 0 satisﬁes m 1 Z T 1 X f˜ (x) = f(x, s) dh(s) + I (x(0)) = f (x) + I (x(0)), x ∈ X. 0 T T k 0 0 0 k=1 By the periodic averaging theorem for measure functional diﬀerential equations (see Theorem 3.5.2), there ε ε is a constant J > 0 such that kx (t) − y (t)k ≤ Jε for every ε ∈ (0, ε0] and t ∈ [0, L/ε]. We now proceed to the periodic averaging theorem for impulsive functional dynamic equations on time scales.

Deﬁnition 5.8.2. Let T > 0 be a real number. A time scale T is called T -periodic if t ∈ T implies t + T ∈ T and µ(t) = µ(t + T ).

81 Theorem 5.8.3. Assume that T is a T -periodic time scale, [t0 − r, t0 + σ]T a time scale interval, t0 ∈ T, n ε0 > 0, L > 0, B ⊂ R , X = G([−r, 0],B), m ∈ N, t1, . . . , tm ∈ T are right-dense points such that n t0 ≤ t1 < t2 < ··· < tm < t0 + T . Let Ik : B → R , k ∈ {1, 2, . . . , m} be bounded and Lipschitz-continuous functions. For every integer k > m, define tk and Ik by the recursive formulas tk = tk−m + T and n n Ik = Ik−m. Consider a pair of bounded functions f : X × [t0, ∞)T → R , g : X × [t0, ∞)T × (0, ε0] → R such that the following conditions are satisfied: R b 1. The integral 0 f(yt, t)∆t exists for every b > 0 and y ∈ G([−r, b],B). 2. f is Lipschitz-continuous with respect to the first variable. 3. f is T -periodic in the second variable. 4. The integral 1 Z t0+T f0(x) = f(x, s)∆s T t0 exists for every x ∈ X. Denote m 1 X I (y) = I (y), y ∈ B. 0 T k k=1

Let φ ∈ G([t0 − r, t0]T,B) and suppose for every ε ∈ (0, ε0], the initial value problems Z t Z t ∗ 2 ∗ X x(t) = x(t0) + ε f(xs, s)∆s + ε g(xs, s, ε)∆s + ε Ik(y(tk)), t ∈ [t0, t0 + L/ε]T, t t 0 0 k∈N, tk

x(t) = φ(t), t ∈ [t0 − r, t0]T, Z t y(t) = y(t0) + ε (f0(ys) + I0(y(s))) ds, t0 y = φ∗ t0 t0

ε ε have solutions x :[t0 − r, t0 + L/ε]T → B and y :[t0 − r, t0 + L/ε] → B, respectively. Then there exists a constant J > 0 such that kxε(t) − yε(t)k ≤ Jε, for every ε ∈ (0, ε0] and t ∈ [t0, t0 + L/ε]T.

Proof. Without loss of generality, we can assume that t0 = 0; otherwise, consider a shifted problem with the time scale Te = {t − t0; t ∈ T} and functions fe(x, t) = f(x, t + t0) and ge(x, t, ε) = g(x, t + t0, ε). For every t ∈ [t0, ∞), x ∈ X and ε ∈ (0, ε0], let f ∗(x, t) = f(x, t∗) and g∗(x, t, ε) = g(x, t∗, ε).

∗ Also, let h(t) = t for every t ∈ [t0, ∞). Since T is T -periodic, it follows that h(t + T ) − h(t) = T, t ≥ 0.

From Theorem 5.4.5, we obtain Z T Z T 1 1 ∗ f0(x) = f(x, s)∆s = f (x, s) dh(s) T 0 T 0 for every x ∈ X. R b For every b > 0 and y ∈ G([−r, b],B), the integral 0 f(yt, t)∆t exists. Then, Theorems 5.4.5 and 5.5.1 imply Z b Z b Z b Z b ∗ ∗ ∗ f(yt, t)∆t = f(yt∗ , t ) dh(t) = f(yt, t ) dh(t) = f (yt, t) dh(t), 0 0 0 0

82 i.e. the last integral exists as well. It follows from Theorem 5.5.2 that for every ε ∈ (0, ε0] and t ∈ [0, L/ε], we have

Z t Z t ε ∗ ε ∗ ∗ ε ∗ 2 ∗ ε ∗ X ε ∗ (x ) (t) = (x ) (0) + ε f ((x )s, s) dh(s) + ε g ((x )s, s, ε) dh(s) + ε Ik((x ) (tk)), 0 0 k∈N, tk

Finally, by Theorem 5.8.1, there exists a constant J > 0 such that k(xε)∗(t) − yε(t)k ≤ Jε for every ε ∗ ε ε ∈ (0, ε0] and t ∈ [0, L/ε]. We conclude the proof by observing that (x ) (t) = x (t) for t ∈ [0, L/ε]T.

83 Chapter 6

Measure functional diﬀerential equations with inﬁnite delay

6.1 Introduction

Measure functional differential equations with finite delay have the form Z t y(t) = y(t0) + f(ys, s) dg(s), (6.1.1) t0 n where y is an unknown function with values in R and the symbol ys denotes the function ys(τ) = y(s+τ) defined on [−r, 0], r ≥ 0 being a fixed number corresponding to the length of the delay. The integral on the right-hand side of (6.1.1) is the Kurzweil-Stieltjes integral with respect to a nondecreasing function g. Measure functional differential equations have been introduced in Chapter 4. In the special case g(s) = s, equation (6.1.1) reduces to the classical functional differential equation Z t y(t) = y(t0) + f(ys, s) ds, (6.1.2) t0 which has been studied by many authors (see e.g. [36]). On the other hand, the general form (6.1.1) includes other familiar types of equations such as functional differential equations with impulses or functional dynamic equations on time scales (see Chapters 4 and 5). For example, consider the impulsive functional differential equation 0 y (t) = f(yt, t), t ∈ [t0, ∞)\{t1, t2,...}, + (6.1.3) ∆ y(ti) = Ii(y(ti)), i ∈ N, where the impulses take place at preassigned times t1, t2,... ∈ [t0, ∞), and their action is described by n n the operators Ii : R → R , i ∈ N; the solution is assumed to be left-continuous at every point ti. The corresponding integral form is

Z t X y(t) = y(t0) + f(ys, s) ds + Ii(y(ti)), t ∈ [t0, ∞), t0 i∈N, ti

84 The aim of this chapter is to discuss measure functional differential equations with infinite delay, i.e., equations of the form (6.1.1), where ys now denotes the function ys(τ) = y(s + τ) defined on (−∞, 0]. The case g(s) = s corresponds to classical functional differential equations with infinite delay, which have been studied by numerous authors (see e.g. [19, 35, 39] and the references there). The general case when g is a nondecreasing function includes certain other types of functional equations, such as the impulsive functional differential equation (6.1.3) with infinite delay. One particular example is the impulsive Volterra integro-differential equation

Z t 0 y (t) = a(y(s), s) ds, t ∈ [0, ∞)\{t1, t2,...}, 0 + ∆ y(ti) = Ii(y(ti)), i ∈ N, which has the form (6.1.3) with Z 0 f(x, t) = a(x(τ), t + τ) dτ −t for a function x :(−∞, 0] → Rn. When dealing with infinite delay, the crucial problem is the choice of the phase space, i.e., the domain of the first argument of f. In the classical case (6.1.2), the elements of this phase space are continuous functions. Such a phase space is no longer suitable for a general measure functional differential equation, whose solutions are discontinuous functions. The problem of the choice of phase space is discussed in Section 2. We do not restrict ourselves to a particular phase space; instead, we introduce a certain system of conditions and allow the phase space to be any space satisfying these conditions. A similar axiomatic approach was used by various authors (see e.g. [19, 35, 39] and the references there) to describe the phase space of classical or impulsive functional differential equations with infinite delay. In Section 3, we show that under certain natural assumptions, a measure functional differential equation can be transformed to a generalized ordinary differential equation whose solutions take values in an infinite-dimensional Banach space. Consequently, one can use the existing theory of generalized ordinary differential equations (see e.g. [52, 67]) to obtain new results for measure functional differential equations with infinite delay. The idea of transforming a classical functional differential equation to a generalized ordinary differential equation first appeared in the papers [45, 61] by C. Imaz, F. Oliva, and Z. Vorel. Later, it was extended to impulsive functional differential equations in the paper [24] by M. Federson and S.ˇ Schwabik, and to measure functional differential equations with finite delay in the paper [20] by M. Federson, J. G. Mesquita and A. Slav´ık(cf. Chapter 4). In [1, 26, 25], the correspondence between functional differential equations and generalized ordinary equations was used to obtain various results on boundedness and stability of solutions.

6.2 Phase space description

In general, solutions of measure functional diﬀerential equations are not continuous, but merely regulated functions; recall that a function f :[a, b] → Rn is called regulated, if the limits

lim f(s) = f(t−) ∈ n, t ∈ (a, b] and lim f(s) = f(t+) ∈ n, t ∈ [a, b) s→t− R s→t+ R exist. Regulated functions on open or half-open intervals are defined in a similar way. Given an interval I ⊂ R and a set B ⊂ Rn, we use the symbol G(I,B) to denote the set of all regulated functions f : I → B, and the symbol C(I,B) to denote the set of all continuous functions f : I → B. Our candidate for the phase space of a measure functional differential equation with infinite delay is n a linear space H0 ⊂ G((−∞, 0], R ) equipped with a norm denoted by k · kF. We assume that this normed linear space H0 satisfies the following conditions:

(H1) H0 is complete.

(H2) If y ∈ H0 and t < 0, then yt ∈ H0.

85 + (H3) There exists a locally bounded function κ1 :(−∞, 0] → R such that if y ∈ H0 and t ≤ 0, then ky(t)k ≤ κ1(t)kykF.

(H4) There exists a function κ2 : (0, ∞) → [1, ∞) such that if σ > 0 and y ∈ H0 is a function whose support is contained in [−σ, 0], then

kykF ≤ κ2(σ) sup ky(t)k. t∈[−σ,0]

+ (H5) There exists a locally bounded function κ3 :(−∞, 0] → R such that if y ∈ H0 and t ≤ 0, then

kytkF ≤ κ3(t)kykF.

(H6) If y ∈ H0, then the function t 7→ kytkF is regulated on (−∞, 0]. To establish the correspondence between measure functional differential equations and generalized ordinary differential equations, we also need a suitable space Ha of regulated functions defined on (−∞, a], where a ∈ R. We obtain this space by shifting the functions from H0; more formally, for every τ ∈ R, let n Sτ be the shift operator defined as follows: if y : I → R is an arbitrary function, let I + τ = {t + τ, t ∈ I} n and define Sτ y : I + τ → R by (Sτ y)(t) = y(t − τ). Now, for every a ∈ R, let Ha = {Say, y ∈ H0}. Finally, define a norm k · kF on Ha by letting kykF = kS−aykF for every y ∈ Ha. Note that if y ∈ Ha, then y = Saz for a certain z ∈ H0, and yt = (Saz)t = zt−a ∈ H0 for every t ≤ a. The next lemma shows that the spaces Ha inherit all important properties of H0; the statements follow immediately from the corresponding definitions.

n Lemma 6.2.1. If H0 ⊂ G((−∞, 0], R ) is a space satisfying conditions (H1)–(H6), then the following statements are true for every a ∈ R:

1. Ha is complete.

2. If y ∈ Ha and t ≤ a, then yt ∈ H0.

3. If t ≤ a and y ∈ Ha, then ky(t)k ≤ κ1(t − a)kykF.

4. If σ > 0 and y ∈ Ha+σ is a function whose support is contained in [a, a + σ], then

kykF ≤ κ2(σ) sup ky(t)k. t∈[a,a+σ]

5. If y ∈ Ha+σ and t ≤ a + σ, then kytkF ≤ κ3(t − a − σ)kykF.

6. If y ∈ Ha+σ, then the function t 7→ kytkF is regulated on (−∞, a + σ]. In the rest of this section, we present examples of spaces satisfying conditions (H1)–(H6), which might be used as the phase space H0 for measure functional differential equations with infinite delay. These spaces are modified versions of examples given in [39], where we have replaced continuous functions by regulated functions; other examples presented in [39] can be modified in a similar way.

Example 6.2.2. Probably the simplest example of a normed linear space H0 satisfying conditions (H1)– (H6) is the space BG((−∞, 0], Rn), which consists of all bounded regulated functions on (−∞, 0] and is endowed with the supremum norm

n kyk∞ = sup ky(t)k, y ∈ BG((−∞, 0], R ). t∈(−∞,0]

It is straightforward to check that conditions (H1)–(H5) are satisﬁed; in particular, (H3)–(H5) are true with κ1(t) = κ3(t) = 1 for every t ≤ 0 and κ2(σ) = 1 for every σ > 0. Finally, condition (H6) is a consequence of the following Lemma 6.2.3. n Note also that Ha = BG((−∞, a], R ), i.e., the space of all bounded regulated functions on (−∞, a] with the supremum norm.

86 n Lemma 6.2.3. If y :(−∞, 0] → R is a regulated function, then t 7→ kytk∞ is regulated on (−∞, 0].

Proof. Let us show that limt→t0− kytk∞ exists for every t0 ∈ (−∞, 0]. The function y is regulated, and therefore satisﬁes the Cauchy condition at t0, i.e., given an arbitrary ε > 0, there exists a δ > 0 such that

ky(u) − y(v)k < ε, u, v ∈ (t0 − δ, t0). (6.2.1)

Consider a pair of numbers t1, t2 such that t0 − δ < t1 ≤ t2 < t0. Obviously,

kyt1 k∞ ≤ kyt2 k∞ < kyt2 k∞ + ε.

For every t ∈ [t1, t2], it follows from (6.2.1) that

ky(t)k ≤ ky(t1)k + ky(t2) − y(t1)k < kyt1 k∞ + ε.

Since ky(t)k ≤ kyt1 k∞ < kyt1 k∞ + ε for every t ∈ (−∞, t1], we conclude that

kyt2 k∞ ≤ kyt1 k∞ + ε, and consequently

kyt1 k∞ − kyt2 k∞ < ε, t1, t2 ∈ (t0 − δ, t0), i.e. the Cauchy condition for the existence of limt→t0− kytk∞ is satisﬁed. The existence of limt→t0+ kytk∞ for t0 ∈ (−∞, 0) can be proved similarly.

Remark 6.2.4. In connection with the previous lemma, note that if y :(−∞, 0] → Rn is a regulated 2 function, then t 7→ yt need not be regulated on (−∞, 0]. For example, let y(t) = sin t , t ∈ (−∞, 0]. Then for every h > 0, we have lim supt→−∞ |y(t + h) − y(t)| = 2, and therefore kyt+h − ytk∞ = 2 for arbitrarily small values of h. The next example is a phase space that can be used when dealing with unbounded functions; it consists of functions that do not grow faster than a certain exponential function as t → −∞. n Example 6.2.5. For an arbitrary γ ≥ 0, let Gγ ((−∞, 0], R ) be the space of all regulated functions n γt y :(−∞, 0] → R such that supt∈(−∞,0] ke y(t)k is ﬁnite. This space is endowed with the norm γt n kykγ = sup ke y(t)k, y ∈ Gγ ((−∞, 0], R ). t∈(−∞,0]

n n The operator T : Gγ ((−∞, 0], R ) → BG((−∞, 0], R ) deﬁned by (T y)(t) = eγty(t)

n n n is an isometric isomorphism between Gγ ((−∞, 0], R ) and BG((−∞, 0], R ), and thus Gγ ((−∞, 0], R ) is a complete space. −γt Again, it is not diﬃcult to check that conditions (H1)–(H5) are satisﬁed with κ1(t) = κ3(t) = e and n κ2(σ) = 1. To verify condition (H6), let y ∈ Gγ ((−∞, 0], R ) and note that γs −γt γ(s+t) −γt kytkγ = sup ke y(s + t)k = e sup ke y(s + t)k = e kztk∞, s∈(−∞,0] s∈(−∞,0]

γs where z(s) = e y(s) for every s ≤ 0. By Lemma 6.2.3, the function t 7→ kztk∞ is regulated, and thus −γt t 7→ e kztk∞ is also regulated. Remark 6.2.6. Consider the measure functional diﬀerential equation Z t y(t) = y(t0) + f(ys, s) dg(s), t0 where g(s) = s. In this special case, the solutions are continuous and it might be more appropriate to choose a space H0 that contains continuous functions only. In particular, one can modify the previous two examples to obtain the space BC((−∞, 0], Rn) consisting of all bounded continuous functions, or the n n γt space Cγ ((−∞, 0], R ) consisting of all continuous functions y :(−∞, 0] → R such that limt→−∞ e y(t) exists and is ﬁnite. More information about phase spaces consisting of continuous functions can be found in [39].

87 6.3 Relation to generalized ordinary diﬀerential equations

In this section, we establish the correspondence between measure functional differential equations with infinite delay and generalized ordinary differential equations. In the previous chapters, we were dealing with generalized equations whose right-hand sides F were elements of the class F(G, h, ω) or F(G, h). In this chapter, it will be more convenient to separate the two conditions from the definition of F(G, h):

(F1) There exists a nondecreasing function h :[a, b] → R such that F : O × [a, b] → X satisﬁes

kF (x, s2) − F (x, s1)k ≤ |h(s2) − h(s1)|

for every x ∈ O and s1, s2 ∈ [a, b].

(F2) There exists a nondecreasing function k :[a, b] → R such that F : O × [a, b] → X satisﬁes

kF (x, s2) − F (x, s1) − F (y, s2) + F (y, s1)k ≤ |k(s2) − k(s1)| · kx − yk

for every x, y ∈ O and s1, s2 ∈ [a, b]. According to the following lemma, solutions of a generalized ordinary differential equation whose right- hand side satisfies condition (F1) are regulated functions (the proof follows directly from the definition of the Kurzweil integral and can be found in [67], Lemma 3.12).

Lemma 6.3.1. Let X be a Banach space. Consider a set O ⊂ X, an interval [a, b] ⊂ R and a function F : O × [a, b] → X, which satisfies condition (F1). If x :[a, b] → O is a solution of the generalized ordinary differential equation dx = DF (x, t), dτ then kx(s2) − x(s1)k ≤ h(t2) − h(t1) for each pair s1, s2 ∈ [a, b], and x is a regulated function. n Let H0 ⊂ G((−∞, 0], R ) be a Banach space satisfying conditions (H1)–(H6), t0 ∈ R, σ > 0, O ⊂ Ht0+σ and P = {yt; y ∈ O, t ∈ [t0, t0 + σ]} ⊂ H0. Consider a nondecreasing function g :[t0, t0 + σ] → R and n a function f : P × [t0, t0 + σ] → R . We will show that under certain assumptions, a measure functional differential equation of the form Z t y(t) = y(t0) + f(ys, s) dg(s), t ∈ [t0, t0 + σ] (6.3.1) t0 is equivalent to a generalized ordinary differential equation of the form dx = DF (x, t), t ∈ [t , t + σ], (6.3.2) dτ 0 0 n where x takes values in O, and F : O × [t0, t0 + σ] → G((−∞, t0 + σ], R ) is given by  0, −∞ < ϑ ≤ t ,  0 R ϑ f(xs, s) dg(s), t0 ≤ ϑ ≤ t ≤ t0 + σ, F (x, t)(ϑ) = t0 (6.3.3) R t  f(xs, s) dg(s), t ≤ ϑ ≤ t0 + σ t0 for every x ∈ O and t ∈ [t0, t0 + σ]. It will turn out that the relation between the solution x of (6.3.2) and the solution y of (6.3.1) is described by ( y(ϑ), ϑ ∈ (−∞, t], x(t)(ϑ) = y(t), ϑ ∈ [t, t0 + σ], where t ∈ [t0, t0 + σ].

88 n Deﬁnition 6.3.2. Let I ⊂ R be an interval, t0 ∈ I, and O a set whose elements are functions f : I → R . We say that O has the prolongation property for t ≥ t0, if for every y ∈ O and every t ∈ I ∩ [t0, ∞), the functiony ¯ : I → Rn given by ( y(s), s ∈ (−∞, t] ∩ I, y¯(s) = y(t), s ∈ [t, ∞) ∩ I is an element of O.

n For example, consider an arbitrary set B ⊂ R , an interval I ⊂ R, and t0 ∈ I. Then both the set G(I,B) of all regulated functions f : I → B and the set C(I,B) of all continuous functions f : I → B have the prolongation property for t ≥ t0. The following theorem, which is a special case of Theorem 1.16 in [67], conﬁrms that the values of F + deﬁned by (6.3.3) are indeed regulated functions on (−∞, t0 + σ]. We use the symbol ∆ y(t) to denote y(t+) − y(t).

Theorem 6.3.3. Let f :[a, b] → Rn and g :[a, b] → R be a pair of functions such that g is regulated and R b a f dg exists. Then the function Z t u(t) = f(s) dg(s), t ∈ [a, b], a is regulated and satisfies ∆+u(t) = f(t)∆+g(t) for every t ∈ [a, b). To justify the relation between measure functional differential equations and generalized ordinary differential equations, we assume that the following three conditions are satisfied:

R t0+σ (A) The integral f(yt, t) dg(t) exists for every y ∈ O. t0 + (B) There exists a function M :[t0, t0 + σ] → R , which is Kurzweil-Stieltjes integrable with respect to g, such that

Z b Z b

f(yt, t) dg(t) ≤ M(t) dg(t) a a

whenever y ∈ O and [a, b] ⊆ [t0, t0 + σ]. + (C) There exists a function L :[t0, t0 + σ] → R , which is Kurzweil-Stieltjes integrable with respect to g, such that

Z b Z b (f(y , t) − f(z , t)) dg(t) ≤ L(t)ky − z k dg(t) t t t t F a a

whenever y, z ∈ O and [a, b] ⊆ [t0, t0 + σ] (we are assuming that the integral on the right-hand side exists). Let us remark that the assumptions (B) and (C) are weaker than similar conditions used in Chapters 4 and 5, where it was assumed that M and L are constants. In the special case g(t) = t, our conditions coincide with those given in [24]. The next lemma demonstrates the relation between conditions (A), (B), (C) and (F1), (F2). We point out that (F2) is not really needed to establish the relation between the two types of equations. However, we include it here for reader’s convenience since it is often needed in proofs of various facts concerning generalized ordinary diﬀerential equations.

Lemma 6.3.4. Let O ⊂ Ht0+σ and P = {yt; y ∈ O, t ∈ [t0, t0 + σ]}. Assume that g :[t0, t0 + σ] → R n is a nondecreasing function, f : P × [t0, t0 + σ] → R satisfies condition (A), and F : O × [t0, t0 + σ] → n G((−∞, t0 + σ], R ) given by (6.3.3) has values in Ht0+σ. Then the following statements are true: • If f satisfies condition (B), then F satisfies condition (F1) with Z t h(t) = κ2(σ) M(s) dg(s), t ∈ [t0, t0 + σ]. t0

89 • If f satisﬁes condition (C), then F satisﬁes condition (F2) with

! Z t k(t) = κ2(σ) sup κ3(s) L(s) dg(s), t ∈ [t0, t0 + σ]. s∈[−σ,0] t0

Proof. By condition (A), the integrals in the deﬁnition of F are guaranteed to exist. Assume that t0 ≤ s1 < s2 ≤ t0 + σ. We have  0, −∞ < τ ≤ s ,  1 R τ F (y, s2)(τ) − F (y, s1)(τ) = f(ys, s) dg(s), s1 ≤ τ ≤ s2, s1 R s2  f(ys, s) dg(s), s2 ≤ τ ≤ t0 + σ s1 for every y ∈ O. Condition (B) and the fourth statement of Lemma 6.2.1 imply

kF (y, s2) − F (y, s1)kF ≤ κ2(σ) sup kF (y, s2)(τ) − F (y, s1)(τ)k = τ∈[t0,t0+σ]

Z τ Z s2

= κ2(σ) sup f(ys, s) dg(s) ≤ κ2(σ) M(s) dg(s) = h(s2) − h(s1). τ∈[s1,s2] s1 s1 Similarly, condition (C) and the fourth statement of Lemma 6.2.1 imply that for every y, z ∈ O, we have

kF (y, s2) − F (y, s1) − F (z, s2) + F (z, s1)kF

≤ κ2(σ) sup kF (y, s2)(τ) − F (y, s1)(τ) − F (z, s2)(τ) + F (z, s1)(τ)k = τ∈[t0,t0+σ]

Z τ Z s2

= κ2(σ) sup (f(ys, s) − f(zs, s)) dg(s) ≤ κ2(σ) L(s)kys − zskF dg(s). τ∈[s1,s2] s1 s1 By the ﬁfth statement of Lemma 6.2.1, the last expression is smaller than or equal to ! Z s2 κ2(σ) sup κ3(s − t0 − σ) ky − zkF L(s) dg(s) = (k(s2) − k(s1)) · ky − zkF. s∈[t0,t0+σ] s1

The following statement is a slightly modiﬁed version of Lemma 4.3.5 (the original proof for equations with ﬁnite delay and g(t) = t can be found in [24]).

Lemma 6.3.5. Assume that O is a subset of Ht0+σ, P = {yt; y ∈ O, t ∈ [t0, t0 + σ]}, g :[t0, t0 + σ] → R n is a nondecreasing function, f : P × [t0, t0 + σ] → R satisﬁes condition (A), and F : O × [t0, t0 + σ] → n G((−∞, t0 + σ], R ) given by (6.3.3) has values in Ht0+σ. If x :[t0, t0 + σ] → O is a solution of dx = DF (x, t) dτ and x(t0) ∈ Ht0+σ is a function which is constant on [t0, t0 + σ], then

x(v)(ϑ) = x(v)(v), t0 ≤ v ≤ ϑ ≤ t0 + σ, (6.3.4)

x(v)(ϑ) = x(ϑ)(ϑ), t0 ≤ ϑ ≤ v ≤ t0 + σ. (6.3.5)

Proof. Consider the case when t0 ≤ v ≤ ϑ. Since x is a solution of dx = DF (x, t), dτ we have Z v x(v)(v) = x(t0)(v) + DF (x(τ), t) (v), t0

90 Z v x(v)(ϑ) = x(t0)(ϑ) + DF (x(τ), t) (ϑ). t0

Using the fact that x(t0)(ϑ) = x(t0)(v), we obtain

Z v Z v x(v)(ϑ) − x(v)(v) = DF (x(τ), t) (ϑ) − DF (x(τ), t) (v). t0 t0

It follows from the existence of the integral R v DF (x(τ), t) that for every ε > 0, there is a tagged t0 partition {(τi, [si−1, si]), i = 1, . . . , k} of [t0, v] such that

k Z v X (F (x(τi), si) − F (x(τi), si−1)) − DF (x(τ), t) < ε. i=1 t0 F By the deﬁnition of F in (6.3.3), we have

F (x(τi), si)(ϑ) − F (x(τi), si−1)(ϑ) = F (x(τi), si)(v) − F (x(τi), si−1)(v) for every i ∈ {1, . . . , k}, and consequently

Z v k X kx(v)(ϑ) − x(v)(v)k ≤ DF (x(τ), t) (ϑ) − (F (x(τi), si) − F (x(τi), si−1))(ϑ) t0 i=1

k k X X + (F (x(τi), si) − F (x(τi), si−1))(ϑ) − (F (x(τi), si) − F (x(τi), si−1))(v) i=1 i=1

k Z v X + (F (x(τi), si) − F (x(τi), si−1))(v) − DF (x(τ), t) (v) < (κ1(ϑ − t0 − σ) + κ1(v − t0 − σ))ε i=1 t0 (we have used the third statement of Lemma 6.2.1). This proves (6.3.4), since ε > 0 can be arbitrarily small. For ϑ ≤ v, we have Z v x(v)(ϑ) = x(t0)(ϑ) + DF (x(τ), t) (ϑ), t0

Z ϑ ! x(ϑ)(ϑ) = x(t0)(ϑ) + DF (x(τ), t) (ϑ), t0

Z v x(v)(ϑ) − x(ϑ)(ϑ) = DF (x(τ), t) (ϑ). ϑ

If {(τi, [si−1, si]), i = 1, . . . , k} is an arbitrary tagged partition of [ϑ, v], it follows from (6.3.3) that

F (x(τi), si)(ϑ) − F (x(τi), si−1)(ϑ) = 0, i ∈ {1, . . . , k}.

R v Consequently, ϑ DF (x(τ), t)(ϑ) = 0, and (6.3.5) is proved.

We are now ready to prove two theorems describing the relationship between measure functional differential equations whose solutions take values in Rn, and generalized ordinary differential equations whose solutions take values in Ht0+σ. The basic ideas of both proofs are similar to the proofs given in Chapter 4 for equations with finite delay. However, as we have already noted, our conditions (B) and (C) are weaker, and thus the results presented here are more general even in the case of finite delay.

91 Theorem 6.3.6. Assume that O is a subset of Ht0+σ having the prolongation property for t ≥ t0, P = n {xt; x ∈ O, t ∈ [t0, t0 +σ]}, φ ∈ P , g :[t0, t0 +σ] → R is a nondecreasing function, f : P ×[t0, t0 +σ] → R n satisﬁes conditions (A), (B), (C), and F : O ×[t0, t0 +σ] → G((−∞, t0 +σ], R ) given by (6.3.3) has values in Ht0+σ. If y ∈ O is a solution of the measure functional diﬀerential equation Z t y(t) = y(t0) + f(ys, s) dg(s), t ∈ [t0, t0 + σ], t0

yt0 = φ, then the function x :[t0, t0 + σ] → O given by ( y(ϑ), ϑ ∈ (−∞, t], x(t)(ϑ) = y(t), ϑ ∈ [t, t0 + σ] is a solution of the generalized ordinary diﬀerential equation dx = DF (x, t), t ∈ [t , t + σ]. dτ 0 0 R v Proof. We have to show that, for every v ∈ [t0, t0 + σ], the integral DF (x(τ), t) exists and t0 Z v x(v) − x(t0) = DF (x(τ), t). t0 Let an arbitrary ε > 0 be given. Since the function

Z t h(t) = κ2(σ) M(s) dg(s), t ∈ [t0, t0 + σ] t0

+ is nondecreasing, it can have only a ﬁnite number of points t ∈ [t0, v] such that ∆ h(t) ≥ ε; denote these + points by t1, . . . , tm. Consider a gauge δ :[t0, t0 + σ] → R such that t − t δ(τ) < min k k−1 , k = 2, . . . , m , τ ∈ [t , t + σ], 2 0 0

δ(τ) < min {|τ − tk|; k = 1, . . . , m} , τ ∈ [t0, t0 + σ]. These conditions imply that if a point-interval pair (τ, [c, d]) satisﬁes [c, d] ⊂ (τ − δ(τ), τ + δ(τ)), then [c, d] contains at most one of the points t1, . . . , tm, and, moreover, τ = tk whenever tk ∈ [c, d].

Since ytk = x(tk)tk , it follows from Theorem 6.3.3 that Z s lim L(s)ky − x(t ) k dg(s) = L(t )ky − x(t ) k ∆+g(t ) = 0 s k s F k tk k tk F k s→tk+ tk for every k ∈ {1, . . . , m}. Thus, the gauge δ might be chosen in such a way that

Z tk+δ(tk) ε L(s)kys − x(tk)skF dg(s) < , k ∈ {1, . . . , m}. tk 2m + 1 By condition (B), we have

Z τ+t Z τ+t

ky(τ + t) − y(τ)k = f(ys, s) dg(s) ≤ M(s) dg(s) ≤ h(τ + t) − h(τ) τ τ

(recall that κ2(σ) ≥ 1), and therefore

+ ky(τ+) − y(τ)k ≤ ∆ h(τ) < ε, τ ∈ [t0, t0 + σ]\{t1, . . . , tm}.

92 Thus, we can assume that the gauge δ is such that

ky(ρ) − y(τ)k ≤ ε for every τ ∈ [t0, t0 + σ]\{t1, . . . , tm} and ρ ∈ [τ, τ + δ(τ)). Let {(τi, [si−1, si]), i = 1, . . . , l} be a δ-ﬁne tagged partition of [t0, v]. Then  0, ϑ ∈ (−∞, si−1],  R ϑ x(s ) − x(s ) (ϑ) = f(ys, s) dg(s), ϑ ∈ [si−1, si], i i−1 si−1 R si  f(ys, s) dg(s), ϑ ∈ [si, t0 + σ] si−1 and  0, ϑ ∈ (−∞, si−1],  R ϑ F (x(τ ), s ) − F (x(τ ), s ) (ϑ) = f(x(τi)s, s) dg(s), ϑ ∈ [si−1, si], i i i i−1 si−1 R si  f(x(τi)s, s) dg(s), ϑ ∈ [si, t0 + σ]. si−1 for every i ∈ {1, . . . , l}. By combination of the these equalities, we obtain x(si) − x(si−1) (ϑ) − F (x(τi), si) − F (x(τi), si−1) (ϑ) =  0, ϑ ∈ (−∞, si−1],  R ϑ = f(ys, s) − f(x(τi)s, s) dg(s), ϑ ∈ [si−1, si], si−1 R si  f(ys, s) − f(x(τi)s, s) dg(s), ϑ ∈ [si, t0 + σ]. si−1 Consequently, kx(si) − x(si−1) − F (x(τi), si) − F (x(τi), si−1) kF ≤ κ2(σ) sup k x(si) − x(si−1) (ϑ) − F (x(τi), si) − F (x(τi), si−1) (ϑ)k = ϑ∈[t0,t0+σ]

Z ϑ = κ2(σ) sup f(ys, s) − f(x(τi)s, s) dg(s) (6.3.6) ϑ∈[si−1,si] si−1

(we have used the fourth statement of Lemma 6.2.1). By the deﬁnition of x, we see that x(τi)s = ys whenever s ≤ τi. Thus,

ϑ ( Z 0, ϑ ∈ [si−1, τi], f(y , s) − f(x(τ ) , s) dg(s) = s i s R ϑ si−1 f(ys, s) − f(x(τi)s, s) dg(s), ϑ ∈ [τi, si]. τi We now use condition (C) to obtain the estimate

Z ϑ Z ϑ Z si f(y , s) − f(x(τ ) , s) dg(s) ≤ L(s)ky − x(τ ) k dg(s) ≤ L(s)ky − x(τ ) k dg(s). s i s s i s F s i s F τi τi τi

Given a particular point-interval pair (τi, [si−1, si]), there are two possibilities:

(i) The intersection of [si−1, si] and {t1, . . . , tm} contains a single point tk = τi.

(ii) The intersection of [si−1, si] and {t1, . . . , tm} is empty. In case (i), it follows from the deﬁnition of the gauge δ that

Z si ε L(s)kys − x(τi)skF dg(s) ≤ , τi 2m + 1

93 and substitution back to Eq. (6.3.6) leads to

κ (σ)ε kx(s ) − x(s ) − F (x(τ ), s ) − F (x(τ ), s )k ≤ 2 . i i−1 i i i i−1 F 2m + 1

In case (ii), if s ∈ [τi, si], then

kys − x(τi)skF ≤ κ2(σ) sup ky(ρ) − x(τi)(ρ)k = κ2(σ) sup ky(ρ) − y(τi)k ≤ κ2(σ)ε, ρ∈[−σ+s,s] ρ∈[τi,s]

(we have used the deﬁnition of the gauge δ). Thus,

Z si Z si L(s)kys − x(τi)skF dg(s) ≤ κ2(σ)ε L(s) dg(s), τi τi and substitution back to Eq. (6.3.6) gives

Z si 2 kx(si) − x(si−1) − F (x(τi), si) − F (x(τi), si−1) kF ≤ κ2(σ) ε L(s) dg(s). τi Combining cases (i) and (ii) and using the fact that case (i) occurs at most 2m times, we obtain

l X Z t0+σ 2m x(v) − x(t ) − F (x(τ ), s ) − F (x(τ ), s ) ≤ εκ (σ) κ (σ) L(s) dg(s) + 0 i i i i−1 2 2 2m + 1 i=1 t0 F

 Z t0+σ < εκ2(σ) κ2(σ) L(s) dg(s) + 1 , t0 which completes the proof.

Theorem 6.3.7. Assume that O is a subset of Ht0+σ having the prolongation property for t ≥ t0, P = n {xt; x ∈ O, t ∈ [t0, t0 +σ]}, φ ∈ P , g :[t0, t0 +σ] → R is a nondecreasing function, f : P ×[t0, t0 +σ] → R n satisfies conditions (A), (B), (C), and F : O ×[t0, t0 +σ] → G((−∞, t0 +σ], R ) given by (6.3.3) has values in Ht0+σ. If x :[t0, t0 + σ] → O is a solution of the generalized ordinary differential equation dx = DF (x, t), t ∈ [t , t + σ] dτ 0 0 with the initial condition ( φ(ϑ − t0), ϑ ∈ (−∞, t0], x(t0)(ϑ) = φ(0), ϑ ∈ [t0, t0 + σ], then the function y ∈ O defined by ( x(t )(ϑ), ϑ ∈ (−∞, t ], y(ϑ) = 0 0 x(ϑ)(ϑ), ϑ ∈ [t0, t0 + σ] is a solution of the measure functional differential equation

Z t y(t) = y(t0) + f(ys, s) dg(s), t ∈ [t0, t0 + σ], t0

yt0 = φ.

Proof. The equality yt0 = φ follows directly from the deﬁnitions of y and x(t0). It remains to prove that if v ∈ [t0, t0 + σ], then Z v y(v) − y(t0) = f(ys, s) dg(s). t0

94 Using Lemma 6.3.5, we obtain

Z v y(v) − y(t0) = x(v)(v) − x(t0)(t0) = x(v)(v) − x(t0)(v) = DF (x(τ), t) (v). t0 Thus, Z v Z v Z v y(v) − y(t0) − f(ys, s) dg(s) = DF (x(τ), t) (v) − f(ys, s) dg(s). (6.3.7) t0 t0 t0 Let an arbitrary ε > 0 be given. Since the function

Z t h(t) = κ2(σ) M(s) dg(s), t ∈ [t0, t0 + σ] t0

+ is nondecreasing, it can have only a ﬁnite number of points t ∈ [t0, v] such that ∆ h(t) ≥ ε; denote these + points by t1, . . . , tm. As in the proof of Theorem 6.3.6, consider a gauge δ :[t0, t0 + σ] → R such that t − t δ(τ) < min k k−1 , k = 2, . . . , m , τ ∈ [t , t + σ], 2 0 0

δ(τ) < min {|τ − tk|; k = 1, . . . , m} , τ ∈ [t0, t0 + σ],

Z tk+δ(tk) ε L(s)kys − x(tk)skF dg(s) < , k ∈ {1, . . . , m}. tk 2m + 1 Finally, assume that the gauge δ satisﬁes

kh(ρ) − h(τ)k ≤ ε, τ ∈ [t0, t0 + σ]\{t1, . . . , tm}, ρ ∈ [τ, τ + δ(τ)).

Let {(τi, [si−1, si]), i = 1, . . . , l} be a δ-ﬁne tagged partition of [t0, v] such that

Z v l X DF (x(τ), t) − F (x(τi), si) − F (x(τi), si−1) < ε t0 i=1 F (the existence of such a partition follows from the deﬁnition of the Kurzweil integral). Using (6.3.7) and the third statement of Lemma 6.2.1, we obtain

Z v Z v Z v

y(v) − y(t0) − f(ys, s) dg(s) = DF (x(τ), t) (v) − f(ys, s) dg(s) t0 t0 t0

l Z v X < εκ1(v − t0 − σ) + F (x(τi), si) − F (x(τi), si−1) (v) − f(ys, s) dg(s) i=1 t0

l Z si X ≤ εκ1(v − t0 − σ) + F (x(τi), si) − F (x(τi), si−1) (v) − f(ys, s) dg(s) . (6.3.8) i=1 si−1 The deﬁnition of F yields

Z si F (x(τi), si) − F (x(τi), si−1) (v) = f(x(τi)s, s) dg(s), si−1 which implies

Z si Z si F (x(τi), si) − F (x(τi), si−1) (v) − f(ys, s) dg(s) = (f(x(τi)s, s) − f(ys, s)) dg(s) . si−1 si−1

95 By Lemma 6.3.5, for every i ∈ {1, . . . , l}, we have x(τi)s = x(s)s = ys for s ∈ [si−1, τi] and ys = x(s)s = x(si)s for s ∈ [τi, si]. Therefore

Z si Z si

(f(x(τi)s, s) − f(ys, s)) dg(s) = (f(x(τi)s, s) − f(ys, s)) dg(s) = si−1 τi

Z si Z si

= (f(x(τi)s, s) − f(x(si)s, s)) dg(s) ≤ L(s)kx(τi)s − x(si)skF dg(s), τi τi where the last inequality follows from condition (C). Again, we distinguish two cases:

(i) The intersection of [si−1, si] and {t1, . . . , tm} contains a single point tk = τi.

(ii) The intersection of [si−1, si] and {t1, . . . , tm} is empty. In case (i), it follows from the deﬁnition of the gauge δ that Z si ε L(s)kys − x(τi)skF dg(s) ≤ , τi 2m + 1 and thus

Z si ε F (x(τ ), s ) − F (x(τ ), s )(v) − f(y , s) dg(s) ≤ . i i i i−1 s 2m + 1 si−1

In case (ii), we make use of the fact that x(si) − x(τi) is zero on (−∞, t0] (this follows from the deﬁnition of F ). Lemmas 6.3.1, 6.3.4 and 6.2.1 lead to the estimate

kx(si)s − x(τi)skF ≤ κ2(σ) sup kx(si)(ρ) − x(τi)(ρ)k ρ∈[s−σ,s] ! !

≤ κ2(σ) sup κ1(ρ − t0 − σ) kx(si) − x(τi)kF ≤ κ2(σ) sup κ1(ρ) (h(si) − h(τi)) ≤ Kε ρ∈[s−σ,s] ρ∈[−2σ,0] for every s ∈ [τi, si], where K = κ2(σ) supρ∈[−2σ,0] κ1(ρ). Thus,

Z si Z si F (x(τi), si) − F (x(τi), si−1) (v) − f(ys, s) dg(s) ≤ Kε L(s) dg(s). si−1 τi Combining cases (i) and (ii) and using the fact that case (i) occurs at most 2m times, we obtain

l Z si X F (x(τi), si) − F (x(τi), si−1) (v) − f(ys, s) dg(s) i=1 si−1 Z t0+σ 2mε Z t0+σ ≤ Kε L(s) dg(s) + < ε K L(s) dg(s) + 1 . t0 2m + 1 t0 Substitution back into (6.3.8) gives

Z v Z t0+σ 

y(v) − y(t0) − f(ys, s) dg(s) < ε κ1(v − t0 − σ) + K L(s) dg(s) + 1 , t0 t0 which completes the proof. Remark 6.3.8. It follows from Lemma 6.3.5 that the deﬁnition ( x(t )(ϑ), ϑ ∈ (−∞, t ], y(ϑ) = 0 0 x(ϑ)(ϑ), ϑ ∈ [t0, t0 + σ] from the previous theorem can be replaced by the single equality

y(ϑ) = x(t0 + σ)(ϑ), ϑ ∈ (−∞, t0 + σ]. This also shows that y is indeed an element of O.

96 Remark 6.3.9. The statements of both Theorem 6.3.6 and Theorem 6.3.7 require that F (x, t) ∈ Ht0+σ for every t ∈ [t0, t0 + σ] and x ∈ O. Although this condition might seem diﬃcult to verify, it is often satisﬁed automatically. Note that F (x, t) is a regulated function whose support is contained in [t0, t0 + σ]. n n Thus, for H0 = BG((−∞, 0], R ) or H0 = Gγ ((−∞, 0], R ) (or when H0 is any other space containing all regulated functions with a compact support), we always have F (x, t) ∈ Ht0+σ.

In the classical case when g(t) = t, the function F is continuous, and F (x, t) ∈ Ht0+σ is always satisﬁed n n for both H0 = BC((−∞, 0], R ) and H0 = Cγ ((−∞, 0], R ).

Remark 6.3.10. Throughout the paper, we have been assuming that our phase space H0 satisﬁes conditions (H1)–(H6). However, assumptions (H5) and (H6) were never used in the proofs of Theorems 6.3.6 and 6.3.7, i.e., the two theorems remain valid if we assume (H1)–(H4) only. Assumption (H6) can be quite useful when dealing with condition (C). For the existence of the integral on the right-hand side of this condition, it is suﬃcient to prove that the function t 7→ L(t)kyt − ztkF is regulated. For example, if L is a regulated function, assumption (H6) guarantees the existence of the integral. Assumption (H5) was used to prove the second part of Lemma 3.4, which will be needed in the proof of Theorem 3.12. Remark 6.3.11. An immediate consequence of the relation between the two types of equations is the following: The initial value problem Z t y(t) = y(t0) + f(ys, s) dg(s), t ∈ [t0, t0 + σ], t0

yt0 = φ has a unique solution if and only if the initial value problem dx = DF (x, t), t ∈ [t , t + σ], dτ 0 0 ( φ(ϑ − t0), ϑ ∈ (−∞, t0], x(t0)(ϑ) = φ(0), ϑ ∈ [t0, t0 + σ], has a unique solution. Indeed, if we assume that the first problem has more than one solution, then Theorem 6.3.6 implies that the second problem has more than one solution. Conversely, assume that the second problem has two different solutions x1, x2, which do not coincide at a point t ∈ (t0, t0 + σ], i.e., x1(t)(τ) 6= x2(t)(τ) for a certain τ ∈ (−∞, t0 + σ]. By the definition of F , this is possible only for τ > t0. Moreover, by Lemma 6.3.5, we can assume that τ ≤ t (otherwise, t can be used instead of τ). According to Theorem 6.3.7, we obtain a pair of solutions y1, y2 of the first initial value problem, and

y1(τ) = x1(τ)(τ) = x1(t)(τ) 6= x2(t)(τ) = x2(τ)(τ) = y2(τ), i.e., the first problem does not have a unique solution. With Theorems 6.3.6 and 6.3.7 at our disposal, we can use existing theorems on generalized differential equations to obtain new results for measure functional differential equations with infinite delay. As an example, we present a theorem on the local existence and uniqueness of solutions for measure equations.

Theorem 6.3.12. Assume that O is an open subset of Ht0+σ having the prolongation property for t ≥ t0, P = {xt; x ∈ O, t ∈ [t0, t0 + σ]}, g :[t0, t0 + σ] → R is a left-continuous nondecreasing function, n n f : P × [t0, t0 + σ] → R satisﬁes conditions (A), (B), (C), and F : O × [t0, t0 + σ] → G((−∞, t0 + σ], R ) given by (6.3.3) has values in Ht0+σ. If φ ∈ P is such that the functions ( φ(t − t0), t ∈ (−∞, t0], x0(t) = φ(0), t ∈ [t0, t0 + σ], and ( φ(t − t0), t ∈ (−∞, t0], x1(t) = + φ(0) + f(φ, t0)∆ g(t0), t ∈ (t0, t0 + σ]

97 n are elements of O, then there exists a δ > 0 and a function y :(−∞, t0 + δ] → R which is a unique solution of the initial value problem

Z t y(t) = y(t0) + f(ys, s) dg(s), t0

yt0 = φ on (−∞, t0 + δ]. Proof. By Theorems 6.3.6 and 6.3.7, the initial value problem is equivalent to

dx = DF (x, t), x(t ) = x , dτ 0 0 where F is given by (6.3.3). By Lemma 6.3.4, F satisfies conditions (F1) and (F2). Thus, by the existence and uniqueness theorem for generalized ordinary differential equations (see e.g. [24, Theorem 2.15] or The- orem 4.5.1), there exists a unique local solution of our initial value problem, provided that the function x1 specified in the statement of the theorem is an element of O.

Remark 6.3.13. It is slightly inconvenient that the above theorem refers to the function F , which represents the right-hand side of the corresponding generalized equation and plays an auxiliary role only.

However, as we mentioned in Remark 6.3.9, the condition F (x, t) ∈ Ht0+σ is often satisfied automatically. We conclude this section by presenting modified versions of Theorems 6.3.6 and 6.3.7 for equations whose solutions are defined on the unbounded interval [t0, ∞).

In this case, the space Ht0+σ has to be replaced by a space H consisting of regulated functions deﬁned on the whole real line. The space H together with its associated norm k · kH can be an arbitrary space having the following properties:

1. H is complete.

2. For every a ∈ [t0, ∞), the space Ha is isometrically embedded into H as follows: if y ∈ Ha, then the function z : R → Rn given by ( y(s), s ∈ (−∞, a], z(s) = y(a), s ∈ (a, ∞)

is an element of H and kzkH = kykF.

The simplest example of a space H satisfying these conditions is the space BG(R, Rn) of all bounded regulated functions with the supremum norm.

Alternatively, we can take an arbitrary γ ≥ 0 and consider the space Gγ,t0 of all regulated functions y : R → Rn such that ! max sup keγ(t−t0)y(t)k, sup keγ(t0−t)y(t)k t∈(−∞,t0] t∈[t0,∞) is a finite number; this number is then defined to be the norm of y. For the unbounded interval [t0, ∞), conditions (A), (B), (C) have to be modified as follows:

R t0+σ (A’) The integral f(yt, t) dg(t) exists for every y ∈ O and every σ > 0. t0

+ (B’) There exists a function M :[t0, ∞) → R , which is locally Kurzweil-Stieltjes integrable with respect to g, such that

Z b Z b

f(yt, t) dg(t) ≤ M(t) dg(t) a a

whenever y ∈ O and [a, b] ⊂ [t0, ∞).

98 + (C’) There exists a function L :[t0, ∞) → R , which is locally Kurzweil-Stieltjes integrable with respect to g, such that

Z b Z b (f(y , t) − f(z , t)) dg(t) ≤ L(t)ky − z k dg(t) t t t t F a a

whenever y, z ∈ O and [a, b] ⊂ [t0, ∞) (we are assuming that the integral on the right-hand side exists).

We can now present modiﬁed versions of Theorem 6.3.6 and 6.3.7 for the interval [t0, ∞).

Theorem 6.3.14. Assume that O is a subset of H having the prolongation property for t ≥ t0, P = n {xt; x ∈ O, t ∈ [t0, ∞)}, φ ∈ P , g :[t0, ∞) → R is a nondecreasing function, f : P × [t0, ∞) → R n satisﬁes conditions (A’), (B’), (C’), and the function F : O × [t0, ∞) → G(R, R ) given by  0, −∞ < ϑ ≤ t ,  0 R ϑ f(xs, s) dg(s), t0 ≤ ϑ ≤ t < ∞, F (x, t)(ϑ) = t0 (6.3.9) R t  f(xs, s) dg(s), t0 ≤ t ≤ ϑ < ∞ t0 has values in H. If y ∈ O is a solution of the measure functional diﬀerential equation Z t y(t) = y(t0) + f(ys, s) dg(s), t ∈ [t0, ∞), t0

yt0 = φ, then the function x :[t0, ∞) → O given by ( y(ϑ), ϑ ∈ (−∞, t], x(t)(ϑ) = y(t), ϑ ∈ [t, ∞). is a solution of the generalized ordinary diﬀerential equation dx = DF (x, t), t ∈ [t , ∞). dτ 0

Theorem 6.3.15. Assume that O is a subset of H having the prolongation property for t ≥ t0, P = n {xt; x ∈ O, t ∈ [t0, ∞)}, φ ∈ P , g :[t0, ∞) → R is a nondecreasing function, f : P × [t0, ∞) → R n satisfies conditions (A’), (B’), (C’), and the function F : O × [t0, ∞) → G(R, R ) given by (6.3.9) has values in H. If x :[t0, ∞) → O is a solution of the generalized ordinary differential equation dx = DF (x, t), t ∈ [t , ∞) dτ 0 with the initial condition ( φ(ϑ − t0), ϑ ∈ (−∞, t0], x(t0)(ϑ) = x(t0)(t0), ϑ ∈ [t0, ∞), then the function y ∈ O defined by ( x(t )(ϑ), ϑ ∈ (−∞, t ], y(ϑ) = 0 0 x(ϑ)(ϑ), ϑ ∈ [t0, ∞). is a solution of the measure functional differential equation Z t y(t) = y(t0) + f(ys, s) dg(s), t ∈ [t0, t0 + σ], t0

yt0 = φ.

99 Let us explain why the previous two theorems are true. In both cases, the fact that x :[t0, ∞) → O is a solution of the generalized ordinary diﬀerential equation dx = DF (x, t) dτ is equivalent to saying that Z t0+σ x(t0 + σ) − x(t0) = DF (x(τ), t) (6.3.10) t0 for every σ ≥ 0, where the integral on the right hand side is taken in the space H with respect to the norm k · kH . For a ﬁxed σ ≥ 0, Eq. (6.3.10) is equivalent to

Z t0+σ x(t0 + σ)(ϑ) − x(t0)(ϑ) = DF (x(τ), t) (ϑ) (6.3.11) t0 for every ϑ ∈ [t0, t0 + σ]; this follows from the fact the both sides are constant functions on [t0 + σ, ∞).

However, since Ht0+σ is isometrically embedded in H, Eq. (6.3.11) is equivalent to

Z t0+σ x¯(t0 + σ) − x¯(t0) = DF (x(τ), t), t0 wherex ¯(t) denotes the restriction of x(t) to (−∞, t0 +σ] and the integral on the right hand side is taken in the space Ht0+σ with respect to the norm k · kF. The whole problem is now reduced to the finite interval [t0, t0 + σ] (where σ ≥ 0 is arbitrary but fixed), where Theorems 6.3.6 and 6.3.7 apply. Using the correspondence between both types of equations described in the previous section, it is possible to utilize the existing theory of generalized ordinary differential equations to obtain results on measure functional differential equations with infinite delay. This approach was already demonstrated in Chapter 4 for equations with finite delay, and can be repeated for equations with infinite delay without any substantial changes. As shown in Chapter 4, functional dynamic equations on time scales represent a special case of measure functional differential equations. Consequently, another possible application of our results is in the investigation of functional dynamic equations with infinite delay; again, one can follow the methods presented in Chapter 4 for equations with finite delay. As described in Chapter 5, the same approach also works for functional differential and dynamic equations with impulses.

100 Chapter 7

Diﬀerentiability of solutions with respect to initial conditions and parameters

7.1 Introduction

In the classical theory of ordinary diﬀerential equations, it is well known that under certain assumptions, solutions of the problem

x0(t) = f(x(t), t), t ∈ [a, b],

x(t0) = x0, are differentiable with respect to the initial condition; that is, if x(t, x0) denotes the value of the solution at t ∈ [a, b], then the function x0 7→ x(t, x0) is differentiable. The key requirement is that the right-hand side f should be differentiable with respect to x. Moreover, the derivative as a function of t is known to satisfy the so-called variational equation, which might be helpful in determining the value of the derivative. Similarly, under suitable assumptions, solutions of a differential equation whose right-hand side depends on a parameter are differentiable with respect to that parameter. These two types of theorems concerning differentiability of solutions with respect to initial conditions and parameters can be found in many differential equations textbooks, see e.g. [50]. Theorems of a similar type are also available for other types of equations, such as differential equations with impulses (see [53]) or dynamic equations on time scales (see [43]). The aim of this chapter is to obtain differentiability theorems for generalized ordinary differential equations. Despite the fact that solutions of generalized equations need not be differentiable or even continuous with respect to t, we show that differentiability of the right-hand side with respect to x (and possibly with respect to parameters) still guarantees that the solutions are differentiable with respect to initial conditions (and parameters, respectively). Consequently, our result unifies and extends existing theorems for other types of equations.

7.2 Generalized diﬀerential equations

Consider a set B ⊂ Rn, an interval [a, b] ⊂ R and a function F : B × [a, b]2 → Rn. In this chapter, we work with generalized ordinary diﬀerential equations of the form

dx = DF (x, τ, t), (7.2.1) dτ

101 which is a shorthand notation for the integral equation Z s x(s) = x(a) + DF (x(τ), τ, t), s ∈ [a, b]. (7.2.2) a In other words, a function x :[a, b] → B is a solution of (7.2.1) if and only if (7.2.2) is satisfied. We emphasize that (7.2.1) is a symbolic notation only and does not mean that x has to be differentiable. Equations of the form (7.2.1) have been introduced by J. Kurzweil in [52]. We have seen in previous chapters that in many situations, it is sufficient to consider the less general type of equation dx = DF (x, t), (7.2.3) dτ where the right-hand side does not depend on τ (in fact, most existing sources devoted to generalized equations focus on this less general type; see e.g. the pioneering paper [51] by J. Kurzweil, or the monograph [67] by S.ˇ Schwabik). The corresponding integral equation has the form Z s x(s) = x(a) + DF (x(τ), t), s ∈ [a, b]. a Under certain conditions, an equation of the form (7.2.1) can be transformed to an equation of the form (7.2.3) (see Chapter 27 in [52]). However, as will be clear in Section 4, the more general type (7.2.1) is quite useful for our purposes. The rest of this section summarizes some basic facts concerning the Kurzweil integral that will be needed later. The following existence theorem can be found in [67, Corollary 1.34] or [52, Chapter 20].

Theorem 7.2.1. If f :[a, b] → Rn is a regulated function and g :[a, b] → R is a nondecreasing function, R b then the integral a f(s) dg(s) exists. The next estimate follows directly from the deﬁnition of the Kurzweil integral.

Lemma 7.2.2. Let U :[a, b]2 → Rn×n be a Kurzweil integrable function. Assume there exists a pair of functions f :[a, b] → R and g :[a, b] → R such that f is regulated, g is nondecreasing, and kU(τ, t) − U(τ, s)k ≤ f(τ)|g(t) − g(s)|, τ, t, s ∈ [a, b].

Then

Z b Z b

DU(τ, t) ≤ f(τ) dg(τ). a a For the ﬁrst part of the following statement, see [52, Corollary 14.18] or [67, Theorem 1.16]; the second part is a direct consequence of Lemma 7.2.2.

Theorem 7.2.3. Assume that U :[a, b]2 → Rn×m is Kurzweil integrable and u :[a, b] → Rn×m is its primitive, i.e., Z s u(s) = u(a) + DU(τ, t), s ∈ [a, b]. a If U is regulated in the second variable, then u is regulated and satisﬁes

u(τ+) = u(τ) + U(τ, τ+) − U(τ, τ), τ ∈ [a, b) u(τ−) = u(τ) + U(τ, τ−) − U(τ, τ), τ ∈ (a, b].

Moreover, if there exists a nondecreasing function h :[a, b] → R such that kU(τ, t) − U(τ, s)k ≤ |h(t) − h(s)|, τ, t, s ∈ [a, b], then ku(t) − u(s)k ≤ |h(t) − h(s)|, t, s ∈ [a, b].

102 The following theorem represents an analogue of Gronwall’s inequality for the Kurzweil-Stieltjes integral; the proof can be found in [67, Corollary 1.43]. Theorem 7.2.4. Let h :[a, b] → [0, ∞) be a nondecreasing left-continuous function, k > 0, l ≥ 0. Assume that ψ :[a, b] → [0, ∞) is bounded and satisﬁes

Z ξ ψ(ξ) ≤ k + l ψ(τ) dh(τ), ξ ∈ [a, b]. a Then ψ(ξ) ≤ kel(h(ξ)−h(a)) for every ξ ∈ [a, b].

7.3 Linear equations

In this section, we consider the homogeneous linear generalized ordinary diﬀerential equation dz = D[A(τ, t)z], (7.3.1) dτ where A :[a, b]2 → Rn×n is a given matrix-valued function and the solution z takes values in Rn. The integral form of this equation is Z s z(s) = z(a) + D[A(τ, t)z(τ)], s ∈ [a, b]. a The case when A does not depend on τ has been studied in numerous works (see e.g. [67, 73]). However, to prove the main result of this chapter, we need some basic facts concerning the more general type (7.3.1). For convenience, let us introduce the following condition:

(A) There exists a nondecreasing function h :[a, b] → R such that kA(τ, t) − A(τ, s)k ≤ |h(t) − h(s)|, τ, t, s ∈ [a, b].

Note that if (A) is satisﬁed, then for every ﬁxed τ ∈ [a, b], the function t 7→ A(τ, t) is regulated (this follows from the Cauchy condition for the existence of one-sided limits). Also, if h is left-continuous, then t 7→ A(τ, t) is left-continuous as well.

Lemma 7.3.1. Assume that A :[a, b]2 → Rn×n satisﬁes (A). Let y, z :[a, b] → Rn be a pair of functions such that Z s z(s) = z(a) + D[A(τ, t)y(τ)], s ∈ [a, b]. a Then z is regulated on [a, b]. Proof. Let U(τ, t) = A(τ, t)y(τ); note that Z s z(s) = z(a) + DU(τ, t), s ∈ [a, b]. a By condition (A), U is regulated in the second variable. By Theorem 7.2.3, z is regulated. A simple consequence of the previous lemma is that every solution of the linear generalized diﬀerential equation (7.3.1) is regulated.

Lemma 7.3.2. Assume that A :[a, b]2 → Rn×n satisﬁes (A) with a left-continuous function h. Then for n every z0 ∈ R , the initial value problem dz = D[A(τ, t)z], z(a) = z dτ 0 has at most one solution.

103 n Proof. Consider a pair of functions z1, z2 :[a, b] → R such that dz i = D[A(τ, t)z ], for i ∈ {1, 2}. dτ i Then Z s

kz1(s) − z2(s)k ≤ kz1(a) − z2(a)k + D[A(τ, t)(z1(τ) − z2(τ))] a Z s ≤ kz1(a) − z2(a)k + kz1(τ) − z2(τ)k dh(τ), s ∈ [a, b] a (the last inequality follows from Lemma 7.2.2). By Gronwall’s inequality from Theorem 7.2.4, we obtain

h(b)−h(a) kz1(s) − z2(s)k ≤ kz1(a) − z2(a)ke , s ∈ [a, b].

Thus, if z1(a) = z2(a), then z1, z2 coincide on [a, b].

The proof of the following lemma is almost identical to the proof of Theorem 3.14 in [67]; however, since our assumptions are diﬀerent, we repeat the proof here for reader’s convenience.

n Lemma 7.3.3. Let zk :[a, b] → R , k ∈ N, be a uniformly bounded sequence of functions, which is pointwise convergent to a function z :[a, b] → Rn. Assume that A :[a, b]2 → Rn×n satisfies (A) and R b R b the integral a D[A(τ, t)zk(τ)] exists for every k ∈ N. Then a D[A(τ, t)z(τ)] exists as well and equals R b limk→∞ a D[A(τ, t)zk(τ)]. Proof. Let Ai denote the i-th row of A. Clearly, it is enough to prove that for every i ∈ {1, . . . , n}, the R b i R b i integral a D[A (τ, t)z(τ)] exists and equals limk→∞ a D[A (τ, t)zk(τ)]. i i Let i ∈ {1, . . . , n} be fixed, U(τ, t) = A (τ, t)z(τ) and Uk(τ, t) = A (τ, t)zk(τ) for every k ∈ N (note that U and Uk are scalar functions). Consider an arbitrary fixed ε > 0. For every τ ∈ [a, b], there is a number p(τ) ∈ N such that ε kz (τ) − z(τ)k < , k ≥ p(τ). k h(b) − h(a) + 1

Let M > 0 be such that kzk(τ)k ≤ M for every k ∈ N, τ ∈ [a, b]. The function

εh(t) µ(t) = , t ∈ [a, b], h(b) − h(a) + 1 is nondecreasing and µ(b) − µ(a) < ε. If τ, t1, t2 ∈ [a, b], t1 ≤ t2, and k ≥ p(τ), we have

i i |Uk(τ, t2) − Uk(τ, t1) − U(τ, t2) + U(τ, t1)| ≤ kA (τ, t2) − A (τ, t1)k · kzk(τ) − z(τ)k

ε(h(t ) − h(t )) ≤ kA(τ, t ) − A(τ, t )k · kz (τ) − z(τ)k ≤ 2 1 = µ(t ) − µ(t ) 2 1 k h(b) − h(a) + 1 2 1 and

M(−h(t2) + h(t1)) ≤ Uk(τ, t2) − Uk(τ, t1) ≤ M(h(t2) − h(t1)). The conclusion now follows from the dominated convergence theorem for the Kurzweil integral ([67, Corol- lary 1.31]).

Lemma 7.3.4. Assume that A :[a, b]2 → Rn×n is Kurzweil integrable and satisﬁes (A). Then for every n R b regulated function y :[a, b] → R , the integral a D[A(τ, t)y(τ)] exists.

104 Proof. Every regulated function is a uniform limit of step functions. Thus, in view of Lemma 7.3.3, it is suﬃcient to prove that the statement is true for every step function y :[a, b] → Rn. Let a = t0 < t1 < ··· < tk = b be a partition of [a, b] such that y is constant on each interval R σ (ti−1, ti). For ti−1 < u < σ < v < ti, the integrability of A implies that the integrals u D[A(τ, t)y(τ)] R v and σ D[A(τ, t)y(τ)] exist and are regulated functions of u, v (this follows from (A) and Lemma 7.3.1). According to Hake’s theorems for the Kurzweil integral (see Theorems 14.20 and 14.22 in [52], or Theo- rem 1.14 and Remark 1.15 in [67]), the integrals R σ D[A(τ, t)y(τ)] and R ti D[A(τ, t)y(τ)] exist as well. ti−1 σ Thus, R ti D[A(τ, t)y(τ)] exists for every i ∈ {1, . . . , k}, which proves the statement. ti−1 Up to a small detail, the proof of the following theorem is the same as the proof of Theorem 23.4 in [52]. Therefore, we provide only a sketch of the proof and leave the details to the reader.

Theorem 7.3.5. Assume that A :[a, b]2 → Rn×n is Kurzweil integrable and satisﬁes (A) with a left- n continuous function h. Then for every z0 ∈ R , the initial value problem dz = D[A(τ, t)z], z(a) = z (7.3.2) dτ 0 has a unique solution z :[a, b] → Rn. Proof. Uniqueness of solutions follows immediately from Lemma 7.3.2. To prove the existence of a solution 1 of (7.3.2) on [a, b], choose a partition a = s0 < s1 < ··· < sk = b of [a, b] such that h(si) − h(si−1+) ≤ 2 n for every i ∈ {1, . . . , k}. Now, it is suﬃcient to prove that for every w0 ∈ R and i ∈ {1, . . . , k}, the initial value problem dz = D[A(τ, t)z], z(s ) = w dτ i−1 0 has a solution on [si−1, si]. This solution can be obtained by the method of successive approximations: Let

v0(s) = w0, s ∈ [si−1, si], Z s vj(s) = w0 + D[A(τ, t)vj−1(τ)], s ∈ [si−1, si], j ∈ N. si−1 The existence of the integral on the right-hand side is guaranteed by Lemma 7.3.4 (this is the only diﬀerence against the proof of Theorem 23.4 in [52]). By Theorem 7.2.3, we have Z s vj(s) = wc0 + D[A(τ, t)vj−1(τ)], s ∈ [si−1, si], j ∈ N, si−1+ where wc0 = w0 + A(si−1, si−1+)w0 − A(si−1, si−1)w0. Using Lemma 7.2.2, it is not diﬃcult to see that

kv1(s) − v0(s)k ≤ kw0k(h(si) − h(si−1)), s ∈ [si−1, si],

kvj+1(s) − vj(s)k ≤ sup kvj(τ) − vj−1(τ)k(h(si) − h(si−1+)), s ∈ (si−1, si], j ∈ N. τ∈[si−1,si] 1 Since h(si) − h(si−1+) ≤ 2 , it follows by induction that

1j kv (s) − v (s)k ≤ kw k (h(s ) − h(s )), s ∈ (s , s ], j ∈ . j+1 j 0 2 i i−1 i−1 i N

∞ n This implies that {vj}j=1 is uniformly convergent to a function v :[si−1, si] → R , which satisﬁes Z s Z s v(s) = lim vj(s) = w0 + lim D[A(τ, t)vj−1(τ)] = w0 + D[A(τ, t)v(τ)] j→∞ j→∞ si−1 si−1 for every s ∈ [si−1, si] (we have used Lemma 7.3.3).

105 Throughout this section, we focused our attention on equations of the form dz = D[A(τ, t)z], z(a) = z , dτ 0 where z takes values in Rn. More generally, it is possible to consider equations where the unknown function n×n n×n has values in R . For example, if Z0 ∈ R is an arbitrary matrix, then dZ = D[A(τ, t)Z],Z(a) = Z dτ 0 is a shorthand notation for the integral equation Z s Z(s) = Z0 + D[A(τ, t)Z(τ)], s ∈ [a, b]. (7.3.3) a For an arbitrary matrix X ∈ Rn×n, let Xi be the i-th column of X. Then it is obvious that (7.3.3) holds if and only if Z s i i i Z (s) = Z0 + D[A(τ, t)Z (τ)], s ∈ [a, b], i ∈ {1, . . . , n}. a We will encounter equations with matrix-valued solutions in the following section.

7.4 Main results

Consider a generalized ordinary differential equation of the form dx = DF (x, τ, t), x(a) = x0(λ), (7.4.1) dτ where the solution x takes values in Rn, and x0 : Rl → Rn is a function which describes the dependence of the initial condition on a parameter λ ∈ Rl. Let x(s, λ) be the value of the solution at s ∈ [a, b]. Our goal is to show that under certain conditions, x(s, λ) is differentiable with respect to λ. Using the definition of the Kurzweil integral, we see that the value of x(s, λ) can be approximated by

k 0 X x (λ) + (F (x(τj, λ), τj, tj) − F (x(τj, λ), τj, tj−1)) , j=1 where a = t0 < t1 < ··· < tk = s is a sufficiently fine partition of [a, s] with tags τj ∈ [tj−1, tj], l j ∈ {1, . . . , k}. Assuming that all expressions are differentiable with respect to λ at λ0 ∈ R , we see that ∂xi n×l the derivative xλ(s, λ0), i.e., the matrix { (s, λ0)}i,j ∈ , should be approximately equal to ∂λj R

k 0 X xλ(λ0) + (Fx(x(τj, λ0), τj, tj)xλ(τj, λ0) − Fx(x(τj, λ), τj, tj−1)xλ(τj, λ0)) . j=1 Now, the right-hand side is an approximation to Z s 0 xλ(λ0) + D[Fx(x(τ, λ0), τ, t)xλ(τ, λ0)]. a

Thus, it seems reasonable to expect that the derivative Z(t) = xλ(t, λ0), t ∈ [a, b], is a solution of the linear equation dz = DG(z, τ, t), z(a) = x0 (λ ), (7.4.2) dτ λ 0 where G(z, τ, t) = Fx(x(τ, λ0), τ, t)z. This provides a motivation for the following theorem. Note that even in the case when the right-hand side of Eq. (7.4.1) does not depend on τ and has the form F (x, t), the right-hand side of Eq. (7.4.2) has the form G(z, τ, t) = Fx(x(τ, λ0), t)z, i.e., still depends on τ. That is why we had to study the more general type of equations in the previous section. The following proof is based on elementary estimates and Gronwall’s inequality; it is inspired by a proof of Theorem 3.1 in the paper [43], which is concerned with dynamic equations on time scales.

106 n l l 0 Theorem 7.4.1. Let B ⊂ R be an open set, λ0 ∈ R , ρ > 0, Λ = {λ ∈ R ; kλ − λ0k < ρ}, x :Λ → B, F : B × [a, b]2 → Rn. Assume that F is regulated and left-continuous in the third variable, and that for every λ ∈ Λ, the initial value problem dx = DF (x, τ, t), x(a) = x0(λ) (7.4.3) dτ has a solution defined on [a, b]; let x(t, λ) be the value of that solution at t ∈ [a, b]. Moreover, let the following conditions be satisfied: 1. For every fixed pair (t, τ) ∈ [a, b]2, the function x 7→ F (x, τ, t) is continuously differentiable on B.

0 2. The function x is diﬀerentiable at λ0.

3. There exists a left-continuous nondecreasing function h :[a, b] → R such that

kFx(x, τ, t) − Fx(x, τ, s)k ≤ |h(t) − h(s)|, s, t, τ ∈ [a, b], x ∈ B.

4. There exists a continuous increasing function ω : [0, ∞) → [0, ∞) such that ω(0) = 0 and

kFx(x, τ, t)−Fx(x, τ, s)−Fx(y, τ, t)+Fx(y, τ, s)k ≤ ω(kx−yk)·|h(t)−h(s)|, s, t, τ ∈ [a, b], x, y ∈ B.

5. There exists a left-continuous nondecreasing function k :[a, b] → R such that kF (x, τ, t) − F (x, τ, s) − F (y, τ, t) + F (y, τ, s)k ≤ kx − yk · |k(t) − k(s)|, s, t, τ ∈ [a, b], x, y ∈ B.

n 6. There exists a number η > 0 such that if x ∈ R satisﬁes kx − x(t, λ0)k < η for some t ∈ [a, b], then x ∈ B (i.e., the η-neighborhood of the solution t 7→ x(t, λ0) is contained in B).

Then the function λ 7→ x(t, λ) is diﬀerentiable at λ0, uniformly for all t ∈ [a, b]. Moreover, its derivative Z(t) = xλ(t, λ0), t ∈ [a, b], is the unique solution of the generalized diﬀerential equation Z s 0 Z(s) = xλ(λ0) + D[Fx(x(τ, λ0), τ, t)Z(τ)], s ∈ [a, b]. (7.4.4) a Proof. According to the assumptions, we have Z s x(s, λ) = x0(λ) + DF (x(τ, λ), τ, t), λ ∈ Λ, s ∈ [a, b]. a

By Theorem 2.3, every solution x is a regulated and left-continuous function on [a, b]. If ∆λ ∈ Rl is such that k∆λk < ρ, then Z s 0 0 kx(s, λ0 + ∆λ) − x(s, λ0)k ≤ kx (λ0 + ∆λ) − x (λ0)k + DV (τ, t) , a where V (τ, t) = F (x(τ, λ0 + ∆λ), τ, t) − F (x(τ, λ0), τ, t). By assumption 5, we obtain

kV (τ, t1) − V (τ, t2)k ≤ kx(τ, λ0 + ∆λ) − x(τ, λ0)k · |k(t1) − k(t2)|, and consequently (using Lemma 7.2.2) Z s 0 0 kx(s, λ0 + ∆λ) − x(s, λ0)k ≤ kx (λ0 + ∆λ) − x (λ0)k + kx(τ, λ0 + ∆λ) − x(τ, λ0)k dk(τ). a for every s ∈ [a, b]. Gronwall’s inequality from Theorem 7.2.4 implies

0 0 k(b)−k(a) kx(s, λ0 + ∆λ) − x(s, λ0)k ≤ kx (λ0 + ∆λ) − x (λ0)ke , s ∈ [a, b].

Thus we see that for ∆λ → 0, x(s, λ0 + ∆λ) approaches x(s, λ0) uniformly for all s ∈ [a, b].

107 By assumption 3, the function A(τ, t) = Fx(x(τ, λ0), τ, t) satisﬁes condition (A). By Theorem 7.3.5, Eq. (7.4.4) has a unique solution Z :[a, b] → Rn×n. By Lemma 7.3.1, the solution is regulated. Conse- quently, there exists a constant M > 0 such that kZ(t)k ≤ M for every t ∈ [a, b]. For every ∆λ ∈ Rl such that k∆λk < ρ, let x(r, λ + ∆λ) − x(r, λ ) − Z(r)∆λ ξ(r, ∆λ) = 0 0 , r ∈ [a, b]. k∆λk

Our goal is to prove that if ∆λ → 0, then ξ(r, ∆λ) → 0 uniformly for r ∈ [a, b]. Let ε > 0 be given. There exists a δ > 0 such that if ∆λ ∈ Rl and k∆λk < δ, then

kx(t, λ0 + ∆λ) − x(t, λ0)k < min(ε, η), t ∈ [a, b], and kx0(λ + ∆λ) − x0(λ ) − x0 (λ )∆λk 0 0 λ 0 < ε. k∆λk Observe that x0(λ + ∆λ) − x0(λ ) − x0 (λ )∆λ ξ(a, ∆λ) = 0 0 λ 0 , k∆λk x(r, λ + ∆λ) − x(a, λ + ∆λ) x(r, λ ) − x(a, λ ) (Z(r) − Z(a))∆λ ξ(r, ∆λ) − ξ(a, ∆λ) = 0 0 − 0 0 − k∆λk k∆λk k∆λk Z r = DU(τ, t), a where F (x(τ, λ + ∆λ), τ, t) − F (x(τ, λ ), τ, t) − F (x(τ, λ ), τ, t)Z(τ)∆λ U(τ, t) = 0 0 x 0 . k∆λk For every τ ∈ [a, b] and u ∈ [0, 1], we have

kux(τ, λ0 + ∆λ) + (1 − u)x(τ, λ0) − x(τ, λ0)k ≤ kx(τ, λ0 + ∆λ) − x(τ, λ0)k < η.

By assumption 6, the point ux(τ, λ0 +∆λ)+(1−u)x(τ, λ0) is an element of B. In other words, the segment connecting x(τ, λ0 + ∆λ) and x(τ, λ0) is contained in B. Thus we can use the mean-value theorem for vector-valued functions (see e.g. [50, Lemma 8.11]) to examine the following diﬀerence:

F (x(τ, λ + ∆λ), τ, t) − F (x(τ, λ ), τ, t) − F (x(τ, λ + ∆λ), τ, s) + F (x(τ, λ ), τ, s) U(τ, t) − U(τ, s) = 0 0 0 0 k∆λk

(F (x(τ, λ ), τ, t) − F (x(τ, λ ), τ, s))(x(τ, λ + ∆λ) − x(τ, λ )) − x 0 x 0 0 0 k∆λk (F (x(τ, λ ), τ, t) − F (x(τ, λ ), τ, s))(x(τ, λ + ∆λ) − x(τ, λ ) − Z(τ)∆λ) + x 0 x 0 0 0 k∆λk

1 Z 1 = (Fx(ux(τ, λ0 + ∆λ) + (1 − u)x(τ, λ0), τ, t) − Fx(ux(τ, λ0 + ∆λ) + (1 − u)x(τ, λ0), τ, s)) du k∆λk 0

Z 1 ! − (Fx(x(τ, λ0), τ, t) − Fx(x(τ, λ0), τ, s)) du · (x(τ, λ0 + ∆λ) − x(τ, λ0)) 0

+(Fx(x(τ, λ0), τ, t) − Fx(x(τ, λ0), τ, s))ξ(τ, ∆λ) (In the second integral above, we are simply integrating a constant function.) If k∆λk < δ, then (by assumption 4) kFx(ux(τ, λ0 + ∆λ) + (1 − u)x(τ, λ0), τ, t) − Fx(ux(τ, λ0 + ∆λ) + (1 − u)x(τ, λ0), τ, s) − Fx(x(τ, λ0), τ, t)

108 +Fx(x(τ, λ0), τ, s)k ≤ ω(kux(τ, λ0 + ∆λ) + (1 − u)x(τ, λ0) − x(τ, λ0)k)|h(t) − h(s)|

= ω(ku(x(τ, λ0 + ∆λ) − x(τ, λ0))k)|h(t) − h(s)| ≤ ω(ε)|h(t) − h(s)|, and thus (using assumption 3)

kx(τ, λ + ∆λ) − x(τ, λ )k kU(τ, t) − U(τ, s)k ≤ ω(ε)|h(t) − h(s)| 0 0 + |h(t) − h(s)| · kξ(τ, ∆λ)k k∆λk

kx(τ, λ + ∆λ) − x(τ, λ ) − Z(τ)∆λk + kZ(τ)∆λk ≤ |h(t) − h(s)| ω(ε) 0 0 + kξ(τ, ∆λ)k k∆λk ≤ |h(t) − h(s)| (ω(ε)(kξ(τ, ∆λ)k + M) + kξ(τ, ∆λ)k) . Consequently, by Lemma 7.2.2, Z r Z r

kξ(r, ∆λ) − ξ(a, ∆λ)k = DU(τ, t) ≤ (ω(ε)(kξ(τ, ∆λ)k + M) + kξ(τ, ∆λ)k) dh(τ) a a Z r = ω(ε)M(h(r) − h(a)) + (1 + ω(ε)) kξ(τ, ∆λ)k dh(τ) a Z r ≤ ω(ε)M(h(b) − h(a)) + (1 + ω(ε)) kξ(τ, ∆λ)k dh(τ). a It follows that kξ(r, ∆λ)k ≤ kξ(r, ∆λ) − ξ(a, ∆λ)k + kξ(a, ∆λ)k Z r ≤ ε + ω(ε)M(h(b) − h(a)) + (1 + ω(ε)) kξ(τ, ∆λ)k dh(τ). a Finally, Gronwall’s inequality leads to the estimate

kξ(r, ∆λ)k ≤ (ε + ω(ε)M(h(b) − h(a)))e(1+ω(ε))(h(r)−h(a))

≤ (ε + ω(ε)M(h(b) − h(a)))e(1+ω(ε))(h(b)−h(a)).

Since limε→0+ ω(ε) = 0, we see that if ∆λ → 0, then ξ(r, ∆λ) → 0 uniformly for r ∈ [a, b]. In the simplest case when l = n,Λ ⊂ B and x0(λ) = λ for every λ ∈ Λ, the previous theorem says that solutions of dx = DF (x, τ, t), x(a) = λ dτ are diﬀerentiable with respect to λ, and the derivative Z(t) = xλ(t, λ0), t ∈ [a, b], is the unique solution of the generalized diﬀerential equation Z s Z(s) = I + D[Fx(x(τ, λ0), τ, t)Z(τ)], s ∈ [a, b]. a

Remark 7.4.2. In Theorem 7.4.1, we are assuming the existence of a ρ > 0 such that for every λ ∈ Rl satisfying kλ − λ0k < ρ, the initial value problem (7.4.3) has a solution t 7→ x(t, λ) deﬁned on [a, b] and taking values in B. For equations whose right-hand side F does not depend on τ, this assumption can be replaced by the following simple condition:

kF (x, t) − F (x, s)k ≤ |k(t) − k(s)|, s, t ∈ [a, b], x ∈ B. (7.4.5)

Let us explain why this condition is suﬃcient. (We are still assuming that conditions 1–6 from Theo- rem 7.4.1 are satisﬁed. In particular, we are assuming that the initial value problem (7.4.3) has a solution corresponding to λ = λ0.) Observe that if c ∈ [a, b) and ky − x(c+, λ0)k < η/2, then y ∈ B. (Choose δ > 0 such that kx(c+, λ0) − x(c + δ, λ0)k < η/2; then ky − x(c + δ, λ0)k ≤ ky − x(c+, λ0)k + kx(c+, λ0) − x(c + δ, λ0)k < η, and thus y ∈ B by assumption 6.)

109 Following the ﬁrst part of proof of Theorem 7.4.1, we observe that there is a δ > 0 such that if kλ − λ0k < δ and if the solution t 7→ x(t, λ) exists on [a, c] ⊆ [a, b], then

η 1 kx(t, λ) − x(t, λ )k < · min 1, , t ∈ [a, c]. 0 4 k(b) − k(a) + 1

l If λ ∈ R satisﬁes kλ − λ0k < δ and the solution t 7→ x(t, λ) exists on [a, c] ⊆ [a, b] with c ∈ [a, b), then

k (x(c, λ) + F (x(c, λ), c+) − F (x(c, λ), c)) − (x(c, λ0) + F (x(c, λ0), c+) − F (x(c, λ0), c)) k

≤ kx(c, λ) − x(c, λ0)k + kF (x(c, λ), c+) − F (x(c, λ), c) − F (x(c, λ0), c+) + F (x(c, λ0), c)k

< η/4 + kx(c, λ) − x(c, λ0)k(k(c+) − k(c)) < η/4 + kx(c, λ) − x(c, λ0)k(k(b) − k(a)) < η/2

(we have used assumption 5). By Theorem 7.2.3, x(c+, λ0) = x(c, λ0) + F (x(c, λ0), c+) − F (x(c, λ0), c). Thus the previous inequality and assumption 6 imply x(c, λ) + F (x(c, λ), c+) − F (x(c, λ), c) ∈ B. All assumptions of the local existence theorem for generalized diﬀerential equations (see [67, Theo- rem 4.2]) are satisﬁed (we need (7.4.5) at this moment), and thus the solution t 7→ x(t, λ) can be extended to a larger interval [a, d], d ∈ (c, b]. Consequently, for kλ − λ0k < δ, the solution must exist on the whole interval [a, b].

Remark 7.4.3. Assume that the set B from Theorem 7.4.1 is convex. Then it is easy to see that assumption 5 in this theorem is redundant. Indeed, the mean-value theorem for vector-valued functions and assumption 3 lead to the estimate

kF (x, τ, t) − F (x, τ, s) − F (y, τ, t) + F (y, τ, s)k

Z 1 ≤ kFx(ux + (1 − u)y, τ, t) − Fx(ux + (1 − u)y, τ, s)k du · kx − yk ≤ |h(t) − h(s)| · kx − yk 0 for all s, t, τ ∈ [a, b], x, y ∈ B, i.e., assumption 5 is satisﬁed with k = h.

With the help of Theorem 7.4.1, it is easy to obtain an even more general theorem for equations where not only the initial condition, but also the right-hand side of the equation depends on the parameter λ. The proof is inspired by a similar proof of Theorem 8.49 in [50].

n l l 0 Theorem 7.4.4. Let B ⊂ R be an open set, λ0 ∈ R , ρ > 0, Λ = {λ ∈ R ; kλ − λ0k < ρ}, x :Λ → B, F : B × [a, b]2 × Λ → Rn. Assume that F is regulated and left-continuous in the third variable, and that for every λ ∈ Λ, the initial value problem

dx = DF (x, τ, t, λ), x(a) = x0(λ) dτ has a solution in B; let x(t, λ) be the value of that solution at t ∈ [a, b]. Moreover, let the following conditions be satisﬁed:

1. For every ﬁxed pair (t, τ) ∈ [a, b]2, the function (x, λ) 7→ F (x, τ, t, λ) is continuously diﬀerentiable on B × Λ.

0 2. The function x is diﬀerentiable at λ0.

3. There exists a left-continuous nondecreasing function h :[a, b] → R such that

kFx(x, τ, t, λ) − Fx(x, τ, s, λ)k ≤ |h(t) − h(s)|,

kFλ(x, τ, t, λ) − Fλ(x, τ, s, λ)k ≤ |h(t) − h(s)| for all s, t, τ ∈ [a, b], x ∈ B, λ ∈ Λ.

110 4. There exists a continuous increasing function ω : [0, ∞) → [0, ∞) such that ω(0) = 0 and

kFx(x, τ, t, λ1)−Fx(x, τ, s, λ1)−Fx(y, τ, t, λ2)+Fx(y, τ, s, λ2)k ≤ ω(kx−yk+kλ1 −λ2k)·|h(t)−h(s)|,

kFλ(x, τ, t, λ1)−Fλ(x, τ, s, λ1)−Fλ(y, τ, t, λ2)+Fλ(y, τ, s, λ2)k ≤ ω(kx−yk+kλ1 −λ2k)·|h(t)−h(s)|

for all s, t, τ ∈ [a, b], x, y ∈ B, λ1, λ2 ∈ Λ.

5. There exists a left-continuous nondecreasing function k :[a, b] → R such that

kF (x, τ, t, λ1) − F (x, τ, s, λ1) − F (y, τ, t, λ2) + F (y, τ, s, λ1)k ≤ (kx − yk + kλ1 − λ2k) · |k(t) − k(s)|

for all s, t, τ ∈ [a, b], x, y ∈ B, λ1, λ2 ∈ Λ.

Z s 0 Z(s) = xλ(λ0) + D[Fx(x(τ, λ0), τ, t, λ0)Z(τ) + Fλ(x(τ, λ0), τ, t, λ0)], s ∈ [a, b]. a

Proof. Let B˜ = B ×Λ. Without loss of generality, assume that all ﬁnite-dimensional spaces we are working with are equipped with the L1 norm. In particular, when (x, λ) ∈ B˜, then

n l X X k(x, λ)k = |xi| + |λj| = kxk + kλk. i=1 j=1

Deﬁne F˜ : B˜ × [a, b]2 → Rn+l by

n+l F˜((x, λ), τ, t) = (F (x, τ, t, λ), 0,..., 0) ∈ R , x ∈ B, λ ∈ Λ, t, τ ∈ [a, b], and y0 :Λ → B˜ by y0(λ) = (x0(λ), λ), λ ∈ Λ. From these deﬁnitions, it is clear that for every λ ∈ Λ, the function

y(t, λ) = (x(t, λ), λ), t ∈ [a, b], is a solution of the initial value problem

dy = DF˜(y, τ, t), y(a) = y0(λ) dτ

(note that by the deﬁnition of F˜, the last l components of every solution are constant on [a, b]). Without loss of generality, assume that η < ρ. If (x, λ) ∈ B˜ is such that k(x, λ) − (x(t, λ0), λ0)k < η for some t ∈ [a, b], then kx − x(t, λ0)k < η and kλ − λ0k < η, i.e., x ∈ B and λ ∈ Λ. In other words, the η-neighborhood of the solution t 7→ y(t, λ0) is contained in B. The derivative of F˜ with respect to y is the (n + l) × (n + l) matrix

˜ ˜  ∂F1 (y, τ, t) ··· ∂F1 (y, τ, t) ∂y1 ∂yn+l ˜  . . .  Fy(y, τ, t) =  . .. .   ˜ ˜  ∂Fn+l (y, τ, t) ··· ∂Fn+l (y, τ, t) ∂y1 ∂yn+l

111  ∂F1 (x, τ, t, λ) ··· ∂F1 (x, τ, t, λ) ∂F1 (x, τ, t, λ) ··· ∂F1 (x, τ, t, λ) ∂x1 ∂xn ∂λ1 ∂λl  ......   ......     ∂Fn (x, τ, t, λ) ··· ∂Fn (x, τ, t, λ) ∂Fn (x, τ, t, λ) ··· ∂Fn (x, τ, t, λ) =  ∂x1 ∂xn ∂λ1 ∂λl  ,  0 ··· 0 0 ··· 0     ......   ......  0 ··· 0 0 ··· 0 where y = (x, λ) ∈ B × Λ and t, τ ∈ [a, b]. 0 Similarly, the derivative of y with respect to λ at λ0 is the (n + l) × l matrix

0 0  ∂x1 ∂x1  (λ0) ··· (λ0) ∂λ1 ∂λl  . . .   . .. .   0 0   ∂xn ∂xn  0  (λ0) ··· (λ0) yλ(λ0) =  ∂λ1 ∂λl  .  1 ··· 0     . . .   . .. .  0 ··· 1

Using assumptions 3, 4 and 5, it is not diﬃcult to see that F˜ and F˜y satisfy assumptions 3, 4 and 5 of Theorem 7.4.1. For example, let s, t, τ ∈ [a, b], y1, y2 ∈ B˜, where y1 = (x1, λ1) and y2 = (x2, λ2). Then

kF˜y(y1, τ, t) − F˜y(y1, τ, s) − F˜y(y2, τ, t) + F˜y(y2, τ, s)k

= kFx(x1, τ, t, λ1) − Fx(x1, τ, s, λ1) − Fx(x2, τ, t, λ2) + Fx(x2, τ, s, λ2)k

+kFλ(x1, τ, t, λ1) − Fλ(x1, τ, s, λ1) − Fλ(x2, τ, t, λ2) + Fλ(x2, τ, s, λ2)k

≤ 2ω(kx1 − x2k + kλ1 − λ2k) · |h(t) − h(s)| = ω(ky1 − y2k) · |2h(t) − 2h(s)|, which verifies assumption 4 of Theorem 7.4.1. Now, according to Theorem 7.4.1, the function λ 7→ y(t, λ) is differentiable at λ0, uniformly for all t ∈ [a, b], and its derivative Z˜(t) = yλ(t, λ0), t ∈ [a, b], is the unique solution of the generalized differential equation Z s ˜ 0 ˜ ˜ Z(s) = yλ(λ0) + D[Fy(y(τ, λ0), τ, t)Z(τ)], s ∈ [a, b]. a

Let Z(t) = xλ(t, λ0), t ∈ [a, b]; observe that Z is the submatrix of Z˜ corresponding to the ﬁrst n rows. Also, note that the last l rows of Z˜ form the identity matrix. Thus it follows that Z s 0 Z(s) = xλ(λ0) + D[Fx(x(τ, λ0), τ, t, λ0)Z(τ) + Fλ(x(τ, λ0), τ, t, λ0)], s ∈ [a, b]. a 7.5 Relation to other types of equations

In this section, we show that for impulsive differential equations and for dynamic equations on time scales, differentiability of solutions with respect to initial conditions follows from our Theorem 7.4.1. (Similarly, it can be shown that differentiability with respect to parameters follows from Theorem 7.4.4). The reason is that both types of equations can be rewritten as generalized equations, whose right-hand sides do not depend on τ. Assume that r > 0 is a fixed number. We restrict ourselves to the case B = {x ∈ Rn; kxk < r}; we use B to denote the closure of B.

Lemma 7.5.1. Let µ be the Lebesgue-Stieltjes measure generated by a left-continuous nondecreasing function g :[a, b] → R (i.e., µ([c, d)) = g(d) − g(c) for every interval [c, d) ⊂ [a, b]). Assume that f : B × [a, b] → Rm×n satisﬁes the following conditions:

112 • For every x ∈ B, the function s 7→ f(x, s) is measurable on [a, b] with respect to the measure µ. R b • There exists a µ-measurable function M :[a, b] → R such that a M(s) dµ < +∞ and kf(x, s)k ≤ M(s), x ∈ B, s ∈ [a, b].

• For every s ∈ [a, b], the function x 7→ f(x, s) is continuous in B. Consider the function F given by Z Z t F (x, t) = f(x, s) dµ = f(x, s) dg(s), x ∈ B, t ∈ [a, b]. [a,t) a Then the following statements are true:

1. There exists a nondecreasing left-continuous function h :[a, b] → R and a continuous increasing function ω : [0, ∞) → [0, ∞) such that ω(0) = 0 and kF (x, t) − F (x, s)k ≤ |h(t) − h(s)|, s, t ∈ [a, b], x ∈ B, kF (x, t) − F (x, s) − F (y, t) + F (y, s)k ≤ ω(kx − yk)|h(t) − h(s)|, s, t ∈ [a, b], x, y ∈ B.

2. If x :[a, b] → B, Z :[a, b] → Rn×l are regulated functions, then Z b Z b D[F (x(τ), t)Z(τ)] = f(x(τ), τ)Z(τ) dg(τ). (7.5.1) a a Proof. For the first statement, see Proposition 5.9 in [67] and the references given there. Let us prove the second statement. According to Proposition 5.12 in [67], we have Z b Z b DF (x(τ), t) = f(x(τ), τ) dg(τ) (7.5.2) a a for every regulated function x :[a, b] → B. Let [α, β] ⊆ [a, b] and assume that Z :[α, β] → Rn×l is constant on (α, β). Then, by (7.5.2) and Theorem 7.2.3, Z β D[F (x(τ), t)Z(τ)] α Z β−ε Z α+ε Z β ! = lim D[F (x(τ), t)Z(τ)] + D[F (x(τ), t)Z(τ)] + D[F (x(τ), t)Z(τ)] ε→0+ α+ε α β−ε Z β−ε ! = lim f(x(τ), τ)Z(τ) dg(τ) +(F (x(α), α+)−F (x(α), α))Z(α)+(F (x(β), β)−F (x(β), β−))Z(β) ε→0+ α+ε Z β−ε ! = lim f(x(τ), τ)Z(τ) dg(τ) + f(x(α), α)Z(α)(g(α+) − g(α)) + f(x(β), β)Z(β)(g(β) − g(β−)) ε→0+ α+ε Z β = f(x(τ), τ)Z(τ) dg(τ). α This shows that (7.5.1) is satisfied for all step functions Z :[a, b] → Rn×l. For a general regulated ∞ function Z, let {Zk}k=1 be a sequence of step functions that is uniformly convergent to Z. Then Z b Z b D[F (x(τ), t)Z(τ)] = lim D[F (x(τ), t)Zk(τ)] a k→∞ a Z b Z b = lim f(x(τ), τ)Zk(τ) dg(τ) = f(x(τ), τ)Z(τ) dg(τ), k→∞ a a where the first equality follows from Lemma 7.3.3 and the last equality from [67, Corollary 1.32].

113 Let us start by considering differential equations with impulses. Assume that C is an open neighborhood n of B, f : C × [a, b] → R is a continuous function whose derivative fx exists and is continuous on C × [a, b], and I1,...,Ik : C → Rn are continuously differentiable functions. Then it is known (see [67, Chapter 5]) that the impulsive differential equation

0  x (t) = f(x(t), t), t ∈ [a, b]\{t1, . . . , tk}, i  x(ti+) − x(ti) = I (x(ti)), i ∈ {1, . . . , k}, (7.5.3) x(a) = x0(λ),  whose solutions are assumed to be left-continuous, is equivalent to the generalized diﬀerential equation

dx = DF (x, t), t ∈ [a, b], x(a) = x0(λ), dτ where F (x, t) = F 1(x, t) + F 2(x, t) and

Z t k 1 2 X i F (x, t) = f(x, s) ds, F (x, t) = I (x)χ(ti,∞)(t) a i=1

(the symbol χA denotes the characteristic function of a set A ⊂ R). More precisely, x :[a, b] → B is a solution of the impulsive equation (7.5.3) if and only if it is a solution of the generalized equation (see [67, Theorem 5.20]). Now, F 1 and F 2 are diﬀerentiable with respect to x and

Z t k 1 2 X i Fx (x, t) = fx(x, s) ds, Fx (x, t) = Ix(x)χ(ti,∞)(t). a i=1

By the ﬁrst part of Lemma 7.5.1 (where we take g(s) = s and µ is the Lebesgue measure), we obtain the existence of functions h1 :[a, b] → R and ω1 : [0, ∞) → [0, ∞) such that

1 1 kFx (x, t) − Fx (x, s)k ≤ |h1(t) − h1(s)|, s, t ∈ [a, b], x ∈ B,

1 1 1 1 kFx (x, t) − Fx (x, s) − Fx (y, t) + Fx (y, s)k ≤ ω1(kx − yk)|h1(t) − h1(s)|, s, t ∈ [a, b], x, y ∈ B. i By continuity, there exists a K ≥ 1 such that kIx(x)k ≤ K for all x ∈ B, i ∈ {1, . . . , k}. Let h2(t) = Pk 1 k K i=1 χ(ti,∞)(t) for t ∈ [a, b], and let ω2 be the common modulus of continuity of the mappings I ,...,I on B. Then a simple calculation reveals that

2 2 kFx (x, t) − Fx (x, s)k ≤ |h2(t) − h2(s)|, s, t ∈ [a, b], x ∈ B,

2 2 2 2 kFx (x, t) − Fx (x, s) − Fx (y, t) + Fx (y, s)k ≤ ω2(kx − yk)|h2(t) − h2(s)|, s, t ∈ [a, b], x, y ∈ B. 1 2 Consequently, the function Fx = Fx + Fx satisfies assumptions 3, 4 of Theorem 7.4.1 with h = h1 + h2 and ω = ω1 + ω2. By Remark 7.4.3, assumption 5 is satisfied with k = h. It follows that solutions of the impulsive equation (7.5.3) are differentiable with respect to λ, uniformly on [a, b]. (Actually, the whole procedure still works under weaker hypotheses on f; the crucial thing is to ensure that f and fx satisfy the assumptions of Lemma 7.5.1.) To obtain an equation for the derivative Z(s) = xλ(s, λ0), we make use of the fact that

Z t ˜ Fx(x, t) = fx(x, s) dg(s), t ∈ [a, b], a

Pk where g(s) = s + i=1 χ(ti,∞)(s), and ( ˜ fx(x, t) if t ∈ [a, b]\{t1, . . . , tk}, fx(x, t) = i Ix(x) if t = ti for some i ∈ {1, . . . , k}

114 (see e.g. Remark 4.3.10). By Theorem 7.4.1 and the second part of Lemma 7.5.1, Z s Z s 0 0 ˜ Z(s) = xλ(λ0) + D[Fx(x(τ, λ0), t)Z(τ)] = xλ(λ0) + fx(x(τ, λ0), τ)Z(τ) dg(τ), s ∈ [a, b]. a a This integral equation can be rewritten back (see again Remark 4.3.10) as the impulsive equation

0 Z (t) = fx(x(t, λ0), t)Z(t), t ∈ [a, b]\{t1, . . . , tk}, i Z(ti+) − Z(ti) = Ix(x(ti, λ0))Z(ti), i ∈ {1, . . . , k}, 0 x(a) = xλ(λ0), which agrees with the result from [53]. Next, let us turn our attention to dynamic equations on time scales. Let T be a time scale, a, b ∈ T, a < b. We use the notation [a, b]T = [a, b] ∩ T. For every t ∈ [a, b], let ∗ t = inf{s ∈ [a, b]T; s ≥ t}.

n ∗ n Given an arbitrary function f :[a, b]T → R , we deﬁne a function f :[a, b] → R by f ∗(t) = f(t∗), t ∈ [a, b].

n n ∗ n Similarly, for every function f : C × [a, b]T → R , where C ⊂ R , let f : C × [a, b] → R be deﬁned by f ∗(x, t) = f(x, t∗), t ∈ [a, b], x ∈ C.

n Assume that C is an open neighborhood of B and f : C × [a, b]T → R satisﬁes the following conditions:

• For every t ∈ [a, b]T, the function x 7→ f(x, t) is continuously diﬀerentiable on C.

• For every continuous function x :[a, b]T → B, the functions t 7→ f(x(t), t) and t 7→ fx(x(t), t) are rd-continuous.

• fx is bounded in B × [a, b]T.

A consequence of these conditions is that f is bounded in B×[a, b]T (use the estimate kf(x, t)k ≤ kf(x, t)− f(0, t)k + kf(0, t)k and apply the mean-value theorem in the ﬁrst norm). Under these assumptions, it is known (see Theorem 2.3.7) that the dynamic equation

∆ 0 x (t) = f(x(t), t), t ∈ [a, b]T, x(a) = x (λ) (7.5.4) is equivalent to the generalized ordinary diﬀerential equation dx = DF (x, t), t ∈ [a, b], x(a) = x0(λ), (7.5.5) dτ where Z t F (x, t) = f ∗(x, s) dg(s), a ∗ and g(s) = s for every s ∈ [a, b]. More precisely, if x :[a, b]T → B is a solution of (7.5.4), then the function x∗ :[a, b] → B is a solution of (7.5.5). Conversely, every solution y :[a, b] → B of (7.5.5) has the ∗ form y = x , where x :[a, b]T → B is a solution of (7.5.4). We have Z t ∗ Fx(x, t) = fx (x, s) dg(s). a By the ﬁrst part of Lemma 7.5.1, there exist functions h :[a, b] → R and ω : [0, ∞) → [0, ∞) such that

kFx(x, t) − Fx(x, s)k ≤ |h(t) − h(s)|, s, t ∈ [a, b], x ∈ B,

kFx(x, t) − Fx(x, s) − Fx(y, t) + Fx(y, s)k ≤ ω(kx − yk)|h(t) − h(s)|, s, t ∈ [a, b], x, y ∈ B.

115 Thus, Fx satisfies assumptions 3, 4 of Theorem 7.4.1. By Remark 7.4.3, assumption 5 is satisfied with k = h. This means that solutions of the dynamic equation (7.5.4) are differentiable with respect to λ, uniformly on [a, b]T. ∗ Let Z(s) = xλ(s , λ0) be the corresponding derivative at λ0. By Theorem 7.4.1 and the second part of Lemma 7.5.1, Z s Z s 0 ∗ 0 ∗ ∗ Z(s) = xλ(λ0) + D[Fx(x(τ , λ0), t)Z(τ)] = xλ(λ0) + fx (x(τ , λ0), τ)Z(τ) dg(τ), s ∈ [a, b]. a a Consequently (see Theorem 2.2.5), Z s 0 Z(s) = xλ(λ0) + fx(x(τ, λ0), τ)Z(τ) ∆τ, s ∈ [a, b]T, a and therefore ∆ 0 Z (t) = fx(x(t, λ0), t)Z(t), t ∈ [a, b]T,Z(a) = xλ(λ0), which agrees with the result obtained in [43]. Besides impulsive differential equations and dynamic equations on time scales, our differentiability results are also applicable to the so-called measure differential equations of the form

Z t Z t x(t) = x(a) + f(x(s), s) ds + g(x(s), s) du(s), t ∈ [a, b], a a where u is a left-continuous function with bounded variation. It was shown in [67, Chapter 5] that under certain assumptions, this equation is equivalent to the generalized ordinary differential equation whose right-hand side is Z t Z t F (x, t) = f(x, s) ds + g(x, s) du(s). a a As in the previous section, a simple application of Lemma 7.5.1 shows that under suitable assumptions on f and g, the hypotheses of Theorem 7.4.1 are satisfied, i.e., solutions of measure differential equations are differentiable with respect to initial conditions. An interesting open question is whether the results of this chapter can be extended to generalized equations whose solutions take values in infinite-dimensional Banach spaces. Numerous authors have already investigated equations of this type (see e.g. [52, 59]). For example, it is known that under certain assumptions, measure functional differential equations are equivalent to generalized ordinary differential equations with vector-valued solutions (see Chapters 4, 6 and the references there). Therefore, differentiability results for the latter type of equations would be directly applicable in the study of functional differential equations.

116 Chapter 8

Linear measure functional diﬀerential equations with inﬁnite delay

8.1 Introduction

In this chapter, we deal with linear functional equations of the form

Z t Z t y(t) = y(a) + `(ys, s) dg(s) + p(s) dg(s), t ∈ [a, b], (8.1.1) a a where the functions y, `, and p take values in Rn, ` is linear in the first variable, and both integrals are the Kurzweil-Stieltjes integrals with respect to a nondecreasing function g :[a, b] → R. As is usual in the theory of functional differential equations, the symbol ys stands for the function ys(θ) = y(s + θ), θ ∈ (−∞, 0]. Equation (8.1.1) represents a special case of the measure functional differential equation

Z t y(t) = y(a) + f(ys, s) dg(s), t ∈ [a, b] (8.1.2) a introduced in Chapter 4 for the case of finite delay; equations of this type with infinite delay were studied in Chapter 6. For g(s) = s, equation (8.1.2) reduces to the classical functional differential equation studied by numerous authors (see e.g. [36]). Moreover, it was shown in Chapters 5 and 6 that impulsive functional differential equations as well as functional dynamic equations on time scales are special cases of the measure functional differential equation (8.1.2). Our main tool in the study of equation (8.1.1) is the theory of generalized ordinary differential equations. The relation between functional differential equations and generalized ordinary differential equations in infinite-dimensional Banach spaces was first described by C. Imaz, F. Oliva and Z. Vorel in [45, 61]. Later, a similar correspondence was established for impulsive functional differential equations by M. Federson and S.ˇ Schwabik in [24], and for measure functional differential equations with finite and infinite delay in Chapters 4 and 6, respectively. For equations with infinite delay, an important issue is the choice of the phase space; this topic is discussed in Section 2. In Section 3, we summarize some basic facts of the Kurzweil integration theory needed for our purposes and prove a new convergence theorem for the Kurzweil-Stieltjes integral. Section 4 describes the correspondence between linear measure functional differential equations and generalized linear ordinary differential equations. In Section 5, we prove a global existence-uniqueness theorem for linear measure functional differential equations. Section 6 contains the main results: a new continuous dependence theorem for generalized linear ordinary differential equations (inspired by the work of G. A. Monteiro and M. Tvrdýin [59]), and its counterpart for functional equations. Finally, in Section 7, we present an application of the previous results to impulsive functional differential equations.

117 Our results confirms that the theory of generalized ordinary differential equations plays an important role in the study of functional differential equations. Moreover, by focusing on linear equations, we are able to obtain much stronger results than in the nonlinear case (even for equations with a finite delay).

8.2 Axiomatic description of the phase space

In contrast to classical functional diﬀerential equations, the solutions of measure functional diﬀerential equations are no longer continuous but merely regulated functions. Given an interval [a, b] ⊂ R and a Banach space X, recall that a function f :[a, b] → X is called regulated if the limits

lim f(s) = f(t−) ∈ X, t ∈ (a, b] and lim f(s) = f(t+) ∈ X, t ∈ [a, b) s→t− s→t+

exist. It is well known that every regulated function f :[a, b] → X is bounded; the symbol kfk∞ stands for the supremum norm of f. Regulated functions on open or half-open intervals are defined in a similar way. Given an interval I ⊂ R, we use the symbol G(I,X) to denote the set of all regulated functions f : I → X. For equations with infinite delay, one of the crucial problems is the choice of a suitable phase space. In the axiomatic approach, we do not choose a fixed phase space, but instead deal with all spaces satisfying a given set of axioms. Consequently, there is no need to prove similar results repeatedly for different phase spaces. For classical functional differential equations with infinite delay, the axiomatic approach is well described in the paper [35] of J. K. Hale and J. Kato, as well as in the monograph [39] by S. Hino, S. Murakami, and T. Naito. Our candidate for the phase space of a linear measure functional differential equation is a space H0 ⊂ n G((−∞, 0], R ) equipped with a norm denoted by k · kF. We assume that H0 satisfies the following conditions:

(HL1) H0 is complete.

(HL2) If y ∈ H0 and t < 0, then yt ∈ H0.

+ (HL3) There exists a locally bounded function κ1 :(−∞, 0] → R such that if y ∈ H0 and t ≤ 0, then ky(t)k ≤ κ1(t)kykF. + (HL4) There exist functions κ2 : [0, ∞) → [1, ∞) and λ : [0, ∞) → R such that if u ≤ t ≤ 0 and y ∈ H0, then

kytkF ≤ κ2(t − u) sup ky(s)k + λ(t − u)kyukF. s∈[u,t]

+ (HL5) There exists a locally bounded function κ3 :(−∞, 0] → R such that if y ∈ H0 and t ≤ 0, then

kytkF ≤ κ3(t)kykF.

Our conditions (HL1)–(HL5) are almost identical to conditions (H1)–(H5) in Chapter 6, except that (HL4) is stronger than (H4). Indeed, assume that σ > 0 and y ∈ H0 is a function whose support is contained in [−σ, 0]. Using (HL4) with t = 0 and u = −σ, we obtain

kykF ≤ κ2(σ) sup ky(t)k, t∈[−σ,0]

which is precisely condition (H4). On the other hand, there is an additional condition (H6) in Chapter 6, which is however not strictly necessary (see Remark 6.3.10) and we omit it here.

Remark 8.2.1. One can replace (HL3) by the following condition, which is known from the axiomatic theory of classical functional diﬀerential equations with inﬁnite delay (see [39]): There exists a constant

β > 0 such that if y ∈ H0 and t ≤ 0, then ky(t)k ≤ βkytkF. Indeed, combining this assumption with (HL5), we obtain ky(t)k ≤ βkytkF ≤ βκ3(t)kykF, i.e., (HL3) is satisﬁed with κ1 = βκ3.

118 n The following example of a phase space is a simple modification of the space Cϕ((−∞, 0], R ), which is well known from the classical theory of functional differential equations with infinite delay (see [39]). Example 8.2.2. Consider the space

n n Gϕ((−∞, 0], R ) = {y ∈ G((−∞, 0], R ); y/ϕ is bounded}, n where ϕ :(−∞, 0] → R is a fixed continuous positive function. The norm of a function y ∈ Gϕ((−∞, 0], R ) is defined as ky(t)k kykϕ = sup . t∈(−∞,0] ϕ(t) Assume that ϕ(s + t) γ1(t) = sup < ∞, t ≥ 0, (8.2.1) s∈(−∞,−t] ϕ(s) ϕ(s + t) γ2(t) = sup < ∞, t ≤ 0, (8.2.2) s∈(−∞,0] ϕ(s) and that γ2 is a locally bounded function. (In the typical case when ϕ is nonincreasing, the first condition is satisfied automatically.) The following calculations show that under these hypotheses, the space n Gϕ((−∞, 0], R ) satisfies conditions (HL1)–(HL5). n • The mapping y 7→ y/ϕ is an isometric isomorphism between the space Gϕ((−∞, 0], R ) and the space BG((−∞, 0], Rn) of all bounded regulated functions on (−∞, 0], which is endowed with the n supremum norm. The latter space is complete, and thus Gϕ((−∞, 0], R ) is complete, too. n • For every t < 0 and y ∈ Gϕ((−∞, 0], R ), we have

kyt(s)k ky(t + s)k ϕ(t + s) sup ≤ sup sup ≤ kykϕγ2(t), s∈(−∞,0] ϕ(s) s∈(−∞,0] ϕ(t + s) s∈(−∞,0] ϕ(s)

which shows that (HL2) is true and (HL5) is satisﬁed with κ3(t) = γ2(t). • Since ky(t)k ky(s)k ky(t)k ≤ ϕ(t) ≤ ϕ(t) sup , ϕ(t) s∈(−∞,0] ϕ(s)

we see that (HL3) is satisﬁed with κ1(t) = ϕ(t). • For an arbitrary u ≤ t ≤ 0, we have ky(t + s)k ky(s)k ky(s)k ky(s)k kytkϕ = sup = sup ≤ sup + sup . s∈(−∞,0] ϕ(s) s∈(−∞,t] ϕ(s − t) s∈(−∞,u] ϕ(s − t) s∈[u,t] ϕ(s − t) We estimate the ﬁrst term as follows: ky(s)k ky(s)k ϕ(s − u) sup ≤ sup sup s∈(−∞,u] ϕ(s − t) s∈(−∞,u] ϕ(s − u) s∈(−∞,u] ϕ(s − t) ϕ(s + t − u) = kyukϕ sup ≤ kyukϕγ1(t − u) s∈(−∞,u−t] ϕ(s) For the second term, we have the following estimate: ky(s)k sup ky(s)k sup ky(s)k sup ≤ s∈[u,t] = s∈[u,t] . s∈[u,t] ϕ(s − t) infs∈[u,t] ϕ(s − t) infs∈[u−t,0] ϕ(s) Thus, sups∈[u,t] ky(s)k kytkϕ ≤ kyukϕγ1(t − u) + , infs∈[u−t,0] ϕ(s)

which shows that (HL4) is satisﬁed with κ2(σ) = 1/(infs∈[−σ,0] ϕ(s)) and λ(σ) = γ1(σ), for σ ∈ [0, ∞).

119 n For example, when ϕ(t) = 1 for every t ∈ (−∞, 0], then Gϕ((−∞, 0], R ) coincides with the space BG((−∞, 0], Rn) of all bounded regulated functions on (−∞, 0] and endowed with the supremum norm (see Example 6.2.2). Another important special case, which is a phase space commonly used for dealing with unbounded functions, is obtained by taking an arbitrary γ ≥ 0 and letting ϕ(t) = e−γt (see Example 6.2.5).

Besides the phase space H0, we also need suitable spaces Ha of regulated functions deﬁned on (−∞, a], where a ∈ R. We obtain these spaces by shifting the functions from H0. More precisely, for every a ∈ R, n denote Ha = {y ∈ G((−∞, a], R ); ya ∈ H0}. Finally, deﬁne a norm k · kF on Ha by letting kykF = kyakF for every y ∈ Ha.

Example 8.2.3. Let H0 be one of the phase spaces Gϕ described in Example 8.2.2. Then Ha consists of all regulated functions y :(−∞, a] → Rn such that

ky(t)k sup < ∞. t∈(−∞,a] ϕ(t − a)

In this case, the value of the supremum equals kykF.

The following lemma is a straightforward consequence of (HL1)–(HL5).

n Lemma 8.2.4. If H0 ⊂ G((−∞, 0], R ) is a space satisfying conditions (HL1)–(HL5), then the following statements are true for every a ∈ R:

1. Ha is complete.

2. If y ∈ Ha and t ≤ a, then yt ∈ H0.

3. If t ≤ a and y ∈ Ha, then ky(t)k ≤ κ1(t − a)kykF.

4. If y ∈ Ha and u ≤ t ≤ a, then

kytkF ≤ κ2(t − u) sup ky(s)k + λ(t − u)kyukF. s∈[u,t]

5. If y ∈ Ha and t ≤ a, then kytkF ≤ κ3(t − a)kykF.

The next lemma analyzes the particular case when a function y ∈ Hb is deﬁned as the prolongation of a function in H0.

Lemma 8.2.5. Let φ ∈ H0, a, b ∈ R, with a < b, be given and consider a function x˜ ∈ Hb of the form ( φ(ϑ − a), ϑ ∈ (−∞, a], x˜(ϑ) = φ(0), ϑ ∈ [a, b].

Then, kx˜kF ≤ (κ2(b − a)κ1(0) + λ(b − a))kφkF.

Proof. Using the deﬁnition of the norm in Hb, Lemma 8.2.4 and (HL3), we obtain

kx˜kF = kx˜bkF ≤ κ2(b − a) sup kx˜(s)k + λ(b − a)kx˜akF s∈[a,b]

= κ2(b − a)kφ(0)k + λ(b − a)kφkF ≤ (κ2(b − a)κ1(0) + λ(b − a))kφkF.

120 8.3 Kurzweil integration

Consider a Banach space X and let k · kX denote its norm. As usual, the symbol L(X) denotes the space of all bounded linear operators on X. R b The integrals which occur in this chapter represent special cases of the Kurzweil integral a DU(τ, t), where U :[a, b] × [a, b] → X. R b We are particularly interested in Stieltjes-type integrals. The Kurzweil-Stieltjes integral a f dg of a function f :[a, b] → Rn with respect to a function g :[a, b] → R is obtained by letting U(τ, t) = f(τ)g(t). This is the integral which appears in the definition of a measure functional differential equation. R b Secondly, we need the abstract Kurzweil-Stieltjes integral a d[A] g, where A :[a, b] → L(X) and g :[a, b] → X (see [70]). This integral corresponds to the choice U(τ, t) = A(t)g(τ), and it will appear in the definition of a generalized linear ordinary differential equation. Let us recall that a function f :[a, b] → X has bounded variation on [a, b], if

m X var[a,b] f = sup kf(sj) − f(sj−1)kX < ∞, j=1 where the supremum is taken over all divisions D : a = s0 < s1 < . . . < sm = b of the interval [a, b]. Let BV ([a, b],X) denote the set of all functions f :[a, b] → X with bounded variation. It is worth mentioning that BV ([a, b],X) ⊂ G([a, b],X). The next theorem (see [70, Proposition 15]) provides a simple criterion for the existence of the abstract Kurzweil-Stieltjes integral.

Theorem 8.3.1. If A ∈ BV ([a, b], L(X)) and g :[a, b] → X is a regulated function, then the integral R b a d[A]g exists and we have Z b

d[A]g ≤ (var[a,b] A)kgk∞. a X The following property of the indeﬁnite Kurzweil-Stieltjes integral (see [67, Theorem 1.16]) implies that solutions of measure functional diﬀerential equations are regulated functions.

n R b Theorem 8.3.2. Let f :[a, b] → R and g :[a, b] → R be such that g is regulated and the integral a f dg exists. Then the function Z t u(t) = f dg, t ∈ [a, b], a is regulated. Moreover, if g is left-continuous, then so is the function u.

The next convergence theorem for the Kurzweil-Stieltjes integral is inspired by a similar result from ∞ [59, Theorem 2.2]. Instead of requiring the uniform convergence of the sequence {Ak}k=1 to A0, we show that a weaker assumption is suﬃcient.

Theorem 8.3.3. Let Ak ∈ BV ([a, b], L(X)), gk ∈ G([a, b],X) for k ∈ N0. Assume that the following conditions are satisﬁed:

• limk→∞ kgk − g0k∞ = 0.

• There exists a constant γ > 0 such that var[a,b] Ak ≤ γ for every k ∈ N.

• limk→∞ supt∈[a,b] k[Ak(t) − A0(t)]xkX = 0 for every x ∈ X. Then Z t Z t

lim sup d[Ak]gk − d[A0]g0 = 0. k→∞ t∈[a,b] a a X

121 Proof. Let ε > 0 be given. Since g0 is regulated, there exists a step function g :[a, b] → X such that kg0 − gk∞ < ε (see [41, Theorem I.3.1]). Also, there exist a division a = t0 < t1 < ··· < tm = b of [a, b] and elements c1, . . . , cm ∈ X such that g(t) = cj for every t ∈ (tj−1, tj). Let k0 ∈ N be such that kgk − g0k∞ < ε,

k(Ak − A0)(τ)g(tj)kX < ε/m, j ∈ {0, . . . , m},

k(Ak − A0)(τ)cjkX < ε/m, j ∈ {1, . . . , m} for every k ≥ k0 and τ ∈ [a, b]. For an arbitrary t ∈ [a, b], we have

Z t Z t Z t Z t Z t

d[Ak]gk − d[A0]g0 ≤ d[Ak](gk − g) + d[Ak − A0]g + d[A0](g − g0) . a a X a X a X a X

For k ≥ k0, the ﬁrst and third term on the right-hand side can be estimated using Theorem 8.3.1: Z t

d[Ak](gk − g) ≤ γkgk − gk∞ ≤ γ(kgk − g0k∞ + kg0 − gk∞) < 2γε, a X Z t

d[A0](g − g0) ≤ (var[a,b] A0)kg − g0k∞ < (var[a,b] A0)ε. a X R t Since g is a step function, we can calculate the integral a d[Ak − A0]g in the second term (see [70, Proposition 14]), and obtain the following estimate:

Z t m X d[Ak − A0]g ≤ k[Ak − A0](tj−1+)g(tj−1) − [Ak − A0](tj−1)g(tj−1)kX

a X j=1 m X + k[Ak − A0](tj−)cj − [Ak − A0](tj−1+)cjkX j=1 m X + k[Ak − A0](tj)g(tj) − [Ak − A0](tj−)g(tj)kX ≤ 6ε. j=1

Consequently, Z t Z t

sup d[Ak]gk − d[A0]g0 < ε(2γ + var[a,b] A0 + 6) t∈[a,b] a a X for every k ≥ k0, which completes the proof. Remark 8.3.4. For the so-called interior integral, a result similar to Theorem 8.3.3 was proved by C. H¨onig in [41, Theorem I.5.8].

8.4 Linear measure functional diﬀerential equations and generalized linear ODEs

A generalized linear ordinary diﬀerential equation is an integral equation of the form

Z t x(t) =x ˜ + d[A] x + h(t) − h(a), t ∈ [a, b], (8.4.1) a where A :[a, b] → L(X), h ∈ G([a, b],X), andx ˜ ∈ X. R b We say that a function x :[a, b] → X is a solution of (8.4.1) on the interval [a, b], if the integral a d[A] x exists and equality (8.4.1) holds for all t ∈ [a, b].

122 Equations of the form (8.4.1), which represent a special case of generalized ordinary differential equations introduced by J. Kurzweil in [51], have been studied by numerous authors. The situation when X is a general Banach space was for the first time investigated by S.ˇ Schwabik in [71, 72]. In this section, we clarify the relation between linear measure functional differential equations and generalized linear ordinary differential equations. Consider the functional equation

Z t Z t y(t) = y(a) + `(ys, s) dg(s) + p(s) dg(s), t ∈ [a, b], a a

n n where g :[a, b] → R is nondecreasing, ` : H0 × [a, b] → R is linear in the ﬁrst variable and p :[a, b] → R . We will show that, under certain assumptions, this functional equation is equivalent to the generalized equation (8.4.1), where X = Hb and the functions A, h are deﬁned as follows: n For every t ∈ [a, b], A(t): Hb → G((−∞, b], R ) is the operator given by

0, −∞ < ϑ ≤ a,  R ϑ (A(t)y)(ϑ) = a `(ys, s) dg(s), a ≤ ϑ ≤ t ≤ b, (8.4.2) R t a `(ys, s) dg(s), t ≤ ϑ ≤ b, and h(t) ∈ G((−∞, b], Rn) is the function given by

0, −∞ < ϑ ≤ a,  R ϑ h(t)(ϑ) = a p(s) dg(s), a ≤ ϑ ≤ t ≤ b, (8.4.3) R t a p(s) dg(s), t ≤ ϑ ≤ b. We introduce the following system of conditions, which will be useful later (in particular, conditions (A) and (E) guarantee that the integrals in (8.4.2) and (8.4.3) exist):

R b (A) The integral a `(yt, t) dg(t) exists for every y ∈ Hb. (B) There exists a function M :[a, b] → R+, which is Kurzweil-Stieltjes integrable with respect to g, such that Z v Z v

(`(yt, t) − `(zt, t)) dg(t) ≤ M(t)kyt − ztkF dg(t) u u

whenever y, z ∈ Hb and [u, v] ⊆ [a, b].

(C) For every y ∈ Hb, A(b)y is an element of Hb.

(D) Hb has the prolongation property for t ≥ a, i.e., for every y ∈ Hb and t ∈ [a, b), the function y¯ :(−∞, b] → Rn given by ( y(s), s ∈ (−∞, t], y¯(s) = y(t), s ∈ [t, b]

is an element of Hb. R b (E) The integral a p(t) dg(t) exists. (F) There exists a function N :[a, b] → R+, which is Kurzweil-Stieltjes integrable with respect to g, such that Z v Z v

p(t) dg(t) ≤ N(t) dg(t) u u whenever [u, v] ⊆ [a, b].

(G) h(b) is an element of Hb.

We start with the following auxiliary statements.

123 n Lemma 8.4.1. Assume that ` : H0 × [a, b] → R is linear in the ﬁrst variable and conditions (A), (B) are satisﬁed. Let α˜ = supt∈[a,b] κ3(t − b). Then Z v Z v

`(yt, t) dg(t) ≤ kykF α˜ · M(t) dg(t) u u whenever y ∈ Hb and [u, v] ⊆ [a, b].

Proof. For every y ∈ Hb and t ≤ b, we have

kytkF ≤ κ3(t − b)kykF ≤ α˜kykF (the ﬁrst inequality follows from Lemma 8.2.4). To ﬁnish the proof, it is enough to apply (B) with z ≡ 0.

n Corollary 8.4.2. Assume that ` : H0 ×[a, b] → R is linear in the ﬁrst variable and conditions (A), (B) are + satisﬁed. For every bounded O ⊂ Hb, there exists a function K :[a, b] → R , which is Kurzweil-Stieltjes integrable with respect to g, such that Z v Z v

`(yt, t) dg(t) ≤ K(t) dg(t) u u whenever y ∈ O and [u, v] ⊆ [a, b]. The properties of the function A defined in (8.4.2), such as the fact that it has bounded variation on [a, b], are described in the next lemma. n Lemma 8.4.3. Assume that ` : H0 × [a, b] → R is linear in the first variable and conditions (A)–(D) are n satisfied. For every t ∈ [a, b], let A(t): Hb → G((−∞, b], R ) be given by (8.4.2). Then, the function A takes values in L(Hb) and has bounded variation on [a, b]. Moreover, Z b var[a,b] A ≤ κ2(b − a) α˜ · M(s) dg(s), a where α˜ = supt∈[a,b] κ3(t − b).

Proof. It is clear that for every t ∈ [a, b], A(t) is a linear operator deﬁned on Hb. Using (C), (D), and the deﬁnition of A, we see that A(t)y ∈ Hb for every y ∈ Hb and t ∈ [a, b]. Note that (A(t)y)a ≡ 0 for every y ∈ Hb. Thus, by Lemma 8.2.4, we have

Z ϑ kA(t)yk = k(A(t)y) k ≤ κ (b − a) sup k(A(t)y)(ϑ)k = κ (b − a) sup `(y , s) dg(s) . F b F 2 2 s ϑ∈[a,b] ϑ∈[a,t] a

By Lemma 8.4.1, A(t) is a bounded linear operator on Hb and Z t

kA(t)kL(Hb) ≤ κ2(b − a) α˜ · M(s) dg(s). a

To show that A :[a, b] → L(Hb) has bounded variation, consider a ≤ u < v ≤ b and y ∈ Hb. By Lemmas 8.2.4 and 8.4.1,

k[A(v) − A(u)]ykF ≤ κ2(b − a) sup k([A(v) − A(u)]y)(ϑ)k ϑ∈[a,b]

Z ϑ

= κ2(b − a) sup `(ys, s) dg(s) ϑ∈[u,v] u Z v ≤ κ2(b − a) kykF α˜ · M(s) dg(s). u Hence, Z v

kA(v) − A(u)kL(Hb) ≤ κ2(b − a) α˜ · M(s) dg(s), u which concludes the proof.

124 The following two theorems describe the relation between linear measure functional differential equations and generalized linear ordinary differential equations. Similar results for nonlinear equations were already obtained in Chapter 6; therefore, it is sufficient to verify that the assumptions from Chapter 6 are satisfied.

n Theorem 8.4.4. Assume that g :[a, b] → R is a nondecreasing function, ` : H0 × [a, b] → R is linear in the first variable, φ ∈ H0, and conditions (A)–(G) are satisfied. If y ∈ Hb is a solution of the measure functional differential equation

Z t Z t y(t) = y(a) + `(ys, s) dg(s) + p(s) dg(s), t ∈ [a, b], a a ya = φ, then the function x :[a, b] → Hb given by ( y(ϑ), ϑ ∈ (−∞, t], x(t)(ϑ) = y(t), ϑ ∈ [t, b], is a solution of the generalized ordinary diﬀerential equation

Z t x(t) = x(a) + d[A]x + h(t) − h(a), t ∈ [a, b], (8.4.4) a where A, h are given by (8.4.2), (8.4.3).

Proof. Consider the set O = {x(t); t ∈ [a, b]} ⊂ Hb. Clearly, O has the prolongation property for t ≥ a. Observe that for every t ∈ [a, b], the support of x(t) − x(a) is contained in [a, b]. Thus, by Lemma 8.2.4, we have

kx(t)kF ≤ kx(t) − x(a)kF + kx(a)kF ≤ κ2(b − a) sup kx(t)(τ) − x(a)(τ)k + kx(a)kF τ∈[a,b]

≤ κ2(b − a) sup kx(b)(τ) − x(a)(τ)k + kx(a)kF. τ∈[a,b]

The right-hand side does not depend on t, which means that the set O is bounded. Let f(y, t) = `(y, t) + p(t) for every t ∈ [a, b] and y ∈ H0. For every y ∈ O and [u, v] ⊆ [a, b], Corollary 8.4.2 and condition (F) lead to the estimate

Z v Z v Z v Z v

f(ys, s) dg(s) ≤ `(ys, s) dg(s) + p(s) dg(s) ≤ (K(s) + N(s)) dg(s). u u u u

The function F given by F (y, t) = A(t)y + h(t) for every t ∈ [a, b], y ∈ Hb is well defined thanks to (A) and (E), and has values in Hb by (C), (D), and (G). This shows that all assumptions of Theorem 6.3.6 are satisfied. Consequently, x is a solution of the generalized ordinary differential equation whose right-hand side is F ; however, this equation coincides with (8.4.4).

n Theorem 8.4.5. Assume that g :[a, b] → R is a nondecreasing function, ` : H0 × [a, b] → R is linear in the first variable, φ ∈ H0, and conditions (A)–(G) are satisfied. Let A, h be given by (8.4.2), (8.4.3). If x :[a, b] → Hb is a solution of the generalized ordinary differential equation

Z t x(t) = x(a) + d[A]x + h(t) − h(a), t ∈ [a, b], a with the initial condition ( φ(ϑ − a), ϑ ∈ (−∞, a], x(a)(ϑ) = φ(0), ϑ ∈ [a, b],

125 then the function y ∈ Hb deﬁned by ( x(a)(ϑ), ϑ ∈ (−∞, a], y(ϑ) = x(ϑ)(ϑ), ϑ ∈ [a, b] is a solution of the measure functional diﬀerential equation Z t Z t y(t) = y(a) + `(ys, s) dg(s) + p(s) dg(s), t ∈ [a, b], a a ya = φ.

Proof. From the deﬁnition of A, it follows that the functions x(t), where t ∈ [a, b], coincide on (−∞, a]. As in the proof of the previous theorem, the set O = {x(t), t ∈ [a, b]} is bounded, has the prolongation property for t ≥ a (this follows from Lemma 6.3.5), and the functions f, F given by f(y, t) = `(y, t) + p(t) and F (y, t) = A(t)y + h(t) satisfy all assumptions of Theorem 6.3.7; consequently, y is a solution of the given measure functional diﬀerential equation.

8.5 Existence and uniqueness of solutions

A local existence and uniqueness theorem for nonlinear measure functional differential equations with a finite delay was obtained in Theorem 4.5.3. A generalized version for equations with infinite delay was proved in Theorem 6.3.12. For linear equations, it is possible to prove a much stronger global existence and uniqueness theorem; this is the content of the present section. The following theorem guarantees the existence and uniqueness of a solution of the generalized linear ordinary differential equation, and corresponds to a special case of Proposition 2.8 from [71]. Theorem 8.5.1. Consider a Banach space X and let A ∈ BV ([a, b], L(X)) be a left-continuous function. Then, for every x˜ ∈ X and every h ∈ G([a, b],X), the equation Z t x(t) =x ˜ + d[A] x + h(t) − h(a), t ∈ [a, b] a has a unique solution x on [a, b]. Moreover, x is a regulated function. With this in mind and using the relation established in the previous section, we derive the following result.

Theorem 8.5.2. Assume that g :[a, b] → R is a nondecreasing left-continuous function, ` : H0 × [a, b] → n R is linear in the first variable, φ ∈ H0, conditions (A)–(G) are satisfied, and the function x0 :(−∞, b] → Rn given by ( φ(ϑ − a), ϑ ∈ (−∞, a], x0(ϑ) = φ(0), ϑ ∈ [a, b], is an element of Hb. Then, the measure functional differential equation Z t Z t y(t) = y(a) + `(ys, s) dg(s) + p(s) dg(s), t ∈ [a, b], a a ya = φ, has a unique solution on [a, b]. Proof. Let A, h be given by (8.4.2), (8.4.3). It follows from Theorem 8.3.2 that A, h are regulated left- continuous functions. Moreover, by Lemma 8.4.3, A has bounded variation on [a, b]. Thus, Theorem 8.5.1 ensures the existence of a solution of the generalized linear ordinary differential equation Z t x(t) = x0 + d[A]x + h(t) − h(a), t ∈ [a, b], a

126 where x takes values in the Banach space X = Hb. By Theorem 8.4.5, there exists a corresponding solution of the given measure functional differential equation. If this equation had two different solutions, then, by Theorem 8.4.4, the corresponding generalized ordinary differential equation would have two different solutions, which is a contradiction. Thus, the solution has to be unique.

8.6 Continuous dependence theorems

In this section, we use the theory of generalized linear ordinary differential equations (especially the results from [59]) to prove a new continuous dependence result for linear measure functional differential equations with infinite delay. We remark that a continuous dependence theorem for nonlinear measure functional differential equations with a finite delay was obtained in Theorem 4.6.3. Although this theorem is also applicable to linear equations, our result is much stronger. The following continuous dependence theorem (including its proof) is almost identical to Theorem 3.4 from [59]; however, our assumptions are weaker since we do not require that kAk − A0k∞ → 0. (On the other hand, [59, Theorem 3.4] does not require that Ak, k ∈ N0, are left-continuous functions.) Also, the result from the next theorem is more general than [2, Proposition A.3] when restricted to the linear case.

Theorem 8.6.1. Let X be a Banach space. Consider Ak ∈ BV ([a, b], L(X)), hk ∈ G([a, b],X) and x˜k ∈ X, for k ∈ N0. Assume that the following conditions are satisﬁed:

• For every k ∈ N0, Ak is a left-continuous function.

• limk→∞ kx˜k − x˜0kX = 0.

• limk→∞ supt∈[a,b] k[Ak(t) − A0(t)]xkX = 0 for every x ∈ X.

• limk→∞ khk − h0k∞ = 0.

• There exists a constant γ > 0 such that var[a,b] Ak ≤ γ for every k ∈ N.

Then, for every k ∈ N0, the equation Z t xk(t) = xek + d[Ak] xk + hk(t) − hk(a), t ∈ [a, b] a has a unique solution xk on [a, b], and limk→∞ kxk − x0k∞ = 0. Proof. Existence and uniqueness of solutions follow from Theorem 8.5.1. Next, observe that

lim kxk − x0k∞ ≤ lim khk − h0k∞ + lim kxk − hk − x0 + h0k∞ = lim kwkk∞, k→∞ k→∞ k→∞ k→∞ where wk = xk − hk − x0 + h0, k ∈ N. Note that wk(a) =x ˜k − hk(a) − x˜0 + h0(a), k ∈ N, and therefore limk→∞ kwk(a)kX = 0. A simple calculation reveals that Z t Z t wk(t) =x ˜k + d[Ak]xk − hk(a) − x˜0 − d[A0]x0 + h0(a) a a Z t Z t = wk(a) + d[Ak]xk − d[A0]x0 a a Z t Z t Z t Z t = wk(a) + d[Ak]wk + d[Ak]hk − d[A0]h0 + d[Ak − A0](x0 − h0) a a a a Z t = wk(a) + d[Ak]wk + zk(t) − zk(a), a

127 where Z t Z t Z t zk(t) = d[Ak]hk − d[A0]h0 + d[Ak − A0](x0 − h0). a a a By Theorem 8.3.3, Z t Z t

lim sup d[Ak]hk − d[A0]h0 = 0, k→∞ t∈[a,b] a a X Z t

lim sup d[Ak − A0](x0 − h0) = 0, k→∞ t∈[a,b] a X and consequently limk→∞ kzkk∞ = 0. According to [59, Lemma 3.2], we have

γ kwk(t)kX ≤ (kwk(a)kX + kzkk∞)e , t ∈ [a, b], which implies that limk→∞ kwkk∞ = 0.

For every k ∈ N0, we consider the linear measure functional diﬀerential equation Z t Z t yk(t) = yk(a) + `k((yk)s, s) dgk(s) + pk(s) dgk(s), t ∈ [a, b], a a (yk)a = φk,

n n where gk :[a, b] → R is nondecreasing, `k : H0 × [a, b] → R is linear in the ﬁrst variable, pk :[a, b] → R n and φk ∈ H0. We deﬁne the corresponding operators Ak(t): Hb → G((−∞, b], R ) by

0, −∞ < ϑ ≤ a,  R ϑ (Ak(t)y)(ϑ) = a `k(ys, s) dgk(s), a ≤ ϑ ≤ t ≤ b, (8.6.1) R t a `k(ys, s) dgk(s), t ≤ ϑ ≤ b,

n and functions hk(t) ∈ G((−∞, b], R ) by

0, −∞ < ϑ ≤ a,  R ϑ (hk(t))(ϑ) = a pk(s) dgk(s), a ≤ ϑ ≤ t ≤ b, (8.6.2) R t a pk(s) dgk(s), t ≤ ϑ ≤ b, for every k ∈ N0 and t ∈ [a, b]. To reﬂect the fact that we are dealing with sequences of functions, we modify conditions (A)–(G) from Section 4 as follows:

R b (A) The integral a `k(yt, t) dgk(t) exists for every k ∈ N0 and y ∈ Hb.

+ (B) For every k ∈ N0, there exists a function Mk :[a, b] → R , which is Kurzweil-Stieltjes integrable with respect to gk, such that Z v Z v

(`k(yt, t) − `k(zt, t)) dgk(t) ≤ Mk(t)kyt − ztkF dgk(t) u u

whenever y, z ∈ Hb and [u, v] ⊆ [a, b].

(C) For every k ∈ N0 and y ∈ Hb, the function Ak(b)y is an element of Hb.

(D) Hb has the prolongation property for t ≥ a. R b (E) For every k ∈ N0, the integral a pk(t) dgk(t) exists.

128 + (F) For every k ∈ N0, there exists a function Nk :[a, b] → R , which is Kurzweil-Stieltjes integrable with respect to gk, such that Z v Z v

pk(t) dgk(t) ≤ Nk(t) dgk(t) u u whenever [u, v] ⊆ [a, b].

(G) For every k ∈ N0, the function hk(b) is an element of Hb. Now, using the correspondence established in Theorems 8.4.4 and 8.4.5, we derive the following result.

Theorem 8.6.2. For every k ∈ N0, suppose that gk :[a, b] → R is a nondecreasing left-continuous n function, and `k : H0 × [a, b] → R is a function linear with respect to the ﬁrst variable. Assume that conditions (A)–(G) as well as the following conditions are satisﬁed:

• For every y ∈ H , lim sup R t ` (y , s)dg (s) − R t ` (y , s) dg (s) = 0. b k→∞ t∈[a,b] a k s k a 0 s 0

• lim sup R t p (s)dg (s) − R t p (s) dg (s) = 0. k→∞ t∈[a,b] a k k a 0 0 R b • There exists a constant γ > 0 such that a Mk(s)dgk(s) ≤ γ for all k ∈ N.

Further, consider a sequence of functions φk ∈ H0, k ∈ N0, such that

lim kφk − φ0k = 0, k→∞ F and such that for every k ∈ N0, the function ( φk(ϑ − a), ϑ ∈ (−∞, a], x˜k(ϑ) = φk(0), ϑ ∈ [a, b], is an element of Hb. n Then, for each k ∈ N0, there exists a solution yk :(−∞, b] → R of the measure functional diﬀerential equation R t R t ) yk(t) = yk(a) + `k((yk)s, s) dgk(s) + pk(s) dgk(s), t ∈ [a, b], a a (8.6.3) (yk)a = φk, ∞ and the sequence {yk}k=1 converges uniformly to y0 on [a, b].

Proof. Consider Ak, hk, for k ∈ N0, given by (8.6.1), (8.6.2). By Theorem 8.3.2, hk and Ak are regulated left-continuous functions. In addition, by Lemma 8.4.3, Ak ∈ BV ([a, b], L(Hb)) with

Z b var[a,b] Ak ≤ κ2(b − a) α˜ · Mk(s) dgk(s) ≤ κ2(b − a) · α˜ · γ, k ∈ N0, (8.6.4) a whereα ˜ = supt∈[a,b] κ3(t − a). For every k ∈ N0, Theorem 8.5.1 ensures the existence of a unique solution xk :[a, b] → Hb of the generalized ordinary diﬀerential equation

Z t xk(t) = xek + d[Ak] xk + hk(t) − hk(a), t ∈ [a, b]. a

Given y ∈ Hb, the deﬁnition of Ak, k ∈ N0, together with Lemma 8.2.4, implies

kAk(t)y − A0(t)ykF ≤ κ2(b − a) sup k[Ak(t)y − A0(t)y](ϑ)k ϑ∈[a,b] Z ϑ Z ϑ

= κ2(b − a) sup `k(ys, s)dgk(s) − `0(ys, s) dg0(s) ϑ∈[a,b] a a

129 for every t ∈ [a, b]. Consequently,

Z ϑ Z ϑ sup k[A (t) − A (t)] yk ≤ κ (b − a) sup ` (y , s)dg (s) − ` (y , s) dg (s) , k 0 F 2 k s k 0 s 0 t∈[a,b] ϑ∈[a,b] a a and we conclude that

lim sup k[Ak(t) − A0(t)] ykF = 0, y ∈ Hb. k→∞ t∈[a,b] Analogously, we have

Z ϑ Z ϑ kh (t) − h (t)k ≤ κ (b − a) sup p (s)dg (s) − p (s) dg (s) , t ∈ [a, b], k 0 F 2 k k 0 0 ϑ∈[a,b] a a and thus limk→∞ khk − h0k∞ = 0 holds.

To show that limk→∞ kx˜k − x˜0kF = 0, it is enough to notice that, by Lemma 8.2.5, we have

kx˜k − x˜0kF ≤ (κ2(b − a)κ1(0) + λ(b − a))kφk − φ0kF.

In summary, all hypotheses of Theorem 8.6.1 are satisﬁed, which proves that limk→∞ kxk − x0k∞ = 0. By Theorems 8.4.4 and 8.4.5, for each k ∈ N0, the function ( xk(a)(ϑ), ϑ ∈ (−∞, a], yk(ϑ) = xk(ϑ)(ϑ), ϑ ∈ [a, b] is the unique solution of Eq. (8.6.3). For t ∈ [a, b], we can use Lemma 8.2.4 to see that ˜ kyk(t) − y0(t)k = kxk(t)(t) − x0(t)(t)k ≤ κ1(t − b)kxk(t) − x0(t)kF ≤ β sup kxk(τ) − x0(τ)kF, τ∈[a,b]

˜ ∞ where β = supσ∈[a,b] κ1(σ − a). Thus, the sequence {yk}k=1 is uniformly convergent to y0. Remark 8.6.3. By the fourth part of Lemma 8.2.4,

kyk − y0kF = k(yk − y0)bkF ≤ κ2(b − a) sup kyk(s) − y0(s)k + λ(b − a)kφk − φ0kF, s∈[a,b]

∞ i.e., the sequence of solutions {yk}k=1 from Theorem 8.6.2 converges to y0 also in the k · kF norm.

8.7 Application to functional diﬀerential equations with impulses

As an example, we show how our results apply to functional diﬀerential equations with impulses. For simplicity, we restrict ourselves to the case when the phase space H0 coincides with one of the spaces n n Gϕ from Example 8.2.2. Recall that Gϕ((−∞, 0], R ) = {y ∈ G((−∞, 0], R ); y/ϕ is bounded}, and for H0 = Gϕ, the norm of a function y ∈ Ha is given by

ky(t + a)k ky(t)k kykF = sup = sup . t∈(−∞,0] ϕ(t) t∈(−∞,a] ϕ(t − a)

We consider linear impulsive functional diﬀerential equations of the form

0 y (t) = `(yt, t) + p(t), a.e. in [a, b], + (8.7.1) ∆ y(ti) = Aiy(ti) + bi, i ∈ {1, . . . , k},

n n where ` : H0 × [a, b] → R is linear in the ﬁrst variable, p :[a, b] → R , a ≤ t1 < ··· < tk < b, n×n n + A1,...,Ak ∈ R , b1, . . . , bk ∈ R , and, as usual, ∆ y(s) = y(s+) − y(s), s ∈ [a, b). In addition, assume the following conditions are satisﬁed:

130 R b (1) The Lebesgue integral a `(yt, t) dt exists for every y ∈ Hb. (2) There exists a Lebesgue integrable function M :[a, b] → R+ such that Z v Z v

(`(yt, t) − `(zt, t)) dt ≤ M(t)kyt − ztkF dt u u

whenever y, z ∈ Hb and [u, v] ⊆ [a, b]. R b (3) The Lebesgue integral a p(t) dt exists. (4) There exists a Lebesgue integrable function N :[a, b] → R+ such that Z v Z v

p(t) dt ≤ N(t) dt u u whenever [u, v] ⊆ [a, b].

The solutions of Eq. (8.7.1) are assumed to be left-continuous on [a, b], and absolutely continuous on [a, t1], (t1, t2],..., (tk, b]. The equivalent integral form of Eq. (8.7.1) is

Z t X y(t) = y(a) + (`(ys, s) + p(s)) ds + (Aiy(ti) + bi). (8.7.2) a i; ti

Lemma 8.7.1. Let k ∈ N, a ≤ t1 < t2 < ··· < tk < b, and

k X g(s) = s + χ(ti,∞)(s), s ∈ [a, b] i=1

(the symbol χA denotes the characteristic function of a set A ⊂ R). Consider an arbitrary function ˜ ˜ f :[a, b] → R and let f :[a, b] → R be such that f(s) = f(s) for every s ∈ [a, b]\{t1, . . . , tk}. R b ˜ R b Then the integral a f(s) dg(s) exists if and only if the integral a f(s) ds exists; in that case, we have Z t Z t ˜ X ˜ f(s) dg(s) = f(s) ds + f(ti), t ∈ [a, b]. a a i∈{1,...,k}, ti

Z v Z v Z v Z v X X ˜ p˜(t) dg(t) = p(t) dt + bi ≤ N(t) dt + kbik = N(t) dg(t), u u u u i; ti∈[u,v) i; ti∈[u,v)

131 where ( N(t) if t ∈ [a, b]\{t , . . . , t }, N˜(t) = 1 k kbik if t = ti for some i ∈ {1, . . . , k}.

This veriﬁes assumption (F). Similarly, for every y, z ∈ Hb, we get

Z v Z v ˜ ˜ X (`(yt, t) − `(zt, t)) dg(t) = (`(yt, t) − `(zt, t)) dt + Ai(y(ti) − z(ti)) u u i; ti∈[u,v) Z v Z v X ˜ ≤ M(t)kyt − ztkF dt + kAikκ1(0)kyti − zti kF = M(t)kyt − ztkF dg(t), u u i; ti∈[u,v) where ( M(t) if t ∈ [a, b]\{t , . . . , t }, M˜ (t) = 1 k kAikκ1(0) if t = ti for some i ∈ {1, . . . , k}. Hence, assumption (B) from Section 4 is satisfied. The remaining assumptions (C), (D), and (G) are fulfilled, too: (C) and (G) follow from the fact that A(b)y and h(b) are regulated functions with compact support contained in [a, b], and hence elements of Hb. Also, it is clear that our space Hb has the prolongation property for t ≥ a. n Finally, let φ ∈ H0. We claim that the function y :(−∞, b] → R given by ( φ(ϑ − a), ϑ ∈ (−∞, a], y(ϑ) = φ(0), ϑ ∈ [a, b] is an element of Hb. (Recall that assumptions of this type appear both in the existence-uniqueness theorem and in the continuous dependence theorem.) The claim follows from the estimate ky(t)k ky(t)k ky(t)k sup ≤ sup + sup , t∈(−∞,b] ϕ(t − b) t∈(−∞,a] ϕ(t − b) t∈[a,b] ϕ(t − b) since the second supremum is finite, and ! ! ! ky(t)k ky(t)k ϕ(t − a) ϕ(t − a) sup ≤ sup sup = kφkF sup t∈(−∞,a] ϕ(t − b) t∈(−∞,a] ϕ(t − a) t∈(−∞,a] ϕ(t − b) t∈(−∞,a] ϕ(t − b) is finite, too (we need (8.2.1) here). Our calculations lead to the following existence-uniqueness theorem, which is an immediate consequence of Theorem 8.5.2. (For a local existence-uniqueness theorem for nonlinear impulsive functional differential equations with a finite delay, see [24, Theorem 5.1].)

Theorem 8.7.2. Assume that H0 = Gϕ and conditions (1)–(4) are satisfied. Then for every φ ∈ Gϕ, the impulsive functional differential equation 0 y (t) = `(yt, t) + p(t), a.e. in [a, b], + ∆ y(ti) = Aiy(ti) + bi, i ∈ {1, . . . , k}, ya = φ has a unique solution on [a, b]. We now proceed to continuous dependence and consider a sequence of impulsive functional differential equations of the form 0 yj(t) = `j((yj)t, t) + pj(t), a.e. in [a, b], + i i ∆ yj(ti) = Ajyj(ti) + bj, i ∈ {1, . . . , k}, n n 1 k n×n where `j : H0 × [a, b] → R is linear in the first variable, pj :[a, b] → R , Aj ,...,Aj ∈ R , and 1 k n bj , . . . , bj ∈ R for every j ∈ N0. For linear equations, the following result is much stronger than the continuous dependence result stated in [24, Theorem 4.1].

132 Theorem 8.7.3. Assume that H0 = Gϕ and the following conditions are satisﬁed:

R b • The Lebesgue integral a `j(yt, t) dt exists for every j ∈ N0 and y ∈ Hb.

+ • For every j ∈ N0, there exists a Lebesgue integrable function Mj :[a, b] → R such that Z v Z v

(`j(yt, t) − `j(zt, t)) dt ≤ Mj(t)kyt − ztkF dt u u

whenever y, z ∈ Hb and [u, v] ⊆ [a, b].

R b • For every j ∈ N0, the Lebesgue integral a pj(t) dt exists.

+ • For every j ∈ N0, there exists a Lebesgue integrable function Nj :[a, b] → R such that Z v Z v

pj(t) dt ≤ Nj(t) dt u u

whenever [u, v] ⊆ [a, b].

• For every y ∈ H , lim sup R t ` (y , s) ds − R t ` (y , s) ds = 0. b j→∞ t∈[a,b] a j s a 0 s

• lim sup R t p (s) ds − R t p (s) ds = 0. j→∞ t∈[a,b] a j a 0

R b • There exists a constant γ > 0 such that a Mj(s) ds ≤ γ for all j ∈ N.

i i i i • For every i ∈ {1, . . . , k}, limj→∞ Aj = A0 and limj→∞ bj = b0.

Further, consider a sequence of functions φj ∈ H0, j ∈ N0, such that limj→∞ kφj − φ0kF = 0. n Then, for each j ∈ N0, there exists a solution yj :(−∞, b] → R of the impulsive functional diﬀerential equation 0 yj(t) = `j((yj)t, t) + pj(t), a.e. in [a, b], + i i ∆ yj(ti) = Ajyj(ti) + bj, i ∈ {1, . . . , k}, (yj)a = φj,

∞ and the sequence {yj}j=1 converges uniformly to y0 on [a, b]. Proof. The theorem is a consequence of Theorem 8.6.2. Indeed, our sequence of impulsive equations is equivalent to the sequence of measure functional diﬀerential equations

R t ˜ R t yj(t) = yj(a) + a `j((yj)s, s) dg(s) + a p˜j(s) dg(s), t ∈ [a, b],

(yj)a = φj,

˜ where the functions `j,p ˜j, and g are deﬁned as in the beginning of the present section. Our previous ˜ calculations show that assumptions (A)–(G) from Section 5 (with `j, pj replaced by `j,p ˜j) are satisﬁed. In particular, we have the estimate

Z v Z v ˜ ˜ ˜ (`j(yt, t) − `j(zt, t)) dg(t) ≤ Mj(t)kyt − ztkF dg(t), u u

˜ ˜ i where Mj(t) = Mj(t) if t ∈ [a, b]\{t1, . . . , tk}, and Mj(t) = kAjkκ1(0) if t = ti for some i ∈ {1, . . . , k}. Hence, Z b Z b k k ˜ X i X i Mj(s) dg(s) = Mj(s) ds + kAjkκ1(0) ≤ γ + κ1(0) sup kAjk a a i=1 i=1 j∈N

133 for every j ∈ N. By Lemma 8.7.1, we have

Z t Z t X j 0 (˜pj(s) − p˜0(s)) dg(s) ≤ (pj(s) − p0(s)) ds + (bi − bi ) , a a i; ti

lim sup p˜j(s) dg(s) − p˜0(s) dg(s) = 0. j→∞ t∈[a,b] a a

Similarly, for every y ∈ Hb, we have

Z t Z t ˜ ˜ X j 0 (`j(ys, s) − `0(ys, s)) dg(s) ≤ (`j(ys, s) − `0(ys, s)) ds + (Ai − Ai )y(ti) . a a i; ti

Pk j 0 Since the last term can be majorized by i=1 kAi − Ai k · ky(ti)k, we obtain Z t Z t ˜ ˜ lim sup `j(ys, s) dg(s) − `0(ys, s) dg(s) = 0. j→∞ t∈[a,b] a a Thus, we have verified that all assumptions of Theorem 8.6.2 are satisfied. The results of this chapter are also applicable to linear functional dynamic equations on time scales. As shown in Chapter 4, functional dynamic equations represent another special case of measure functional differential equations.

Let T be a time scale, i.e., a nonempty closed subset of R. Given a, b ∈ T, a < b, let [a, b]T = [a, b] ∩ T. Using the notation from Chapter 2, let t∗ = inf{s ∈ T; s ≥ t} for every t ∈ (−∞, b]. Recall that in the time scale calculus, the usual derivative f 0 is replaced by the ∆-derivative f ∆ (see [8] for the basic definitions). Therefore, we consider the linear functional dynamic equation ∆ ∗ y (t) = `(yt , t) + q(t), t ∈ [a, b]T, (8.7.4) n n ∗ where ` : H0 × [a, b] → R is linear in the first variable and q :[a, b]T → R . Here, the symbol yt should ∗ ∗ be interpreted as (y )t; as explained in Chapter 4, the reason for working with yt instead of just yt is that the first argument of ` has to be a function defined on the whole interval (−∞, 0]. For example, a linear delay dynamic equation of the form

k ∆ X y (t) = pi(t) y(τi(t)) + q(t), i=1 n where τi : T → T, pi :[a, b]T → R and τi(t) ≤ t for every t ∈ [a, b]T, i ∈ {1, . . . , k}, is a special case of Eq. (8.7.4) corresponding to the choice

k X `(y, t) = pi(t) y(τi(t) − t). i=1 Now, under certain assumptions on ` and q, the linear functional dynamic equation (8.7.4) is equivalent to the measure functional diﬀerential equation Z t Z t ∗ ∗ z(t) = z(a) + `(zs, s ) dg(s) + q(s ) dg(s), t ∈ [a, b], a a where g(s) = s∗, s ∈ [a, b]. More precisely, a function z :[a, b] → Rn is a solution of the last equation if ∗ n and only if z(t) = y(t ), t ∈ [a, b], where y :[a, b]T → R is a solution of (8.7.4). Therefore, the existence-uniqueness theorem as well as the continuous dependence theorem for linear functional dynamic equations of the form (8.7.4) are simple consequences of our results for measure functional diﬀerential equations. Moreover, the same approach also works for functional dynamic equations with impulses (see Chapter 5). Since the whole procedure would be very similar to the one described in Section 7, we omit the details.

134 Chapter 9

Well-posedness results for abstract generalized diﬀerential equations

9.1 Introduction

In the previous chapters, we have described the relationship between various types of functional differential equations and generalized differential equations whose solutions take values in infinite-dimensional Banach spaces. Unfortunately, the existing theory for abstract generalized equations is not as powerful as in the finite-dimensional case. The only exception is the class of linear equations, where the results are quite satisfactory (see Chapter 8 and [33, 59]). Our goal is to rectify this situation and obtain new results concerning well-posedness of solutions to abstract nonlinear generalized differential equations under reasonably weak assumptions on the right-hand sides. In the theory of classical ordinary differential equations, it is well known that Picard’s theorem on the local existence and uniqueness of solutions of the problem

0 x (t) = f(x(t), t), t ∈ [a, b], x(a) = x0, (9.1.1) can be improved in the following way: Instead of working with a locally Lipschitz continuous right-hand side, it is enough to assume that f is a continuous function satisyﬁng

kf(x, t) − f(y, t)k ≤ ω(kx − yk), where ω : [0, ∞) → [0, ∞) is a continuous increasing function such that ω(0) = 0 and

Z u dr lim = ∞ (9.1.2) v→0+ v ω(r) for every u > 0. This existence-uniqueness result is known as Osgood’s theorem (in [62], W. F. Osgood identified (9.1.2) to be a sufficient condition for uniqueness). The situation in finite-dimensional spaces is rather simple, since the existence of a local solution follows from Peano’s theorem, and condition (9.1.2) together with Bihari’s inequality guarantee uniqueness. Remarkably, Peano’s theorem is no longer valid in infinite-dimensional Banach spaces, but Osgood’s theorem remains true (see e.g. [15, Theorem 3.2] and [76]). In this case, condition (9.1.2) is necessary to prove both existence and uniqueness. The proof uses the fact that a continuous right-hand side can be uniformly approximated by locally Lipschitz continuous right-hand sides; the corresponding initial-value problems have unique solutions, which are uniformly convergent to a solution of the original equation. Things become more complicated when we switch to abstract generalized differential equations, whose right-hand sides are usually assumed to be continuous in x, but may be discontinuous in t. A local existence-uniqueness theorem, whose proof is based on the contraction mapping theorem, can be found in [24]; this is the analogue of Picard’s theorem for generalized equations. In Section 3, we prove a more

135 general Osgood-type existence and uniqueness theorem; again, the idea is to approximate the right-hand side by functions such that the theorem from [24] is applicable. In Section 4, we discuss continuous dependence of solutions to generalized differential equations with respect to initial values and parameters. We obtain two new theorems, which generalize several results available in the literature. We also provide an example showing why we cannot expect the infinite- dimensional theorems to hold under exactly the same assumptions as in the finite-dimensional case. In Section 5, we use the previous theory to study the well-posedness for nonlinear measure functional differential equations of the form

Z t y(t) = y(t0) + f(ys, s) dg(s). t0

These equations were studied in Chapters 3, 4, 5, 6 and 8, as well as in the papers [4, 23]. Fairly general results concerning the well-posedness for linear equations with finite delay have been obtained in Chapter 8, while the nonlinear case with finite delay was considered in Chapter 4. Here we show that the theory from Sections 3 and 4 leads to new well-posedness theorems for nonlinear equations with infinite delay, which significantly improve the results from Chapter 4. Functional differential equations with impulses, which represent an important special case of measure functional differential equations, are briefly discussed in Section 6.

9.2 Preliminaries

In this section, we collect some basic facts about regulated functions, Kurzweil integration, and generalized ordinary diﬀerential equations. The following proposition, which characterizes relatively compact sets in the space of regulated functions, was proved in [30, Theorem 2.18]. For an arbitrary interval I ⊂ R, we use the symbol G(I,X) to denote the space of all bounded regulated functions G : I → X (if I is a compact interval, then every regulated function G : I → X is necessarily bounded). The space G(I,X) is equipped with the supremum norm

kfk∞ = sup kf(t)k, f ∈ G(I,X). t∈I

Theorem 9.2.1. For every set A ⊂ G([a, b], Rn), the following conditions are equivalent: 1. A is relatively compact.

2. The set {x(a); x ∈ A} is bounded, there is an increasing continuous function η : [0, ∞) → [0, ∞) with η(0) = 0, and an increasing function K :[a, b] → R such that

kx(t2) − x(t1)k ≤ η(K(t2) − K(t1))

whenever x ∈ A and [t1, t2] ⊂ [a, b].

We now present two inequalities for the Kurzweil integral. Both follow easily from the deﬁnition of the integral, and are special cases of [67, Corollary 1.36].

Lemma 9.2.2. Let B ⊂ X, G = B × [a, b]. Assume that F : G → X satisﬁes

kF (x, t2) − F (x, t1)k ≤ h1(t2) − h1(t1), x ∈ B, [t1, t2] ⊂ [a, b],

R b where h1 :[a, b] → R is a nondecreasing function. If x :[a, b] → B and the integral a DF (x(τ), t) exists, then

Z b

DF (x(τ), t) ≤ h1(b) − h1(a). a

136 dx Consequently, if x :[a, b] → X is a solution of a generalized equation dτ = DF (x, t) whose right-hand side is an element of F(G, h1, h2, ω), we have the estimate

kx(t2) − x(t1)k ≤ h1(t2) − h1(t1), [t1, t2] ∈ [a, b].

Lemma 9.2.3. Let B ⊂ X, G = B × [a, b]. Assume that F : G → X belongs to the class F(G, h1, h2, ω). If x, y :[a, b] → B are arbitrary functions, then

Z b Z b

D[F (x(τ), t) − F (y(τ), t)] ≤ ω(kx(t) − y(t)k) dh2(t), a a provided that the integrals on both sides exist. R b The next three lemmas provide suﬃcient conditions for the existence of the integral a DF (x(τ), t). The proof of the ﬁrst statement can be found in the proof of [67, Corollary 3.15]. Lemma 9.2.4. Let B ⊂ X, G = B × [a, b]. Assume that F : G → X belongs to the class F(G, h, ω). If x :[a, b] → B is a step function, i.e., if there exists a partition

a = s0 < s1 < ··· < sm = b and elements c1, . . . , cm ∈ X such that

x(s) = ci, s ∈ (si−1, si), i ∈ {1, . . . , m}, R b then the integral a DF (x(τ), t) exists and equals

m ! X F (cj, sj−) − F (cj, sj−1+) + F (x(sj−1), sj−1+) − F (x(sj−1), sj−1) + F (x(sj), sj) − F (x(sj), sj−) . j=1 The next lemma generalizes [2, Proposition 2.13], which is a special case of our statement corresponding to ω(r) = r. (A ﬁnite-dimensional version can be found in [67, Corollary 3.15]; unfortunately, the proof no longer works in inﬁnite dimension.) Lemma 9.2.5. Let B ⊂ X, G = B × [a, b]. Assume that F : G → X belongs to the class F(G, h, ω). If x :[a, b] → B is the uniform limit of a sequence of step functions xk :[a, b] → B, k ∈ N, then the integral R b R b a DF (x(τ), t) exists and equals limk→∞ a DF (xk(τ), t). R b Proof. By Lemma 9.2.4, the integral a DF (xk(τ), t) exists for every k ∈ N. Let us prove the existence of R b limk→∞ a DF (xk(τ), t). For each pair i, j ∈ N, Lemma 9.2.3 implies

Z b Z b

D[F (xi(τ), t) − F (xj(τ), t)] ≤ ω(kxi(t) − xj(t)k) dh(t) ≤ ω (kxi − xjk∞)(h(b) − h(a)). a a The right-hand side approaches zero as i, j → ∞, and therefore the Cauchy condition for the existence of R b limk→∞ a DF (xk(τ), t) is satisﬁed. Choose an arbitrary ε > 0. There exists a k ∈ N such that

Z b Z b ε

DF (xk(τ), t) − lim DF (xl(τ), t) < , a l→∞ a 3 ε ω (kx − x k )(h(b) − h(a)) < . k ∞ 3 Also, there exists a gauge δ on [a, b] such that

m X Z b ε (F (x (τ ), s ) − F (x (τ ), s )) − DF (x (τ), t) < k i i k i i−1 k 3 i=1 a

137 for every δ-ﬁne tagged partition of [a, b] with division points s0, s1, . . . , sm and tags τ1, . . . , τm. For these partitions, we get m X Z b (F (x(τi), si) − F (x(τi), si−1)) − lim DF (xl(τ), t) l→∞ i=1 a

m X ≤ (F (x(τi), si) − F (x(τi), si−1) − F (xk(τi), si) + F (xk(τi), si−1)) i=1

m X Z b Z b Z b + (F (xk(τi), si) − F (xk(τi), si−1)) − DF (xk(τ), t) + DF (xk(τ), t) − lim DF (xl(τ), t) l→∞ i=1 a a a m X 2ε 2ε ≤ ω(kx(τ ) − x (τ )k)(h(s ) − h(s )) + ≤ ω (kx − x k )(h(b) − h(a)) + < ε, i k i i i−1 3 k ∞ 3 i=1 R b R b which proves that a DF (x(τ), t) exists and equals liml→∞ a DF (xl(τ), t). Since every regulated function is the uniform limit of step functions, we get the following corollary.

Lemma 9.2.6. Let B ⊂ X, G = B × [a, b]. Assume that F : G → X belongs to the class F(G, h1, h2, ω). R b If x :[a, b] → B is a regulated function, then the integral a DF (x(τ), t) exists. The proof of the next lemma is almost identical to the proof of [59, Proposition 3.8] (which is concerned with the special case when Fk(t, x) = Ak(t), k ∈ N0), but we include it here for reader’s convenience.

Lemma 9.2.7. Let B ⊂ X, G = B × [a, b], and consider a sequence of functions Fk : B × [a, b] → X, k ∈ N0, such that lim Fk(x, t) = F0(x, t), (x, t) ∈ G. k→∞

Assume that Fk ∈ F(G, h, ω) for every k ∈ N0. Then

lim Fk(x, t+) = F0(x, t+), (x, t) ∈ B × [a, b), k→∞ lim Fk(x, t−) = F0(x, t−), (x, t) ∈ B × (a, b]. k→∞

Moreover, for every ﬁxed x ∈ B, the sequence of functions t 7→ Fk(x, t), k ∈ N, is uniformly convergent to the function t 7→ F0(x, t) on [a, b].

Proof. For every x ∈ B and t ∈ [a, b), we have Fk(x, t+) → F0(x, t+) for k → ∞, because

kFk(x, t+) − F0(x, t+)k ≤ kFk(x, t+) − Fk(x, t + δ)k + kFk(x, t + δ) − F0(x, t + δ)k

+kF0(x, t + δ) − F0(x, t+)k ≤ 2(h(t + δ) − h(t+)) + kFk(x, t + δ) − F0(x, t + δ)k, and the right-hand side can be made arbitrarily small by choosing δ > 0 suﬃciently small and k ∈ N suﬃciently large. Similarly, one can prove that Fk(x, t−) → F0(x, t−) for k → ∞. Now, assume there is a x ∈ B such that the sequence t 7→ Fk(x, t), k ∈ N, is not uniformly convergent ∞ ∞ to t 7→ F0(x, t). Then, there exists an ε > 0, a subsequence {Fkl }l=1, and a sequence {tl}l=1 such that

kFkl (x, tl) − F0(x, tl)k ≥ ε, l ∈ N. (9.2.1)

Moreover, without loss of generality, we can assume that liml→∞ tl = t0 ∈ [a, b]. Then, at least one of the following statements has to be true:

∞ ∞ a) The sequence {tl}l=1 has a subsequence {tlm }m=1 whose terms are all smaller than t0.

∞ ∞ b) The sequence {tl}l=1 has a subsequence {tlm }m=1 whose terms are all greater than t0.

138 (If neither a) nor b) were true, then tl = t0 for inﬁnitely many values of l; together with (9.2.1), this would contradict the fact that Fk(x, t0) → F0(x, t0).) We show that a) leads to a contradiction, and leave the other case up to the reader. We have

ε ≤ kF (x, t ) − F (x, t )k ≤ kF (x, t ) − F (x, t −)k + kF (x, t −) − F (x, t −)k klm lm 0 lm klm lm klm 0 klm 0 0 0 +kF (x, t −) − F (x, t )k ≤ 2(h(t −) − h(t )) + kF (x, t −) − F (x, t −)k. 0 0 0 lm 0 lm klm 0 0 0 However, the expression on the right-hand side approaches zero for m → ∞, which is a contradiction.

The following lemma represents a stronger version of [67, Lemma 8.1]; instead of pointwise convergence, we prove uniform convergence of the indeﬁnite integrals. Also, the original condition that x has bounded variation is replaced by the weaker assumption of regulatedness.

Lemma 9.2.8. Let B ⊂ X, G = B × [a, b], and consider a sequence of functions Fk : G → X, k ∈ N0, such that lim Fk(x, t) = F0(x, t), (x, t) ∈ G. k→∞

Assume that Fk ∈ F(G, h, ω) for every k ∈ N0. If x :[a, b] → B is regulated, then Z s Z s lim DFk(x(τ), t) = DF0(x(τ), t) k→∞ a a uniformly with respect to s ∈ [a, b].

Proof. First, let us verify the statement in the case when x :[a, b] → B is a step function: There exists a partition a = s0 < s1 < ··· < sm = b and elements c1, . . . , cm ∈ X such that x(t) = cj for every t ∈ (sj−1, sj). Choose an arbitrary ε > 0. According to Lemma 9.2.7, there exists a k0 ∈ N such that

kFk(cj, sj−) − F0(cj, sj−)k < ε/(6m), j ∈ {1, . . . , m},

kFk(cj, sj−1+) − F0(cj, sj−1+)k < ε/(6m), j ∈ {1, . . . , m},

kFk(x(sj−1), sj−1+) − F0(x(sj−1), sj−1+)k < ε/(6m), j ∈ {1, . . . , m},

kFk(x(sj), sj−) − F0(x(sj), sj−)k < ε/(6m), j ∈ {1, . . . , m},

kFk(x(sj), s) − F0(x(sj), s)k < ε/(6m), j ∈ {0, . . . , m}, s ∈ [a, b],

kFk(cj, s) − F0(cj, s)k < ε/(6m), j ∈ {1, . . . , m}, s ∈ [a, b], for all k ≥ k0. By Lemma 9.2.4, we have

Z sj DFk(x(τ), t) = Fk(cj, sj−) − Fk(cj, sj−1+) + Fk(x(sj−1), sj−1+) sj−1

−Fk(x(sj−1), sj−1) + Fk(x(sj), sj) − Fk(x(sj), sj−), for all k ∈ N0, j ∈ {1, . . . , m}, and therefore

Z sj Z sj ε DF (x(τ), t) − DF (x(τ), t) < , k ≥ k . k 0 m 0 sj−1 sj−1

When s ∈ (sj−1, sj) for some j ∈ {1, . . . , m}, we have x(s) = cj, Z s DFk(x(τ), t) = Fk(cj, s) − Fk(cj, sj−1+) + Fk(x(sj−1), sj−1+) − Fk(x(sj−1), sj−1), sj−1 and therefore

Z s Z s 4ε ε DF (x(τ), t) − DF (x(τ), t) < < , k ≥ k . k 0 6m m 0 sj−1 sj−1

139 It follows that for every s ∈ [a, b], we have Z s Z s

DFk(x(τ), t) − DF0(x(τ), t) < ε, k ≥ k0. a a Now, consider the general situation when x :[a, b] → B is regulated. Choose an arbitrary ε > 0. There ε exists a step function ϕ :[a, b] → B such that ω(kx − ϕk∞) < 2(h(b)−h(a)+1) . Also, there exists a k0 ∈ N R s R s such that a DFk(ϕ(τ), t) − a DF0(ϕ(τ), t) < ε/2 for all k ≥ k0 and s ∈ [a, b]. Then Z s Z s Z s Z s

DFk(x(τ), t) − DF0(x(τ), t) ≤ DFk(x(τ), t) − DFk(ϕ(τ), t) a a a a Z s Z s Z s Z s

+ DFk(ϕ(τ), t) − DF0(ϕ(τ), t) + DF0(ϕ(τ), t) − DF0(x(τ), t) a a a a Z s ε ε ≤ 2 ω(kx(τ) − ϕ(τ)k) dh(τ) + < 2ω(kx − ϕk∞)(h(s) − h(a)) + < ε a 2 2 holds for all k ≥ k0 and s ∈ [a, b], and the proof is complete. The following lemma summarizes some properties of solutions to generalized diﬀerential equations; it is a consequence of [67, Corollary 3.11] and [67, Lemma 3.12]. Lemma 9.2.9. Let B ⊂ X, G = B × [a, b]. Consider a function F : G → X satisfying

kF (x, t2) − F (x, t1)k ≤ h1(t2) − h1(t1), x ∈ B, [t1, t2] ⊂ [a, b], (9.2.2) where h1 :[a, b] → R is a nondecreasing function. If x :[a, b] → X is a solution of the generalized dx diﬀerential equation dτ = DF (x, t), then x is a regulated function with bounded variation. Moreover, x(s+) = x(s) + F (x(s), s+) − F (x(s), s), s ∈ [a, b), x(s−) = x(s) + F (x(s), s−) − F (x(s), s), s ∈ (a, b].

If h1 is left-continuous, then x is left-continuous as well. dx Consider an equation dτ = DF (x, t), where F satisfies (9.2.2) with a left-continuous function h1. Assume that we have a solution of this equation on [a, b), and would like to extend it to [a, b]. According to Lemma 9.2.9, the extension obtained by letting x(b) = x(b−) is the only candidate for such a solution. In the next lemma, we verify that the limit always exists, and that the extension indeed provides a solution on [a, b]. Lemma 9.2.10. Assume that B ⊂ X, G = B × [a, b], and F : G → X satisfies (9.2.2) with a left- dx continuous function h1. If x :[a, b) → X is a solution of dτ = DF (x, t), then the limit x(b−) exists. If dx x(b−) ∈ B and we extend x to [a, b] by letting x(b) = x(b−), we obtain a solution of dτ = DF (x, t) on [a, b]. Proof. The Cauchy condition for the existence of the left-sided limit x(b−) is satisfied, because

kx(s1) − x(s2)k ≤ |h1(s1) − h1(s2)|, s1, s2 ∈ [a, a + b), and the left-sided limit of h1 does exist. Assume that x(b−) ∈ B. Since

Z b Z s Z b

DF (x(τ), t) − DF (x(τ), t) = DF (x(τ), t) ≤ h1(b) − h1(s), s ∈ [a, b], a a s R s R b it follows that lims→b− a DF (x(τ), t) = a DF (x(τ), t), and therefore Z s Z b x(b) = x(b−) = x(a) + lim DF (x(τ), t) = DF (x(τ), t). s→b− a a

140 The following theorem, which describes the properties of the indefinite Kurzweil-Stieltjes integral, is a special case of [67, Theorem 1.16]. Theorem 9.2.11. Let f :[a, b] → Rn and g :[a, b] → R be a pair of functions such that g is regulated and R b a f dg exists. Then the function Z t h(t) = f(s) dg(s), t ∈ [a, b], a is regulated and satisfies h(t+) = h(t) + f(t)∆+g(t), t ∈ [a, b), h(t−) = h(t) − f(t)∆−g(t), t ∈ (a, b], where ∆+g(t) = g(t+) − g(t) and ∆−g(t) = g(t) − g(t−). The next result is a Bihari-type inequality (i.e., a nonlinear version of the Gronwall inequality) for the Kurzweil-Stieltjes integral, which can be found in [67, Theorem 1.40]. Theorem 9.2.12. Consider functions ψ :[a, b] → [0, +∞), h :[a, b] → R, ω : [0, +∞) → R such that ψ is bounded, h is nondecreasing and left-continuous, ω is continuous, increasing and ω(0) = 0. Suppose there exists a k > 0 such that Z t ψ(t) ≤ k + ω(ψ(s)) dh(s), t ∈ [a, b]. a For an arbitrary u0 > 0, let Z u 1 Ω(u) = dr, u ∈ (0, ∞), u0 ω(r) −1 α = limu→0+ Ω(u) ≥ −∞, and β = limu→+∞ Ω(u) ≤ ∞. Also, let Ω :(α, β) → R be the inverse function to Ω. If Ω(k) + h(b) − h(a) < β, then ψ(t) ≤ Ω−1(Ω(k) + h(t) − h(a)), t ∈ [a, b]. The next proposition is an Osgood-type uniqueness theorem for generalized differential equations. The finite-dimensional version can be found in [67, Theorem 4.11], and its proof remains valid even in the infinite-dimensional case (in fact, it is a fairly straightforward consequence of Theorem 9.2.12). Theorem 9.2.13. Let B ⊂ X, G = B × [a, b]. Consider a function F : G → X such that F ∈ F(G, h, ω), where h :[a, b] → R is left-continuous, and Z u dr lim = ∞ v→0+ v ω(r) for every u > 0. If x˜ ∈ X, [a, b1], [a, b2] ⊂ [a, b], and x1 :[a, b1] → B, x2 :[a, b2] → B are solutions of the initial-value problem dx = DF (x, t), x(a) =x, ˜ dτ then x1(t) = x2(t) for every t ∈ [a, b1] ∩ [a, b2]. Finally, let us recall the following existence-uniqueness theorem for generalized ordinary differential equations whose right-hand sides are elements of F(G, h, ω1) with ω1(r) = r, r ≥ 0. Its proof, which is based on the contraction mapping theorem, can be found in [24, Theorem 2.15]. Theorem 9.2.14. Assume that B ⊂ X is an open set, G = B × [a, b], F : G → X belongs to the class F(G, h, ω1) with a left-continuous function h and ω1(r) = r, r ≥ 0. If x0 ∈ B is such that x0 + F (x0, a+) − F (x0, a) ∈ B, then the initial-value problem dx = DF (x, t), x(a) = x , (9.2.3) dτ 0 has a unique local solution defined on a right neighborhood of a. If x is a local solution of the initial-value problem (9.2.3), it follows from Lemma 9.2.9 that x(a+) = x0 + F (x0, a+) − F (x0, a); this explains the meaning of the condition x0 + F (x0, a+) − F (x0, a) ∈ B.

141 9.3 An Osgood-type existence theorem

In this section, we prove an Osgood-type existence theorem for abstract generalized diﬀerential equations. Unfortunately, the ﬁnite-dimensional proof presented in [67, Theorem 4.2], which makes use of Helly’s choice theorem, is no longer applicable. Our proof is based on the following lemma, which says that a right- hand side F ∈ F(B ×[a, b], h1, h2, ω) can be approximated by a function Fε which is in a certain sense close to F . Moreover, Fε has the property that for every x ∈ B, there is a neighborhood U(x) and a constant L(x) > 0 such that the restriction of Fε to U(x) × [a, b] is an element of F(U(x) × [a, b], h1,L(x)h1, ω1), where ω1(r) = r, r ≥ 0. Lemma 9.3.1. Let B be an open subset of X and G = B × [a, b]. Consider a function F : G → X such that F ∈ F(G, h1, h2, ω). Then, for every ε > 0, there exists a function Fε : G → X with the following properties:

1. k(F − Fε)(x, t2) − (F − Fε)(x, t1)k ≤ ε(h2(t2) − h2(t1)) for all x ∈ B, [t1, t2] ⊂ [a, b].

2. kFε(x, t2) − Fε(x, t1)k ≤ h1(t2) − h1(t1) for all x ∈ B, [t1, t2] ⊂ [a, b]. 3. For every x ∈ B, there exist a neighborhood U(x) and a constant L(x) > 0 such that

kFε(y, t2) − Fε(y, t1) − Fε(z, t2) + Fε(z, t1)k ≤ ky − zkL(x)(h1(t2) − h1(t1))

for all y, z ∈ U(x), [t1, t2] ⊂ [a, b].

Proof. There exists a δ > 0 such that ω(δ) < ε. For every x ∈ B, let Uδ(x) = {y ∈ B; kx − yk < δ/2}. Clearly, the system {Uδ(x); x ∈ B} is an open cover of B. Using the fact that every metric space is paracompact (see e.g. [64] or [7, Corollary 2.2]), we conclude that {Uδ(x); x ∈ B} has a locally finite open refinement {Wj; j ∈ J}; that is, the system {Wj; j ∈ J} is an open cover of B, every Wj is contained in Uδ(x) for a certain x ∈ B, and every x ∈ B has a neighborhood V (x) that intersects only finitely many sets Wj. Let {ϕj; j ∈ J} be a partition of unity subordinated to {Wj; j ∈ J}, i.e., a collection of functions such P that for every j ∈ J, the support of ϕj is contained in Wj, and j∈J ϕj(x) = 1 for every x ∈ B. The existence of such a partition of unity follows from paracompactness again (see e.g. [7, Proposition 2.3]). Moreover, it is known that the functions ϕj can be chosen to be locally Lipschitz continuous; the standard way of achieving this goal (cf. [54, Lemma 1]) is to let ( d(x, ∂Wj), x ∈ Wj ψj(x) = 0, x∈ / Wj

P ψj (x) (where d denotes the distance between a point and a set), ψ(x) = j∈J ψj(x), and finally ϕj(x) = ψ(x) . It is not difficult to verify that the functions ψj are Lipschitz continuous with the Lipschitz constant equal to 1, ψ is locally Lipschitz continuous (because {Wj; j ∈ J} is locally finite), and ϕj are locally Lipschitz continuous. Now, for every j ∈ J, choose an arbitrary wj ∈ Wj, and let X Fε(x, t) = ϕj(x)F (wj, t), x ∈ B, t ∈ [a, b]. j∈J

We claim that Fε possesses the three properties listed in the statement of the theorem. Indeed, for all x ∈ B and [t1, t2] ⊂ [a, b], we have

X k(F − Fε)(x, t2) − (F − Fε)(x, t1)k = ϕj(x)(F (x, t2) − F (wj, t2) − F (x, t1) + F (wj, t1))

j∈J X X ≤ ϕj(x)kF (x, t2) − F (wj, t2) − F (x, t1) + F (wj, t1)k ≤ ϕj(x)ω(kx − wjk)(h2(t2) − h2(t1)). j∈J j∈J

142 Note that ϕj(x) is nonzero only for those j ∈ J such that x ∈ Wj; in this case, both x and wj are elements of Uδ(y). Thus kx − wjk < δ, ω(kx − wjk) < ε, and we obtain the estimate X X ϕj(x)ω(kx − wjk)(h2(t2) − h2(t1)) ≤ ε(h2(t2) − h2(t1)) ϕj(x) = ε(h2(t2) − h2(t1)). j∈J j∈J

Next, note that

X X kFε(x, t2) − Fε(x, t1)k = ϕj(x)(F (wj, t2) − F (wj, t1)) ≤ ϕj(x)(h1(t2) − h1(t1)) = h1(t2) − h1(t1).

j∈J j∈J

Finally, for every x ∈ B, there exists a neighborhood U(x) that intersects only a ﬁnite number of sets from the open cover {Wj; j ∈ J}, say k(x) of them. Without loss of generality, we can assume that U(x) is so small that all functions ϕj whose support intersects U(x) are Lipschitz-continuous in U(x), i.e., satisfy

|ϕj(y) − ϕj(z)| ≤ C(x)ky − zk, y, z ∈ U(x) for a certain constant C(x) ≥ 0. Then

X kFε(y, t2) − Fε(y, t1) − Fε(z, t2) + Fε(z, t1)k = (ϕj(y) − ϕj(z))(F (wj, t2) − F (wj, t1))

j∈J

≤ k(x)C(x)ky − zk(h1(t2) − h1(t1)), y, z ∈ U(x), [t1, t2] ⊂ [a, b]. We are now ready to prove the promised existence-uniqueness theorem for abstract generalized differential equations. It generalizes Theorem 9.2.14, which corresponds to the special case when ω(r) = r. Even in that case, our theorem provides more information since it speciﬁes a lower bound for the length of the interval where the solution is guaranteed to exist. Theorem 9.3.2. Assume that B ⊂ X is an open set, G = B × [a, b], F : G → X belongs to the class F(G, h1, h2, ω), where h1, h2 are left-continuous, and Z u dr lim = ∞ (9.3.1) v→0+ v ω(r) for every u > 0. If x0 ∈ B is such that x0 + F (x0, a+) − F (x0, a) ∈ B, then the initial-value problem dx = DF (x, t), x(a) = x , (9.3.2) dτ 0 has a unique local solution deﬁned on a right neighborhood of a. Moreover, if ∆ > 0 is such that the closed ball

{x ∈ X; kx − (x0 + F (x0, a+) − F (x0, a))k ≤ h1(a + ∆) − h1(a+)} (9.3.3) is contained in B, the solution is guaranteed to exist on [a, a + ∆]. Proof. Let ∆ > 0 be an arbitrary number with the property described in the statement of the theorem (since B is open, such a ∆ always exists). Clearly, it is enough to ﬁnd a unique solution of Eq. (9.3.2) on the interval [a, a + ∆0], where

∆0 = inf{t ∈ [a, a + ∆]; h1(t) = h1(a + ∆)}.

Indeed, if ∆0 < ∆ and we have a solution x deﬁned on [a, a + ∆0], we can extend it to [a, a + ∆] by letting

x(s) = x(a + ∆0) + F (x(a + ∆0), (a + ∆0)+) − F (x(a + ∆0), a + ∆0), s ∈ (a + ∆0, a + ∆].

143 One can easily check that kx(s)−x(a+)k ≤ h1((a+∆0)+)−h1(a+), i.e, x(s) lies in the closed ball (9.3.3). Also, for every x ∈ B, the function t 7→ F (x, t) is constant on (a+∆0, a+∆] (because h1 is constant there). R s According to Lemma 9.2.4, we have DF (x(τ), t) = F (x(a + ∆0), (a + ∆0)+) − F (x(a + ∆0), a + ∆0) a+∆0 for all s ∈ (a + ∆0, a + ∆]. It follows that Z s Z s x(s) = x(a + ∆0) + DF (x(τ), t) = x0 + DF (x(τ), t), s ∈ (a + ∆0, a + ∆], a+∆0 a i.e., x is a solution of Eq. (9.3.2) on [a, a + ∆]; uniqueness follows from Theorem 9.2.13. To show the existence of a unique solution on [a, a + ∆0], it is suﬃcient to prove the existence of a unique solution on [a, a + ∆1] for every ∆1 ∈ (0, ∆0). Then we have a unique solution on [a, a + ∆0), which can be extended to [a, a + ∆0] using Lemma 9.2.10. The assumption x((a + ∆0)−) ∈ B from the lemma will be satisﬁed, because kx(t) − x(a+)k ≤ h1(t) − h1(a+) for all t ∈ [a, a + ∆0), and thus the values x(t), t ∈ [a, a + ∆0) lie in the closed ball (9.3.3). Let ∆1 ∈ (0, ∆0) and note that h1(a + ∆) − h1(a + ∆1) > 0. We claim that for every ε > 0 satisfying

ε(h2(a+) − h2(a)) < h1(a + ∆) − h1(a + ∆1), (9.3.4) there exists a unique solution x :[a, a + ∆1] → X of the initial-value problem dx = DF (x, t), x(a) = x , (9.3.5) dτ ε 0 where Fε is the function obtained from Lemma 9.3.1. To prove this, let C be the set of all c ∈ [a, a + ∆1] such that (9.3.5) has a unique solution x on [a, c]. Clearly, C is nonempty (because a ∈ C) and closed (this is a consequence of Lemma 9.2.10). Our goal is to show that C = [a, a+∆1], which can be accomplished by proving that C is open in [a, a + ∆1]. To this end, it is necessary to show that for every c ∈ [a, a + ∆1) ∩ C, the unique solution x of (9.3.5) deﬁned on [a, c] can be always extended to a larger interval. Let

x˜c = x(c) + Fε(x(c), c+) − Fε(x(c), c).

Using the properties of Fε listed in Lemma 9.3.1 and (9.3.4), we get

kx˜c − (x0 + F (x0, a+) − F (x0, a))k

≤ kx˜c − (x0 + Fε(x0, a+) − Fε(x0, a))k + kFε(x0, a+) − Fε(x0, a) − F (x0, a+) + F (x0, a)k

≤ kx(c) − x(a+)k + kFε(x(c), c+) − Fε(x(c), c)k + ε(h2(a+) − h2(a))

≤ h1(c) − h1(a+) + h1(c+) − h1(c) + h1(a + ∆) − h1(a + ∆1) ≤ h1(a + ∆) − h1(a+), i.e.,x ˜c ∈ B. By Lemma 9.3.1, there is a neighborhood U(˜xc) of the pointx ˜c and a number L(˜xc) > 0 such that Fε ∈ F(U(˜xc) × [a, b], h1,L(x)h1, ω1) ⊂ F(U(˜xc) × [a, b], (1 + L(x))h1, ω1). We would like to dx use Theorem 9.2.14 find a unique local solution of dτ = DFε(x, t) defined on a right neighborhood of c, whose value at c is x(c). If x(c) ∈ U(˜xc), all assumptions of the theorem are satisfied. If x(c) ∈/ U(˜xc), it is enough to find a local solution of the equation dy = DF˜ (y, t), y(c) =x ˜ , dτ ε c where F˜ε(y, c) = Fε(y, c+) and F˜ε(y, t) = Fε(y, t) for all y ∈ U(˜xc), t ∈ [c, b]; note that Fε ∈ F(U(˜xc) × [c, b], (1 + L(x))h1, ω1). Then, extend the solution x of (9.3.5) from [a, c] to a larger interval by letting x(t) = y(t) for t > c. This extended function x has the correct jump at c, and is a solution of (9.3.5) on a right neighborhood of c. 1 Let n0 ∈ be such that (h2(a+) − h2(a)) < h1(a + ∆) − h1(a + ∆1). We have proved that for every N n0 integer n ≥ n0, there exists a function xn :[a, a + ∆1] → X satisfying Z s xn(s) = x0 + DF1/n(xn(τ), t), s ∈ [a, a + ∆1]. (9.3.6) a

144 For each pair of integers m, n ≥ n0, we obtain Z s Z s Z s xn(s)−xm(s) = D[(F1/n −F )(xn(τ), t)]+ D[F (xn(τ), t)−F (xm(τ), t)]+ D[(F −F1/m)(xm(τ), t)]. a a a Lemma 9.2.3 implies Z s Z s

D[F (xn(τ), t) − F (xm(τ), t)] ≤ ω(kxn(t) − xm(t)k) dh2(t), s ∈ [a, a + ∆1], a a and Lemma 9.2.2 gives the estimate Z s h2(s) − h2(a) D[(F1/k − F )(xk(τ), t)] ≤ , s ∈ [a, a + ∆1], k ≥ n0. a k Consequently, Z s 1 1 kxn(s)−xm(s)k ≤ ω(kxn(t)−xm(t)k) dh2(t)+(h2(b)−h2(a)) + , s ∈ [a, a+∆1], m, n ≥ n0. a n m

For an arbitrary u0 > 0, the function Z u 1 Ω(u) = dr, u ∈ (0, ∞), u0 ω(r) is continuous, increasing, α = limu→0+ Ω(u) = −∞, and β = limu→+∞ Ω(u) ≤ ∞. Hence, the inverse function Ω−1 is increasing on its domain (−∞, β). For m, n suﬃciently large, we have 1 1 Ω (h (b) − h (a)) + + h (b) − h (a) < β, 2 2 n m 2 2 and it follows from Theorem 9.2.12 that 1 1 kx (s) − x (s)k ≤ Ω−1 Ω (h (b) − h (a)) + + h (b) − h (a) , s ∈ [a, a + ∆ ]. n m 2 2 n m 2 2 1 As m, n increase, the argument of Ω−1 tends to −∞, and therefore the whole right-hand side approaches zero. Thus {xn}n≥n is a Cauchy sequence in G([a, a + ∆1],X), and has a limit x :[a, a + ∆1] → X. 0 R s R s Observe that limn→∞ a DF1/n(xn(τ), t) = a DF (x(τ), t) for every s ∈ [a, a + ∆1], because Z s Z s Z s

D[F1/n(xn(τ), t) − F (x(τ), t)] ≤ D[(F1/n − F )(xn(τ), t)] + D[F (xn(τ), t) − F (x(τ), t)] a a a Z s h2(s) − h2(a) h2(s) − h2(a) ≤ + ω(kxn(t) − x(t)k) dh2(t) ≤ + ω(kxn − xk∞)(h2(s) − h2(a)), n a n and the last expression approaches zero as n → ∞. dx By letting n → ∞ in Eq. (9.3.6), we see that x is a solution of dτ = DF (x, t) on [a, a + ∆1]; uniqueness follows from Theorem 9.2.13. Remark 9.3.3. We have just proved that the condition

{x ∈ X; kx − (x0 + F (x0, a+) − F (x0, a))k ≤ h1(a + ∆) − h1(a+)} ⊂ B guarantees that the local solution exists on [a, a + ∆]. However, assume that we a priori know that the dx solution of dτ = DF (x, t), x(a) = x0, never leaves a certain set Y ⊂ X. Then, an inspection of the previous proof shows that it is enough to verify the following two conditions:

• If x ∈ Y and kx − (x0 + F (x0, a+) − F (x0, a))k ≤ h1(a + ∆) − h1(a+), then x ∈ B. • If x ∈ Y and t ∈ [a, a + ∆), then x + F (x, t+) − F (x, t) ∈ Y . We will use this observation in the proof of Osgood theorem for measure functional diﬀerential equations.

145 9.4 Continuous dependence

Following the usual convention, we state our continuous dependence theorems for sequences of initial-value problems of the following form: dx k = DF (x , t), t ∈ [a, b], x (a) =x ˜ , k ∈ , (9.4.1) dτ k k k k N0 where Fk → F0 andx ˜k → x˜0 for k → ∞. However, the results may be easily adapted to initial-value problems where the right-hand side as well as the initial condition depend on a parameter λ ∈ Λ (for example, Λ can be a metric space): dx λ = DF (x , t, λ), t ∈ [a, b], x (a) =x ˜(λ), λ ∈ Λ, dτ λ λ where F (x, t, λ) → F (x, t, λ0) andx ˜(λ) → x˜(λ0) for λ → λ0. In this chapter, we are interested in sequences of problems of the form (9.4.1), where all the right- hand sides Fk are elements of F(G, h, ω) for a fixed pair of functions h, ω. In particular, our goal is to investigate infinite-dimensional counterparts of the following two finite-dimensional theorems, which can be found in [67] (see Theorems 8.2 and 8.6 there).

n n Theorem 9.4.1. Let B ⊂ R , G = B × [a, b], and consider a sequence of functions Fk : G → R , k ∈ N0, such that lim Fk(x, t) = F0(x, t), (x, t) ∈ G. k→∞

Assume that Fk ∈ F(G, h, ω) for every k ∈ N0. Finally, suppose there exists a sequence of functions xk :[a, b] → B, k ∈ N0, such that dx k = DF (x , t), t ∈ [a, b], k ∈ , dτ k k N and limk→∞ xk(s) = x0(s) for every s ∈ [a, b]. Then dx 0 = DF (x , t), t ∈ [a, b]. dτ 0 0 n n Theorem 9.4.2. Let B ⊂ R , G = B × [a, b], and consider a sequence of functions Fk : G → R , k ∈ N0, such that lim Fk(x, t) = F0(x, t), (x, t) ∈ G. k→∞

Assume that Fk ∈ F(G, h, ω) for every k ∈ N0, where h is left-continuous. Suppose that x˜0 ∈ B and x0 :[a, b] → B is a unique solution of dx = DF (x, t), x(a) =x ˜ . dτ 0 0

Finally, assume there exists a ρ > 0 such that ky − x0(s)k < ρ implies y ∈ B whenever s ∈ [a, b] (i.e., n the ρ-neighborhood of x0 is contained in B). Then, given an arbitrary sequence x˜k ∈ R , k ∈ N, such that limk→∞ x˜k =x ˜0, there is a k0 ∈ N and a sequence of functions xk :[a, b] → B, k ≥ k0, which satisfy dx k = DF (x , t), t ∈ [a, b], x (a) =x ˜ . (9.4.2) dτ k k k k

Moreover, limk→∞ xk(s) = x0(s) for every s ∈ [a, b]. We start by deriving an infinite-dimensional version of Theorem 9.4.1. The finite-dimensional proof given in [67] makes use of Helly’s selection theorem (which is no longer valid in infinite dimension), but only to conclude that the limit function has bounded variation. Fortunately, this is easy to prove without Helly’s theorem. Moreover, our proof does not depend on this fact because we have generalized Lemma 9.2.8 to regulated functions. Otherwise, the main idea of the proof is similar to the proof from [67].

146 Note the remarkable fact that (thanks to Lemma 9.2.8 and Theorem 9.2.1) the conclusion of our theorem is stronger than in the original ﬁnite-dimensional version: we prove the uniform convergence of ∞ the sequence {xk}k=1. The theorem also generalizes Theorem A.3 from [2], since we do not assume that h is left-continuous and has only ﬁnitely many discontinuities (although these assumptions are not mentioned explicitly in the statement of Theorem A.3, they can be found at the beginning of Appendix A in [2], and are needed in the proof of Theorem A.3).

Theorem 9.4.3. Let B ⊂ X, G = B × [a, b], and consider a sequence of functions Fk : G → X, k ∈ N0, such that lim Fk(x, t) = F0(x, t), (x, t) ∈ G. k→∞

Assume that Fk ∈ F(G, h, ω) for every k ∈ N0. Finally, suppose there exists a sequence of functions xk :[a, b] → B, k ∈ N0, such that dx k = DF (x , t), t ∈ [a, b], k ∈ , dτ k k N and limk→∞ xk(s) = x0(s) for every s ∈ [a, b]. Then dx 0 = DF (x , t), t ∈ [a, b]. dτ 0 0 ∞ Moreover, the sequence {xk}k=1 is uniformly convergent to x0. Proof. We know that Z s xk(s) = xk(a) + DFk(xk(τ), t), s ∈ [a, b], k ∈ N, a and our goal is to prove that Z s x0(s) = x0(a) + DF0(x0(τ), t), s ∈ [a, b]. (9.4.3) a R s R s Clearly, it is enough to show that limk→∞ a DFk(xk(τ), t) = a DF0(x0(τ), t) uniformly with respect to s ∈ [a, b]. The assumption Fk ∈ F(G, h, ω) together with Lemma 9.2.2 imply

kxk(β) − xk(α)k ≤ h(β) − h(α), k ∈ N, [α, β] ⊂ [a, b]. Passing to the limit k → ∞, we obtain

kx0(β) − x0(α)k ≤ h(β) − h(α), [α, β] ⊂ [a, b].

Since h is regulated, it follows that x0 is regulated as well. (In fact, it is clear that x0 has bounded R b R b variation.) By Lemma 9.2.6, the integrals a DF0(x0(τ), t) and a DFk(x0(τ), t) exist. We have Z s Z s Z s Z s DFk(xk(τ), t)− DF0(x0(τ), t) = D[Fk(xk(τ), t)−Fk(x0(τ), t)]+ D[Fk(x0(τ), t)−F0(x0(τ), t)]. a a a a We need to show that both integrals on the right-hand side are convergent to zero for k → ∞, and that the convergence is uniform with respect to s ∈ [a, b]. For the second integral, this is a consequence of Lemma 9.2.8. The ﬁrst integral can be estimated using Lemma 9.2.3: Z s Z s

D[Fk(xk(τ), t) − Fk(x0(τ), t)] ≤ ω(kxk(τ) − x0(τ)k) dh(τ) a a

∞ Observe that the sequence of functions {xk − x0}k=1 is uniformly bounded, because

kxk(s) − x0(s)k ≤ kxk(s) − xk(a)k + kxk(a) − x0(a)k + kx0(a) − x0(s)k ≤ M, k ∈ N, s ∈ [a, b],

147 where M = 2(h(b) − h(a)) + sup kx (a) − x (a)k. Since ω is continuous, the assumptions of the k∈N k 0 dominated convergence theorem for the Kurzweil-Stieltjes integral (see [67, Corollary 1.32]) are satisified, and we get Z s Z s lim ω(kxk(τ) − x0(τ)k) dh(τ) = lim ω(kxk(τ) − x0(τ)k) dh(τ) = 0, s ∈ [a, b]. k→∞ a a k→∞ R s Let ϕk(s) = a ω(kxk(τ) − x0(τ)k) dh(τ), s ∈ [a, b], k ∈ N. We know that ϕk → 0 for k → ∞, and it remains to check that the convergence is uniform on [a, b]. To this end, it is enough to verify that every ∞ subsequence of {ϕk}k=1 has a subsequence which is uniformly convergent to zero. (Then it follows easily ∞ ∞ that {ϕk}k=1 itself is convergent to zero; otherwise, we could find an ε > 0 and a subsequence {ϕkl }l=1 such that kϕkl k∞ ≥ ε, which is a contradiction.) For every k ∈ N, we have ϕk(a) = 0, and Z β |ϕk(β) − ϕk(α)| = ω(kxk(τ) − x0(τ)k) dh(τ) ≤ ω(M)(h(β) − h(α)) α whenever [α, β] ⊂ [a, b]. Therefore, the second condition of Theorem 9.2.1 is satisfied with η(r) = ω(M)r ∞ and K(t) = h(t) + t. By this theorem, every subsequence of {ϕk}k=1 has a subsequence which is uniformly convergent to zero, and the proof is complete.

Our next goal is to obtain an infinite-dimensional version of Theorem 9.4.2. The proof given in [67] is again based on Helly’s selection theorem. This time, there seems to be no simple way to avoid it, and we have to follow a different approach. Moreover, one cannot expect to prove an infinite-dimensional version of Theorem 9.4.2 under the same assumptions as in the finite-dimensional case. The reason is that in Theorem 9.4.2, the assumption Fk ∈ F(G, h, ω) guarantees the local existence of solutions to Eq. (9.4.2) for all sufficiently large k ∈ N (see [67, Chapter 4]). Unfortunately, this is no longer true in a general Banach space. Example 9.4.5 shows that Theorem 9.4.2 fails in infinite dimension; the construction is based on an example from J. Dieudonné’spaper [18], where he demonstrated that Peano’s existence theorem need not hold in infinite-dimensional spaces. First, we need the following inequality. p p p Lemma 9.4.4. If x, y ∈ R, then | |x| − |y|| ≤ |x − y|. √ Proof. The function f(x) = x is concave and f(0) = 0; hence, f is subadditive on [0, ∞). Given an arbitrary pair x, y ∈ R, we have p|x| = p|x − y + y| ≤ p|x − y| + |y| ≤ p|x − y| + p|y|, and therefore p|x|−p|y| ≤ p|x − y|. A similar reasoning leads to the inequality p|y|−p|x| ≤ p|y − x|, which completes the proof.

∞ Example 9.4.5. Let c0 be the space of all real sequences {xn}n=0 such that limn→∞ xn = 0; this space is equipped with the supremum norm. For an arbitrary λ ∈ R, deﬁne the mapping fλ : c0 → c0 by ∞ ∞ p 1 fλ({xn}n=0) = λ |xn| + . n + 1 n=0 Consider the abstract diﬀerential equation

0 x (t) = fλ(x(t)), t ≥ 0, x(0) = 0. (9.4.4)

For λ = 0, the equation has the unique solution x(t) = 0 for t ≥ 0. On the other hand, for λ > 0, the ∞ equation is not even locally solvable. For contradiction, assume that x = {xn}n=0 is a solution deﬁned on [0, b] for some b > 0. Then

 1 x0 (t) = λ p|x (t)| + , t ∈ [0, b], x (0) = 0, n ∈ . n n n + 1 n N0

148 By the comparison theorem for initial-value problems, we get xn(t) ≥ yn(t), where yn is any solution of

0 p yn(t) = λ |yn(t)|, t ∈ [0, b], yn(0) = 0.

2 2 One solution of the last equation is yn(t) = (λt) /4, and therefore xn(t) ≥ (λt) /4. For t > 0, this is in contradiction with the fact that limn→∞ xn(t) = 0. For any λ ≥ 0 and x, y ∈ c0, we have

p 1 p kfλ(x)k = sup λ |xn| + ≤ λ( kxk + 1), (9.4.5) n∈N0 n + 1 p p p p kfλ(x) − fλ(y)k = sup λ| |xn| − |yn|| ≤ sup λ |xn − yn| = λ kx − yk. (9.4.6) n∈N0 n∈N0 These inequalities guarantee that the initial-value problem (9.4.4) is equivalent to

dx = DF (x, t), t ≥ 0, x(0) = 0, (9.4.7) dτ λ R t where Fλ(x, t) = 0 fλ(x) ds = tfλ(x); see [67, Theorem 5.14] (the proof in infinite dimension is the same as the finite-dimensional one, the only exception is that one has to rely on our Lemma 9.2.5 instead of its finite-dimensional counterpart). Hence, Eq. (9.4.7) has a unique global solution for λ = 0, and no local solutions for λ > 0. ∞ Now, let λ0 = 0 and let {λk}k=1 be any sequence of positive real numbers such that limk→∞ λk = 0. Also, suppose that b > 0 and B ⊂ c0 is an arbitrary bounded set containing the zero element. Clearly,

lim Fλ (x, t) = Fλ (x, t), (x, t) ∈ B × [0, b]. k→∞ k 0

Our ﬁnal goal is to verify that Fλk ∈ F(B × [0, b], h, ω) for all k ∈ N0. It follows from (9.4.5) and (9.4.6) that p kFλk (x, t2) − Fλk (x, t1)k = (t2 − t1)kfλk (x)k ≤ (t2 − t1) sup |λk| sup kxk + 1 , k∈N x∈B

kFλk (x, t2) − Fλk (x, t1) − Fλk (y, t2) + Fλk (y, t1)k p = (t2 − t1)kfλk (x) − fλk (y)k ≤ (t2 − t1) sup |λk| kx − yk k∈N whenever x, y ∈ B and [t1, t2] ⊂ [0, b]. This proves that Fλk ∈ F(B × [0, b], h, ω) with p h(t) = t sup |λk| sup kxk + 1 k∈N x∈B √ and ω(r) = r, and shows that Theorem 9.4.2 is no longer true in inﬁnite dimension.

The following lemma will be needed in the proof of our continuous dependence theorem.

Lemma 9.4.6. Let B ⊂ X, G = B × [a, b], and consider a sequence of functions Fk : G → X, k ∈ N0, such that lim Fk(x, t) = F0(x, t), (x, t) ∈ G. k→∞ R u dr Assume that Fk ∈ F(G, h, ω) for every k ∈ N0, where h is left-continuous and limv→0+ v ω(r) = ∞ for every u > 0. Suppose that for every k ∈ N0, xk :[a, b] → B satisﬁes dx k = DF (x , t), t ∈ [a, b], x (a) =x ˜ , dτ k k k k

∞ where limk→∞ x˜k =x ˜0. Then {xk}k=1 is uniformly convergent to x0 on [a, b].

149 R b Proof. For every s ∈ [a, b] and k ∈ N, the integral a DFk(x0(τ), t) exists by Lemma 9.2.6, and we obtain Z s Z s

kx0(s) − xk(s)k ≤ kx˜0 − x˜kk + DF0(x0(τ), t) − DFk(xk(τ), t) a a Z s Z s Z s Z s

≤ kx˜0 − x˜kk + DF0(x0(τ), t) − DFk(x0(τ), t) + DFk(x0(τ), t) − DFk(xk(τ), t) . a a a a

Choose an arbitrary ε > 0. There exists a k0 ∈ N such that kx˜0 − x˜kk < ε/2 and Z s Z s

DF0(x0(τ), t) − DFk(x0(τ), t) < ε/2 a a for every k ≥ k0 (the second statement follows from Lemma 9.2.8). These facts together with Lemma 9.2.3 imply that Z s kx0(s) − xk(s)k ≤ ε + ω(kx0(t) − xk(t)k) dh(t), k ≥ k0, s ∈ [a, b]. a

By letting ψk(s) = kx0(s) − xk(s)k, s ∈ [a, b], the last inequality can be rewritten as Z s ψk(s) ≤ ε + ω(ψk(t)) dh(t), k ≥ k0, s ∈ [a, b]. a

For an arbitrary u0 > 0, the function Z u 1 Ω(u) = dr, u ∈ (0, ∞), u0 ω(r) is continuous, increasing, α = limu→0+ Ω(u) = −∞, and β = limu→+∞ Ω(u) ≤ ∞. Hence, the inverse function Ω−1 is increasing on its domain (−∞, β). Without loss of generality, we can assume that ε is so small that Ω(ε) + h(b) − h(a) < β. It follows from Theorem 9.2.12 that

−1 −1 ψk(s) ≤ Ω (Ω(ε) + h(s) − h(a)) ≤ Ω (Ω(ε) + h(b) − h(a)), s ∈ [a, b], k ≥ k0.

For ε → 0+, we have Ω(ε) + h(b) − h(a) → −∞, and therefore Ω−1(Ω(ε) + h(b) − h(a)) → 0; this completes the proof. We now proceed to an infinite-dimensional counterpart to Theorem 9.4.2. In comparison with that R u dr theorem, we restrict ourselves to the case limv→0+ v ω(r) = ∞ for every u > 0; by Theorem 9.2.13, this guarantees uniqueness of solutions. The proof is similar to the proof of Theorem 8.6 in [67], but the part which was originally based on Helly’s selection theorem is now different and uses Lemma 9.4.6 instead. Also, the new Theorem 9.3.2 is needed in the proof. Again, the conclusion is stronger than in the original finite-dimensional version (we get uniform convergence of solutions instead of pointwise convergence).

Theorem 9.4.7. Let B ⊂ X, G = B × [a, b], and consider a sequence of functions Fk : G → X, k ∈ N0, such that lim Fk(x, t) = F0(x, t), (x, t) ∈ G. k→∞

Assume that Fk ∈ F(G, h, ω) for every k ∈ N0, where h is left-continuous and Z u dr lim = ∞ (9.4.8) v→0+ v ω(r) for every u > 0. Let x0 :[a, b] → B satisfy dx 0 = DF (x , t), x (a) =x ˜ . dτ 0 0 0 0

Finally, assume there exists a ρ > 0 such that ky − x0(s)k < ρ implies y ∈ B whenever s ∈ [a, b].

150 Then, given an arbitrary sequence x˜k ∈ X, k ∈ N, such that limk→∞ x˜k =x ˜0, there is a k0 ∈ N and a sequence of functions xk :[a, b] → B, k ≥ k0, which satisfy dx k = DF (x , t), t ∈ [a, b], x (a) =x ˜ . dτ k k k k Moreover, the sequence {x }∞ is uniformly convergent to x on [a, b]. k k=k0 0 Proof. It follows from the assumptions that

kFk(˜xk, a+) − Fk(˜xk, a) − Fk(˜x0, a+) + Fk(˜x0, a)k ≤ ω(kx˜k − x˜0k)(h(a+) − h(a)) → 0 for k → ∞. According to Lemma 9.2.7, we have Fk(˜x0, a+) → F0(˜x0, a+) for k → ∞. Therefore, we get

Fk(˜x0, a+) − Fk(˜x0, a) → F0(˜x0, a+) − F0(˜x0, a), and consequently

lim (˜xk + Fk(˜xk, a+) − Fk(˜xk, a)) =x ˜0 + F0(˜x0, a+) − F0(˜x0, a) = x0(a+) (9.4.9) k→∞ (the last equality follows from Lemma 9.2.9). Choose a δ > 0 such that h(a + δ) − h(a+) < ρ/2. If y ∈ X satisﬁes ky − x0(a+)k < ρ/2, then

ky − x0(a + δ)k ≤ ky − x0(a+)k + kx0(a+) − x0(a + δ)k < ρ/2 + h(a + δ) − h(a+) < ρ, and therefore y ∈ B, i.e., the ρ/2-neighborhood of x0(a+) is contained in B. This observation together with (9.4.9) imply the existence of a k0 ∈ N such that the values

x˜k + Fk(˜xk, a+) − Fk(˜xk, a), k ≥ k0, together with their ρ/4-neighborhoods, are contained in B. Moreover, we can assume thatx ˜k ∈ B for all k ≥ k0. By Theorem 9.3.2, there exists a ∆ > 0 and a sequence of functions xk :[a, a + ∆] → B, k ≥ k0, such that dx k = DF (x , t), t ∈ [a, a + ∆], x (a) =x ˜ , k ≥ k (9.4.10) dτ k k k k 0 (note that Theorem 9.3.2 requires the set B to be open; if necessary, we can replace our B by the open ρ-neighborhood of x ). According to Lemma 9.4.6, the sequence {x }∞ is uniformly convergent to x . 0 k k=k0 0 Up to this point, we have veriﬁed the statement of the theorem on the interval [a, a + ∆]. For contradiction, assume that the theorem does not hold on the whole interval [a, b], i.e., there exists a c ∈ (a, b) such that the theorem holds on [a, d] for every d < c, but not on [a, d] with d > c. For every k ≥ k0, it follows from Lemma 9.2.2 that

kxk(v) − xk(u)k ≤ h(v) − h(u), [u, v] ⊂ [a, c), k ∈ N.

Now, the existence of lims→c− h(s) implies the existence of lims→c− xk(s) for every k ∈ N. By letting xk(c) = xk(c−), we see that for all k ≥ k0, Eq. (9.4.10) has a unique solution defined on the closed interval [a, c]. According to the Moore-Osgood theorem, we also have limk→∞ xk(c) = lims→c− x0(s) = x0(c). We can now follow the argumentation from the first part of the present proof with a replaced by c to conclude that the theorem holds on an interval [a, d] with d > c, which is a contradiction. We conclude our discussion of continuous dependence theorems for generalized differential equations with two remarks:

• In this section, we were interested in continuous dependence theorems for sequences of equations where all the right-hand sides Fk : G → X, k ∈ N0, are elements of the same class F(G, h, ω). In the ﬁnite-dimensional case, there exists theorems applicable in the situation when Fk ∈ F(G, hk, ω) and ∞ the sequence {hk}k=0 satisﬁes some additional conditions; see [67, Theorem 8.5], [67, Theorem 8.8], [29, Theorem 2.4], and [29, Theorem 2.6]. Therefore, it is natural to ask whether these results

151 remain valid in infinite dimension. It is not difficult to check that the answer is affirmative in case of [67, Theorem 8.5] and [29, Theorem 2.4], which are similar to Theorem 9.4.1, and their original proofs are still applicable without any changes. On the other hand, [67, Theorem 8.8] and [29, Theorem 2.6], which are similar to Theorem 9.4.2, are not true in infinite dimension; again, the reason is that the assumption Fk ∈ F(G, hk, ω) no longer guarantees local existence of solutions when ω : [0, ∞) → [0, ∞) is an arbitrary increasing continuous function (cf. Example 9.4.5). We leave it as an open problem to find out whether the infinite-dimensional counterparts of these two R u dr theorems are valid under the additional assumption limv→0+ v ω(r) = ∞. • In his recent book [52], J. Kurzweil considered abstract nonlinear generalized equations of the form Z s x(s) = x(a) + DF (x(τ), τ, t), s ∈ [a, b], (9.4.11) a where the integral on the right-hand side is the strong Kurzweil integral (see [52, Chapter 14]). On one hand, his equations are more general because F can depend explicitly on τ. On the other hand, strong Kurzweil integrability is a more restrictive property that ordinary Kurzweil integrability. In [52, Lemma 23.8], we find a continuous dependence theorem for equations of the form (9.4.11). In the special case when the right-hand side F does not depend on τ, J. Kurzweil’s four conditions (23.2)–(23.5) reduce to the two inequalities

kF (x, t2) − F (x, t1)k ≤ (1 + kxk)(Φ(t2) − Φ(t1)), x ∈ B, [t1, t2] ⊂ [a, b], (9.4.12)

kF (x, t2) − F (x, t1) − F (y, t2) + F (y, t1)k ≤ kx − yk(Φ(t2) − Φ(t1)), x, y ∈ B, [t1, t2] ⊂ [a, b], (9.4.13) where Φ : [a, b] → R is a nondecreasing left-continuous function. Note that the theorem in [52] is stated for B = X, but remains valid for any B ⊂ X. For example, an ordinary differential equation with a locally Lipschitz-continuous right-hand side is equivalent to a generalized differential equation whose right-hand side F satisfies (9.4.13) only locally. Also, solutions of generalized differential equations are regulated, and therefore bounded on compact intervals. Thus, the situation when B is bounded is quite natural. In this case, it is not difficult to check that the conditions (9.4.12), (9.4.13) hold if and only if F ∈ F(B × [a, b], h). Therefore, J. Kurzweil’s theorem is seemingly similar to our Theorem 9.4.7 with ω(r) = r, but its precise formulation is different: The theorem starts with two right-hand sides F1, F2 : B × [a, b] → X satisfying the inequality

∗ ∗ kF1(x, t2) − F1(x, t1) − F2(x, t2) + F2(x, t1)k ≤ (1 + kxk)(Φ (t2) − Φ (t1)),

∗ for all x ∈ B and [t1, t2] ⊂ [a, b], where Φ :[a, b] → R is a nondecreasing left-continuous function. dxi Then, assuming that x1, x2 :[a, b] → X satisfy dτ = DFi(xi, t) and x1(a) = x2(a), the theorem ∗ ∗ provides an estimate of the form kx1 − x2k∞ ≤ C(Φ (b) − Φ (a)).

9.5 Application to measure functional diﬀerential equations

As an application of our results for abstract diﬀerential equations, let us study the well-posedness for measure functional diﬀerential equations of the form

Z t y(t) = y(t0) + f(ys, s) dg(s), t ∈ [t0, t0 + σ], (9.5.1) t0 where y and f take values in Rn, and the integral on the right-hand side is the Kurzweil-Stieltjes integral with respect to a nondecreasing function g :[t0, t0 + σ] → R. These equations generalize other types of functional equations, such as classical functional differential equations, impulsive functional differential equations, or functional dynamic equations on time scales (see Chapters 4, 5). In Chapter 4, it was shown that there is a one-to-one correspondence between measure functional differential equations with finite delay and generalized ordinary differential equations whose solutions take values in certain infinite-dimensional

152 spaces. In Chapter 6, this correspondence was extended to equations with infinite delay and axiomatically described phase space. For simplicity, we do not discuss general phase spaces as in Chapters 6 and 8, but restrict ourselves to the phase space G((−∞, 0], Rn). The correspondence between measure functional differential equations and generalized ordinary differ- n ential equations is constructed as follows: We take B ⊂ R , O = G((−∞, t0 + σ],B), P = G((−∞, 0],B), n and consider a function f : P × [t0, t0 + σ] → R (note that xt ∈ P whenever x ∈ O and t ∈ [t0, t0 + σ]). Now, under certain assumptions, Eq. (9.5.1) is equivalent (in a sense described below) to the abstract generalized ordinary differential equation

dx = DF (x, t), t ∈ [t , t + σ], (9.5.2) dτ 0 0

n in the Banach space X = G((−∞, t0 + σ], R ), where the solution x takes values in O ⊂ X, and the function F : O × [t0, t0 + σ] → X is given by  0, −∞ < ϑ ≤ t ,  0 R ϑ f(xs, s) dg(s), t0 ≤ ϑ ≤ t ≤ t0 + σ, F (x, t)(ϑ) = t0 (9.5.3) R t  f(xs, s) dg(s), t ≤ ϑ ≤ t0 + σ t0 for every x ∈ O and t ∈ [t0, t0 + σ]. At this moment, we need the following conditions concerning the function f:

R t0+σ (A) The integral f(yt, t) dg(t) exists for every y ∈ O. t0

(B) There exists a function M :[t0, t0 + σ] → [0, ∞), which is Kurzweil-Stieltjes integrable with respect to g, such that

Z b Z b

f(yt, t) dg(t) ≤ M(t) dg(t), y ∈ O, [a, b] ⊆ [t0, t0 + σ]. a a

(C) There exists a function L :[t0, t0 + σ] → [0, ∞), which is Kurzweil-Stieltjes integrable with respect to g, and a continuous increasing function ω : [0, ∞) → [0, ∞) such that ω(0) = 0,

Z b Z b

(f(yt, t) − f(zt, t)) dg(t) ≤ L(t)ω(kyt − ztk∞) dg(t), y, z ∈ O, [a, b] ⊆ [t0, t0 + σ]. a a

The next lemma is a straightforward generalization of Lemma 6.3.4, which corresponds to the special case ω(r) = r.

n Lemma 9.5.1. Assume that B ⊂ R , O = G((−∞, t0 + σ],B), P = G((−∞, 0],B), g :[t0, t0 + σ] → R is n a nondecreasing function, f : P × [t0, t0 + σ] → R satisﬁes conditions (A), (B), (C). Then, the function n F : O × [t0, t0 + σ] → G((−∞, t0 + σ], R ) given by (9.5.3) is an element of F(O × [t0, t0 + σ], h1, h2, ω), R t R t where h1(t) = M(s) dg(s) and h2(t) = L(s) dg(s) for all s ∈ [t0, t0 + σ]. t0 t0

Proof. Condition (A) guarantees that the integrals in the deﬁnition of F exist. When [s1, s2] ⊂ [t0, t0 + σ], we have  0, −∞ < τ ≤ s ,  1 R τ F (y, s2)(τ) − F (y, s1)(τ) = f(ys, s) dg(s), s1 ≤ τ ≤ s2, s1 R s2  f(ys, s) dg(s), s2 ≤ τ ≤ t0 + σ s1 for every y ∈ O. Using condition (B), we get

kF (y, s2) − F (y, s1)k∞ = sup kF (y, s2)(τ) − F (y, s1)(τ)k = τ∈[s1,s2]

153 Z τ Z s2

= sup f(ys, s) dg(s) ≤ M(s) dg(s) = h1(s2) − h1(s1). τ∈[s1,s2] s1 s1 Similarly, condition (C) implies that for every y, z ∈ O, we have

kF (y, s2) − F (y, s1) − F (z, s2) + F (z, s1)k∞ Z τ

= sup kF (y, s2)(τ) − F (y, s1)(τ) − F (z, s2)(τ) + F (z, s1)(τ)k = sup (f(ys, s) − f(zs, s)) dg(s) τ∈[s1,s2] τ∈[s1,s2] s1

Z s2 Z s2 ≤ L(s)ω(kys − zsk∞) dg(s) ≤ ω(ky − zk∞) L(s) dg(s) = ω(ky − zk∞)(h2(s2) − h2(s1)). s1 s1 The next two theorems describe the precise relationship between solutions of the measure functional differential equation (9.5.1) and solutions of the generalized differential equation (9.5.2). The proofs for the special case ω(r) = r can be found in Chapter 6. In the general case, the proofs require some small modifications that are described below.

n Theorem 9.5.2. Assume that B ⊂ R is open, O = G((−∞, t0 + σ],B), P = G((−∞, 0],B), φ ∈ P , n g :[t0, t0 + σ] → R is a nondecreasing function, f : P × [t0, t0 + σ] → R satisﬁes conditions (A), (B), (C), n and F : O × [t0, t0 + σ] → G((−∞, t0 + σ], R ) is given by (9.5.3). If y ∈ O is a solution of the measure functional diﬀerential equation Z t y(t) = y(t0) + f(ys, s) dg(s), t ∈ [t0, t0 + σ], t0

yt0 = φ, then the function x :[t0, t0 + σ] → O given by ( y(ϑ), ϑ ∈ (−∞, t], x(t)(ϑ) = y(t), ϑ ∈ [t, t0 + σ] is a solution of the generalized ordinary diﬀerential equation (9.5.2). Proof. A proof for the special case ω(r) = r can be found in Theorem 6.3.6. In the general case, it is enough to modify the proof from Chapter 6 as follows: Given an ε > 0, there exists an r0 > 0 such that ω(r) ≤ ε for all r ∈ [0, r0]. The function h(t) = R t + M(s) dg(s) has only ﬁnitely many points t ∈ [t0, t0 + σ] such that ∆ h(t) ≥ r0; denote these points by t0 + t1, . . . , tm. Find a gauge δ :[t0, t0 + σ] → R such that t − t δ(τ) < min k k−1 , k = 2, . . . , m , τ ∈ [t , t + σ], 2 0 0

δ(τ) < min {|τ − tk|; k = 1, . . . , m} , τ ∈ [t0, t0 + σ]\{t1, . . . , tm}, Z tk+δ(tk) ε L(s)ω(kys − x(tk)sk∞) dg(s) < , k ∈ {1, . . . , m}, tk 2m + 1

ky(ρ) − y(τ)k ≤ r0, τ ∈ [t0, t0 + σ]\{t1, . . . , tm}, ρ ∈ [τ, τ + δ(τ)). Then proceed as in Chapter 6 with obvious modiﬁcations.

n Theorem 9.5.3. Assume that B ⊂ R is open, O = G((−∞, t0 + σ],B), P = G((−∞, 0],B), φ ∈ P , n g :[t0, t0 + σ] → R is a nondecreasing function, f : P × [t0, t0 + σ] → R satisﬁes conditions (A), (B), (C), n and F : O × [t0, t0 + σ] → G((−∞, t0 + σ], R ) is given by (9.5.3). If x :[t0, t0 + σ] → O is a solution of the generalized ordinary diﬀerential equation (9.5.2) with the initial condition ( φ(ϑ − t0), ϑ ∈ (−∞, t0], x(t0)(ϑ) = φ(0), ϑ ∈ [t0, t0 + σ],

154 then the function y ∈ O deﬁned by ( x(t )(ϑ), ϑ ∈ (−∞, t ], y(ϑ) = 0 0 x(ϑ)(ϑ), ϑ ∈ [t0, t0 + σ] is a solution of the measure functional diﬀerential equation

Z t y(t) = y(t0) + f(ys, s) dg(s), t ∈ [t0, t0 + σ], t0

yt0 = φ.

Proof. A proof for the special case ω(r) = r can be found in Theorem 6.3.7. In the general case, it is enough to modify the proof from Chapter 6 in the same way as described in the proof of Theorem 9.5.2, except that the last condition on the gauge δ should be replaced by

|h(ρ) − h(τ)| ≤ r0, τ ∈ [t0, t0 + σ]\{t1, . . . , tm}, ρ ∈ [τ, τ + δ(τ)).

Then proceed as in Chapter 6 with obvious modiﬁcations.

We now present an Osgood-type existence theorem for measure functional differential equations with infinite delay based on Theorem 9.3.2. Our result generalizes Theorem 4.5.3 and Theorem 6.3.12, which correspond to the special case ω(r) = r. Even in that case, our theorem provides more information since it specifies a lower bound for the length of the interval where the solution is guaranteed to exist.

n Theorem 9.5.4. Assume that B ⊂ R is open, O = G((−∞, t0 + σ],B), P = G((−∞, 0],B), g : n [t0, t0 + σ] → R is nondecreasing and left-continuous function, and f : P × [t0, t0 + σ] → R satisﬁes conditions (A), (B), (C), where the function ω : [0, ∞) → [0, ∞) is such that

Z u dr lim = ∞ v→0+ v ω(r) for every u > 0. + If φ ∈ P is such that φ(0) + f(φ, t0)∆ g(t0) ∈ B, then the initial-value problem

Z t

y(t) = y(t0) + f(ys, s) dg(s), t ∈ [t0, t0 + σ], yt0 = φ, (9.5.4) t0 has a unique local solution deﬁned on a right neighborhood of t0. Moreover, if ∆ > 0 is such that the closed ball

Z t0+∆ n + {x ∈ R ; kx − (φ(0) + f(φ, t0)∆ g(t0))k ≤ M(s) dg(s)} (9.5.5) t0+ is contained in B, the solution is guaranteed to exist on [t0, t0 + ∆].

Proof. According to Lemma 9.5.1, the function F given by (9.5.3) is an element of F(O × [t0, t0 + R t R t σ], h1, h2, ω), where h1(t) = M(s) dg(s) and h2(t) = L(s) dg(s). Since g is left-continuous, h1 t0 t0 and h2 have the same property. By Theorems 9.5.2 and 9.5.3, the initial-value problem (9.5.4) is equivalent to

dx = DF (x, t), t ∈ [t , t + σ], x(t ) = x , (9.5.6) dτ 0 0 0 0 where x0 equals ( φ(ϑ − t0), ϑ ∈ (−∞, t0], x0(ϑ) = φ(0), ϑ ∈ [t0, t0 + σ].

155 The function F is regulated with respect to the second variable (this follows from the fact that F(O × [t0, t0 + σ], h1, h2, ω), where h1 is regulated). Thus for every t ∈ [t0, t0 + σ), the right-sided limit F (x, t+) exists and F (x, t+)(ϑ) = limδ→0+ F (x, t + δ)(ϑ). Using Theorem 9.2.11 and the deﬁnition of F given in (9.5.3), we obtain ( 0, ϑ ∈ (−∞, t], (F (x, t+) − F (x, t))(ϑ) = + f(xt, t)∆ g(t), ϑ ∈ (t, t0 + σ].

By Theorem 9.3.2, equation (9.5.6) has a unique local solution if x0 +F (x0, t0+)−F (x0, t0) ∈ O. Since ( φ(ϑ − t0), ϑ ∈ (−∞, t0], (x0 + F (x0, t0+) − F (x0, t0))(ϑ) = + φ(0) + f(φ, t0)∆ g(t0), ϑ ∈ (t0, t0 + σ],

+ the condition is satisﬁed if and only if φ(0) + f(φ, t0)∆ g(t0) ∈ B. For every x ∈ O and t ∈ [t0, t0 + σ], F (x, t) vanishes on (−∞, t0]. Thus, the solution of (9.5.6) can never leave the set

n Y = {x ∈ G((−∞, t0 + σ], R ); x(ϑ) = φ(ϑ − t0) for ϑ ∈ (−∞, t0]}. By Remark 9.3.3, the solution is guaranteed to exist on [a, a + ∆] if the following statements hold:

• If x ∈ Y and t ∈ [t0, t0 + ∆), then x + F (x, t+) − F (x, t) ∈ Y .

• If x ∈ Y and kx − (x0 + F (x0, t0+) − F (x0, t0))k ≤ h1(t0 + ∆) − h1(t0+), then x ∈ O.

The first condition is clearly satisfied, because F (x, t+) − F (x, t) equals zero on (−∞, t0]. Also, we have R t0+∆ h1(t0 + ∆) − h1(t0+) = M(s) dg(s) and t0+ ( 0, ϑ ∈ (−∞, t0], (x − (x0 + F (x0, t0+) − F (x0, t0)))(ϑ) = + x(ϑ) − (φ(0) + f(φ, t0)∆ g(t0)), ϑ ∈ (t0, t0 + σ], for every x ∈ Y . Hence, the second condition is satisfied if the closed ball (9.5.5) is contained in B. Let us proceed to continuous dependence of solutions to nonlinear measure functional differential equations. The only theorem available in the literature is Theorem 4.6.3, which applies to equations with finite delay and is similar in spirit to Theorem 9.4.3 (i.e., it states that under certain assumptions, the limit of solutions is a solution again). The next result for measure functional differential equations with infinite delay is based on Theorem 9.4.7. Therefore, it is new even in the special case when the delay is finite and ω(r) = r.

n Theorem 9.5.5. Assume that B ⊂ R , O = G((−∞, t0 +σ],B), P = G((−∞, 0],B), g :[t0, t0 +σ] → R is n a nondecreasing left-continuous function, and fk : P × [t0, t0 + σ] → R , k ∈ N0, is a sequence of functions satisfying the following conditions:

R t0+σ 1. The integral fk(yt, t) dg(t) exists for every k ∈ 0, y ∈ O. t0 N

2. There exists a function M :[t0, t0 + σ] → [0, ∞), which is Kurzweil-Stieltjes integrable with respect to g, such that

Z b Z b

fk(yt, t) dg(t) ≤ M(t) dg(t), k ∈ N0, y ∈ O, [a, b] ⊆ [t0, t0 + σ]. a a

3. There exist a function L :[t0, t0 +σ] → [0, ∞), which is Kurzweil-Stieltjes integrable with respect to g, R u dr and a continuous increasing function ω : [0, ∞) → [0, ∞) such that ω(0) = 0, limv→0+ v ω(r) = ∞ for every u > 0, and

Z b Z b

(fk(yt, t) − fk(zt, t)) dg(t) ≤ L(t)ω(kyt −ztk∞) dg(t), k ∈ N0, y, z ∈ O, [a, b] ⊆ [t0, t0 +σ]. a a

156 4. For every y ∈ O, Z t Z t lim fk(ys, s) dg(s) = f0(ys, s) dg(s) k→∞ t0 t0

uniformly with respect to t ∈ [t0, t0 + σ].

Suppose that φ0 ∈ P and y0 :(−∞, t0 + σ] → B satisﬁes Z t

y0(t) = y0(t0) + f0((y0)s, s) dg(s), t ∈ [t0, t0 + σ], (y0)t0 = φ0. t0

Also, assume there exists a ρ > 0 such that ky − y0(s)k < ρ implies y ∈ B whenever s ∈ (−∞, t0 + σ]. Then, given an arbitrary sequence φk ∈ P , k ∈ N, such that limk→∞ kφk − φ0k∞ = 0, there is a k0 ∈ N and a sequence of functions yk :(−∞, t0 + σ] → B, k ≥ k0, which satisfy Z t

yk(t) = yk(t0) + fk((yk)s, s) dg(s), t ∈ [t0, t0 + σ], (yk)t0 = φk. (9.5.7) t0 Moreover, the sequence {y }∞ is uniformly convergent to y on (−∞, t + σ]. k k=k0 0 0 n Proof. For every k ∈ N0, let the function Fk : O × [t0, t0 + σ] → G((−∞, t0 + σ], R ) be given by  0, −∞ < ϑ ≤ t ,  0 R ϑ fk(xs, s) dg(s), t0 ≤ ϑ ≤ t ≤ t0 + σ, Fk(x, t)(ϑ) = t0 R t  fk(xs, s) dg(s), t ≤ ϑ ≤ t0 + σ t0 for every x ∈ O and t ∈ [t0, t0 + σ]. According to Lemma 9.5.1, we have Fk ∈ F(O × [t0, t0 + σ], h1, h2, ω), where h1, h2 are left-continuous functions. Moreover, it follows from assumption 4 that

lim Fk(x, t) = F0(x, t), (x, t) ∈ O × [t0, t0 + σ]. k→∞

For every k ∈ N0, letx ˜k ∈ O be deﬁned as ( φk(ϑ − t0), ϑ ∈ (−∞, t0], x˜k(ϑ) = φk(0), ϑ ∈ [t0, t0 + σ].

By Theorem 9.5.2, the function x0 :[t0, t0 + σ] → O given by ( y0(ϑ), ϑ ∈ (−∞, t], x0(t)(ϑ) = y0(t), ϑ ∈ [t, t0 + σ] satisﬁes dx 0 = DF (x , t), t ∈ [t , t + σ], x (t ) =x ˜ . dτ 0 0 0 0 0 0 0 Also, note that limk→∞ x˜k =x ˜0. Hence, by Theorem 9.4.7, there is a k0 ∈ N and a sequence of functions xk :[t0, t0 + σ] → O, k ≥ k0, which is uniformly convergent to x0 on [t0, t0 + σ], and dx k = DF (x , t), t ∈ [t , t + σ], x (t ) =x ˜ , k ≥ k . dτ k k 0 0 k 0 k 0

For every k ∈ N, let yk ∈ O be given by ( xk(t0)(ϑ), ϑ ∈ (−∞, t0], yk(ϑ) = xk(ϑ)(ϑ), ϑ ∈ [t0, t0 + σ].

By Theorem 9.5.3, the function yk is a solution of the initial-value problem (9.5.7). Also, since the sequence {x }∞ is uniformly convergent to x , it follows that {y }∞ is uniformly convergent to y . k k=k0 0 k k=k0 0

157 9.6 Application to functional diﬀerential equations with impulses

In this section, we consider the well-posedness for impulsive functional diﬀerential equations of the form

0 y (t) = f(yt, t), almost everywhere in [t0, t0 + σ], + ∆ y(ti) = Ii(y(ti)), i ∈ {1, . . . , m}, where the impulses take place at preassigned times t1, . . . , tm ∈ [t0, t0 +σ), and their action is described by n n the operators Ii : R → R , i ∈ {1, . . . , m}; the solution is assumed to be left-continuous at every point ti, and absolutely continuous on every interval whose intersection with {t1, . . . , tm} is empty. The equivalent integral form of the problem is

Z t X y(t) = y(t0) + f(ys, s) ds + Ii(y(ti)), t ∈ [t0, t0 + σ]. (9.6.1) t0 i∈{1,...,m}, ti

Using the following lemma, it is possible to convert impulsive functional diﬀerential equations into measure functional diﬀerential equations. The lemma is a consequence of Lemma 5.2.4.

Lemma 9.6.1. Let m ∈ N, a ≤ t1 < t2 < ··· < tm < b, and

m X g(s) = s + χ(ti,∞)(s), s ∈ [a, b] i=1

(where χA denotes the characteristic function of a set A ⊂ R). Consider an arbitrary pair of functions ˜ ˜ f :[a, b] → R and f :[a, b] → R such that f(s) = f(s) for every s ∈ [a, b]\{t1, . . . , tm}. R b ˜ R b Then the integral a f(s) dg(s) exists if and only if the integral a f(s) ds exists; in that case, we have Z t Z t ˜ X ˜ f(s) dg(s) = f(s) ds + f(ti), t ∈ [a, b]. a a i∈{1,...,m}, ti

According to the last lemma, the impulsive equation (9.6.1) is equivalent to the measure functional diﬀerential equation Z t ˜ y(t) = y(t0) + f(ys, s) dg(s), t ∈ [t0, t0 + σ], (9.6.2) t0 Pm where g(s) = s + i=1 χ(ti,∞)(s) and ( f(x, s) for s ∈ [t , t + σ]\{t , . . . , t }, f˜(x, s) = 0 0 1 m Ii(x(0)) for s = ti, i ∈ {1, . . . , m}.

Now, it is a simple task to obtain an Osgood-type existence theorem for impulsive functional diﬀerential equations with inﬁnite delay.

n Theorem 9.6.2. Let B ⊂ R be open, O = G((−∞, t0 + σ],B), P = G((−∞, 0],B), m ∈ N, t0 ≤ t1 < n n t2 < . . . < tm < t0 + σ. Consider functions f : P × [t0, t0 + σ] → R and I1,...,Im : B → R such that the following conditions are satisﬁed:

R t0+σ 1. The integral f(yt, t) dt exists for every y ∈ O. t0

2. There exists an integrable function M :[t0, t0 + σ] → [0, ∞) such that

Z b Z b

f(yt, t) dt ≤ M(t) dt, y ∈ O, [a, b] ⊆ [t0, t0 + σ]. a a

158 3. There exists an integrable function L :[t0, t0 + σ] → [0, ∞) and a continuous increasing function R u dr ω : [0, ∞) → [0, ∞) such that ω(0) = 0, limv→0+ v ω(r) = ∞ for every u > 0, and

Z b Z b

(f(yt, t) − f(zt, t)) dt ≤ L(t)ω(kyt − ztk∞) dt, y, z ∈ O, [a, b] ⊆ [t0, t0 + σ]. a a

4. There exist constants α1, . . . , αm ≥ 0 such that kIi(x)k ≤ αi for i ∈ {1, . . . , m}, x ∈ B.

5. There exists a constant β > 0 such that kIi(x) − Ii(y)k ≤ βω(kx − yk) for i ∈ {1, . . . , m}, x, y ∈ B.

Let φ ∈ P and assume that either t0 < t1, or t0 = t1 and φ(0) + I1(φ(0)) ∈ B. Then, the initial-value problem

Z t X y(t) = y(t0) + f(ys, s) ds + Ii(y(ti)), t ∈ [t0, t0 + σ], yt0 = φ. (9.6.3) t0 i∈{1,...,m}, ti 0 is such that either t0 < t1 and

m Z t0+∆ n X {x ∈ R ; kx − φ(0)k ≤ M(s) ds + αi} ⊂ B, t0 i=1 or t0 = t1 and m Z t0+∆ n X {x ∈ R ; kx − (φ(0) + I1(φ(0)))k ≤ M(s) ds + αi} ⊂ B, t0 i=2 then the solution is guaranteed to exist on [t0, t0 + ∆]. Proof. It is enough to prove that the equivalent measure functional diﬀerential equation (9.6.2) has a unique local solution satisfying yt0 = φ. Let us verify that all assumptions of Theorem 9.5.5 are satisﬁed. Assume that [a, b] ⊂ [t0, t0 + σ] and y, z ∈ O. From Lemma 9.6.1, we obtain

Z b Z b Z b Z b ˜ X X ˜ f(yt, t) dg(t) = f(yt, t) dt + Ii(y(ti)) ≤ M(t) dt + αi = M(t) dg(t),

a a i∈{1,...,m}, a i∈{1,...,m}, a a≤ti

Z b Z b ˜ ˜ X f(yt, t) − f(zt, t) dg(t) = (f(yt, t) − f(zt, t)) dt + (Ii(y(ti)) − Ii(z(ti)))

a a i∈{1,...,m}, a≤ti

Z b X ≤ L(t)ω(kyt − ztk∞) dt + βω(ky(ti) − z(ti)k) a i∈{1,...,m}, a≤ti

159 where ( L(t) for t ∈ [t , t + σ]\{t , . . . , t }, L˜(t) = 0 0 1 m β for t = ti, i ∈ {1, . . . , m}. ˜ + ˜ + We have either t0 < t1 and φ(0) + f(φ, t0)∆ g(t0) = φ(0) ∈ B, or t0 = t1 and φ(0) + f(φ, t0)∆ g(t0) = φ(0) + I1(φ(0)) ∈ B. Hence, all assumptions of Theorem 9.5.5 are satisﬁed. Finally, the two conditions on the length ∆ follow from the identity Z t0+∆ Z t0+∆ X M˜ (s) dg(s) = M(s) ds + αi. t0+ t0 i∈{1,...,m}, t0

Z b Z b

fk(yt, t) dt ≤ M(t) dt, k ∈ N0, y ∈ O, [a, b] ⊆ [t0, t0 + σ]. a a 3. There exist a continuous increasing function ω : [0, ∞) → [0, ∞) and a Lebesgue integrable function + L :[t0, t0 + σ] → R such that

Z b Z b

(fk(yt, t) − fk(zt, t)) dt ≤ L(t)ω(kyt − ztk∞) dt, k ∈ N0, y, z ∈ O, [a, b] ⊆ [t0, t0 + σ] a a R u dr and limv→0+ v ω(r) = ∞ for every u > 0. 4. For every y ∈ O, Z t Z t lim fk(ys, s) ds = f0(ys, s) ds k→∞ t0 t0

uniformly with respect to t ∈ [t0, t0 + σ]. k 5. There exists a constant α > 0 such that kIi (x)k ≤ α for every i ∈ {1, . . . , m}, k ∈ N0 and x ∈ B. k k 6. There exists a constant β > 0 such that kIi (x) − Ii (y)k ≤ βω(kx − yk) for every i ∈ {1, . . . , m}, k ∈ N0 and x, y ∈ B. k 0 7. For every x ∈ B and k ∈ {1, . . . , m}, limk→∞ Ii (x) = Ii (x).

Suppose that φ0 ∈ P and y0 :(−∞, t0 + σ] → B satisﬁes Z t X 0 y0(t) = y0(t0) + f0((y0)s, s) ds + Ii (y0(ti)), t ∈ [t0, t0 + σ], (y0)t0 = φ0. t0 i∈{1,...,m}, ti

160 Pm Proof. Let g(s) = s + i=1 χ(ti,∞)(s) and ( ˜ fk(x, s), s ∈ [t0, t0 + σ]\{t1, . . . , tm}, fk(x, s) = k Ii (x(0)), s = ti, i ∈ {1, . . . , m} for every k ∈ N0. Then Z t X k yk(t) = yk(t0) + fk((yk)s, s) ds + Ii (yk(ti)), t ∈ [t0, t0 + σ] t0 i∈{1,...,m}, ti

Z b Z b ˜ ˜ fk(yt, t) dg(t) ≤ M(t) dg(t), [a, b] ⊂ [t0, t0 + σ], a a

Z b Z b ˜ ˜ ˜ fk(yt, t) − fk(zt, t) dg(t) ≤ L(t)ω(kyt − ztk∞) dg(t), a a which means that assumptions 2 and 3 are satisﬁed, too. The results obtained in this chapter are also applicable to abstract dynamic equations on time scales, as well as (impulsive) functional dynamic equations on time scales. As shown in the previous chapters, these types of equations can be transformed to measure (functional) diﬀerential equations. Thus, one can easily obtain Osgood-type theorems and continuous dependence theorems for these equations.

161 Chapter 10

Generalized elementary functions

10.1 Introduction

There are many equivalent ways of introducing the classical exponential function; one possibility is to deﬁne the exponential as the unique solution of the initial-value problem z0(t) = z(t), z(0) = 1. More generally, for every continuous function p deﬁned on the real line, the initial-value problem z0(t) = p(t)z(t), z(t0) = 1, which can be written in the equivalent integral form

Z t z(t) = 1 + p(s)z(s) ds, (10.1.1) t0

R t p(s) ds has the unique solution z(t) = e t0 . In this chapter, we replace Eq. (10.1.1) by the more general equation

Z t z(t) = 1 + z(s) dP (s), (10.1.2) t0 where the integral on the right-hand side is the Kurzweil-Stieltjes integral. We define the generalized exponential function edP as the unique solution of Eq. (10.1.2) and study its properties. Obviously, if P is continuously differentiable with P 0 = p, then Eq. (10.1.2) reduces back to Eq. (10.1.1). On the other hand, Eq. (10.1.2) is much more general and makes sense even if P is discontinuous. We point out that Eq. (10.1.2) is a generalized linear differential equation in the sense of J. Kurzweil’s definition [51]. Therefore, we can use the existing theory of generalized differential equations (see [67, 73]) in our study of the generalized exponential function. In Section 2, we start by summarizing some necessary facts about the Kurzweil-Stieltjes integral and generalized linear ordinary differential equations. Next, we prove an existence-uniqueness theorem for equations with complex-valued coefficients and solutions. In Section 3, we define the generalized exponential function and investigate its properties. For example, we show that the product of two exponentials edP edQ gives the exponential of a function denoted by P ⊕ Q; when P , Q are continuous, we have P ⊕ Q = P + Q, but otherwise P ⊕ Q is more complicated and takes into account the discontinuities of P and Q. In Section 4, we use the exponential to introduce the generalized hyperbolic and trigonometric functions. Finally, in Section 5, we demonstrate that our generalized elementary functions include the time scale elementary functions as a special case. At the same time, we show how the usual definitions of the time scale exponential, hyperbolic and trigonometric functions can be extended from rd-continuous to Lebesgue ∆-integrable arguments.

162 10.2 Preliminaries

Throughout this chapter, we work with regulated functions deﬁned on a compact interval [a, b] and use the following notation: ( ( g(t+) − g(t) if t ∈ [a, b), g(t) − g(t−) if t ∈ (a, b], ∆+g(t) = ∆−g(t) = 0 if t = b, 0 if t = a.

Also, we let ∆g(t) = ∆+g(t) + ∆−g(t). The following theorem describes the properties of the indeﬁnite Kurzweil-Stieltjes integral and can be found in [81, Proposition 2.16].

R b Theorem 10.2.1. Consider a pair of functions f, g :[a, b] → R such that g is regulated and a f dg exists. Then, for every t0 ∈ [a, b], the function

Z t h(t) = f dg, t ∈ [a, b] t0 is regulated and satisﬁes

h(t+) = h(t) + f(t)∆+g(t), t ∈ [a, b), h(t−) = h(t) − f(t)∆−g(t), t ∈ (a, b].

If I ⊂ R is an interval, h : I → R is a function which is zero except a countable set {t1, t2,...} ⊂ I, P P and the sum S = i h(ti) is absolutely convergent, we use the notation S = x∈I h(x). The next lemma is taken over from [81, Proposition 2.12].

Lemma 10.2.2. Let f :[a, b] → R be a function which is zero except a countable set {t1, t2,...} ⊂ [a, b] P and i f(ti) is absolutely convergent. Then, for every regulated function g :[a, b] → R, we have

Z b X f dg = f(x)∆g(x). a x∈[a,b]

It is well known that every function h :[a, b] → R with bounded variation has at most countably many P + − + − discontinuities, and x∈[a,b](|∆ h(x)| + |∆ h(x)|) is ﬁnite. Hence, both f = ∆ h and f = ∆ h satisfy the assumptions of the previous lemma; we will use this observation later. The following integration by parts formula for the Kurzweil-Stieltjes integral can be found in [81, Theorem 2.15].

Theorem 10.2.3. If f, g :[a, b] → R are regulated and at least one of them has bounded variation, then

Z b Z b X f dg + g df = f(b)g(b) − f(a)g(a) + (∆−f(x)∆−g(x) − ∆+f(x)∆+g(x)). a a x∈[a,b]

Remark 10.2.4. For our purposes, it will be sometimes more convenient to rewrite the last sum as follows: X (∆−f(x)∆−g(x) − ∆+f(x)∆+g(x)) x∈[a,b]

X X = (∆−f(x)(∆−g(x)+∆+g(x))−(∆+f(x)+∆−f(x))∆+g(x)) = (∆−f(x)∆g(x)−∆f(x)∆+g(x)) x∈[a,b] x∈[a,b]

The next substitution theorem for the Kurzweil-Stieltjes integral was proved in [81, Theorem 2.19].

163 R b Theorem 10.2.5. Assume that h :[a, b] → R is bounded and f, g :[a, b] → R are such that a f dg exists. Then Z b Z x Z b h(x) d f(z) dg(z) = h(x)f(x) dg(x), a a a whenever either side of the equation exists. We need to extend the definition of the Kurzweil-Stieltjes integral to complex-valued functions. Given a pair of functions f, g :[a, b] → C with real parts f1, g1 and imaginary parts f2, g2, we define Z b Z b Z b Z b Z b Z b ! f dg = (f1 + if2) d(g1 + ig2) = f1 dg1 − f2 dg2 + i f1 dg2 + f2 dg1 (10.2.1) a a a a a a whenever the integrals on the right-hand side exist. All results mentioned in this section (integration by parts, substitution, etc.) are still valid for complex- valued functions. We leave the verification of this fact up to the reader: In all cases, it is enough to rewrite the integrals of complex-valued functions using the definition (10.2.1), then apply the corresponding “real- valued” theorem, and finally return back to integration of complex-valued functions. The following statement is the existence and uniqueness theorem for generalized linear differential R b equations (see [67, Theorem 6.5] or [73, Theorem III.1.4]). The Stieltjes-type integral denoted by a d[A] x, where A :[a, b] → Rn×n and x :[a, b] → Rn, should be understood as a special case of J. Kurzweil’s integral R b a DU(τ, t), where U(τ, t) = A(t)x(τ). Theorem 10.2.6. Consider a function A :[a, b] → Rn×n, which has bounded variation on [a, b]. Let + − t0 ∈ [a, b] and assume that I + ∆ A(t) is invertible for every t ∈ [a, t0), and I − ∆ A(t) is invertible for n n every t ∈ (t0, b]. Then, for every x˜ ∈ R , there exists a unique function x :[a, b] → R such that Z t x(t) =x ˜ + d[A] x, t ∈ [a, b]. t0 Moreover, x has bounded variation on [a, b]. Before we introduce the exponential function, we need the following existence and uniqueness theorem for equations with complex-valued coefficients and solutions. The statement can be found in [34, Theorem 3.1], but without proof; for completeness, we include the proof here.

Theorem 10.2.7. Consider a function P :[a, b] → C, which has bounded variation on [a, b]. Let t0 ∈ [a, b] + − and assume that 1 + ∆ P (t) 6= 0 for every t ∈ [a, t0), and 1 − ∆ P (t) 6= 0 for every t ∈ (t0, b]. Then, for every z˜ ∈ C, there exists a unique function z :[a, b] → C such that Z t z(t) =z ˜ + d[P ] z, t ∈ [a, b]. (10.2.2) t0 The function z has bounded variation on [a, b]. If P and z˜ are real, then z is real as well.

Proof. We decompose all complex quantities into real and imaginary parts as follows: P = P1 + iP2, z = z1 + iz2, andz ˜ =z ˜1 + iz˜2. Now, we see that equation (10.2.2) is equivalent to the following system of two equations with real coeﬃcients: Z t Z t z1(t) =z ˜1 + d[P1] z1 − d[P2] z2 t0 t0 Z t Z t z2(t) =z ˜2 + d[P1] z2 + d[P2] z1 t0 t0 The system can be also written in the vector form Z t u(t) =u ˜ + d[A]u, t ∈ [a, b] (10.2.3) t0

164 withu ˜ = (˜z1, z˜2), P1(t) −P2(t) u(t) = (z1(t), z2(t)),A(t) = , t ∈ [a, b]. P2(t) P1(t)

Since P has bounded variation on [a, b], it is clear that A has the same property. The condition 1 + ∆+P (t) 6= 0 implies + + 1 + ∆ P1(t) 6= 0 or ∆ P2(t) 6= 0, t ∈ [a, t0), and similarly 1 − ∆−P (t) 6= 0 implies

− − 1 − ∆ P1(t) 6= 0 or ∆ P2(t) 6= 0, t ∈ (t0, b].

In view of this, we have

+ + 2 + 2 det(I + ∆ A(t)) = (1 + ∆ P1(t)) + (∆ P2(t)) 6= 0, t ∈ [a, t0),

− − 2 − 2 det(I − ∆ A(t)) = (1 − ∆ P1(t)) + (∆ P2(t)) 6= 0, t ∈ (t0, b]. Hence, existence and uniqueness of solution to the equation (10.2.2) follows from Theorem 10.2.6. R t If P andz ˜ are real, the equation for z2 simpliﬁes to z2(t) = d[P1] z2, whose solution is identically t0 zero and therefore z is real.

10.3 Exponential function

We are now ready to introduce the generalized exponential function. Throughout the rest of this chapter, we work with a ﬁxed compact interval [a, b] ⊂ R. The whole theory still works with other types of bounded intervals (open, half-open), as well as unbounded ones; in the latter case, bounded variation functions should be replaced by functions with locally bounded variation.

Definition 10.3.1. Consider a function P :[a, b] → C, which has bounded variation on [a, b]. Let t0 ∈ [a, b] + − and assume that 1 + ∆ P (t) 6= 0 for every t ∈ [a, t0), and 1 − ∆ P (t) 6= 0 for every t ∈ (t0, b]. Then we define the generalized exponential function t 7→ edP (t, t0), t ∈ [a, b], as the unique solution z :[a, b] → C of the generalized linear differential equation

Z t z(t) = 1 + z(s) dP (s). t0

Note that the exponential is a real-valued function whenever P is real. Also, it is clear that for P (s) = s, t−t0 our definition reduces to the classical exponential function: edP (t, t0) = e . Our notion of the exponential function represents a special case of the fundamental matrix corresponding to a system of generalized linear differential equations; this more general concept has been studied in [67, Chapter 6]. In particular, properties 2 to 6 from the next theorem can be found in [67, Theorem 6.15]. However, the results in the rest of this section are completely new, and make a significant use of the fact that we are dealing with scalar equations only. In each of the following statements, we assume that the function P is such that all exponentials appearing in the given identity are well defined. For example, in the fifth statement, it is necessary to assume that 1 + ∆+P (t) 6= 0 for every t ∈ [a, max(s, r)), and 1 − ∆−P (t) 6= 0 for every t ∈ (min(s, r), b].

Theorem 10.3.2. Let P :[a, b] → C be a function with bounded variation. The generalized exponential function has the following properties:

1. If P is constant, then edP (t, t0) = 1 for every t ∈ [a, b].

2. edP (t, t) = 1 for every t ∈ [a, b].

165 3. The function t 7→ edP (t, t0) is regulated on [a, b] and satisﬁes

+ + ∆ edP (t, t0) = ∆ P (t)edP (t, t0), t ∈ [a, b), − − ∆ edP (t, t0) = ∆ P (t)edP (t, t0), t ∈ (a, b], + edP (t+, t0) = (1 + ∆ P (t))edP (t, t0), t ∈ [a, b), − edP (t−, t0) = (1 − ∆ P (t))edP (t, t0), t ∈ (a, b].

4. The function t 7→ edP (t, t0) has bounded variation on [a, b].

5. edP (t, s)edP (s, r) = edP (t, r) for every t, s, r ∈ [a, b].

−1 6. edP (t, s) = (edP (s, t)) for every t, s ∈ [a, b].

7. edP (t, t0) = edP (t, t0) for every t ∈ [a, b], where z denotes the complex conjugate of z ∈ C.

P (t)−P (t0) 8. If P is continuous, then edP (t, t0) = e for every t ∈ [a, b]. Proof. The first two statements are obvious. The third statement is a consequence of Theorem 10.2.1, and the fourth statement follows from Theorem 10.2.7. To prove the fifth statement, note that, given arbitrary r, s ∈ [a, b], we have Z t Z s Z t edP (t, r) = 1 + edP (τ, r) dP (τ) = 1 + edP (τ, r) dP (τ) + edP (τ, r) dP (τ) r r s Z t = edP (s, r) + edP (τ, r) dP (τ), s for every t ∈ [a, b]. Hence, the function y(t) = edP (t, r), t ∈ [a, b], is a solution of the generalized linear differential equation Z t x(t) =x ˜ + x(s) dP (s), t ∈ [a, b], wherex ˜ = edP (s, r). s

On the other hand, it is not hard to see that z(t) = edP (t, s)˜x, t ∈ [a, b], is also a solution of the same equation. By the uniqueness of solutions (see Theorem 10.2.7), we have y(t) = z(t), t ∈ [a, b], which proves the ﬁfth statement. The sixth statement is a direct consequence of previous one. Indeed, for t, s ∈ [a, b], we obtain

edP (t, s)edP (s, t) = edP (t, t) = 1. By the definition of the exponential function, we have Z t edP (t, t0) = 1 + edP (s, t0) dP (s). t0 Taking the complex conjugate of both sides, we get Z t edP (t, t0) = 1 + edP (s, t0) dP (s), t0 which proves the seventh statement. In the eighth statement, assume first that P is continuously differentiable with P 0 = p. The function z(t) = edP (t, t0) is the unique solution of the equation Z t Z t Z s Z t z(t) = 1 + z(s) dP (s) = 1 + z(s) d P (t0) + p(s) ds = 1 + z(s)p(s) ds t0 t0 t0 t0 (we have used the substitution theorem). Differentiation of the equality above gives z0(t) = z(t)p(t) and R t p(s) ds P (t)−P (t0) the unique solution of this equation satisfying z(t0) = 1 is given by z(t) = e t0 = e .

166 Now, consider the general case when P is merely continuous. Because P has bounded variation, we can write P = P 1 − P 2, where P 1,P 2 are nondecreasing and continuous. For j ∈ {1, 2}, there exists j ∞ j a sequence {Pn}n=1 of nondecreasing polynomials which is uniformly convergent to P . (For example, in the well-known constructive proof of the Weierstrass approximation theorem involving the Bernstein polynomials, the approximating polynomials corresponding to a nondecreasing function are nondecreasing; 1 2 Pn(t)−Pn(t0) see [14, Theorem 6.3.3].) For every n ∈ N, the function Pn = Pn − Pn satisﬁes edPn (t, t0) = e . P (t)−P (t0) Hence, limn→∞ edPn (t, t0) = e . On the other hand, thanks to the monotonicity and uniform 1 2 convergence, the functions Pn and Pn , n ∈ N, have uniformly bounded variations, and the same holds for the functions Pn, n ∈ N. Thus, applying the continuous dependence theorem for generalized linear diﬀerential equations (see [34, Theorem 4.1] or [59, Theorem 3.4]), we have limn→∞ edPn (t, t0) = edP (t, t0), which completes the proof. In the next theorem, we show that the product of two exponential functions again leads to an exponential function.

Theorem 10.3.3. Assume that P,Q :[a, b] → C have bounded variation, (1 + ∆+P (t))(1 + ∆+Q(t)) 6= 0 − − for every t ∈ [a, t0), and (1 − ∆ P (t))(1 − ∆ Q(t)) 6= 0 for every t ∈ (t0, b]. Then

edP (t, t0)edQ(t, t0) = ed(P ⊕Q)(t, t0), t ∈ [a, b], where Z t Z t (P ⊕ Q)(t) = P (t) + Q(t) + ∆+Q(s) dP (s) − ∆−P (s) dQ(s), t0 t0 or equivalently X X (P ⊕ Q)(t) = P (t) + Q(t) + ∆+Q(s)∆+P (s) − ∆−Q(s)∆−P (s).

s∈[t0,t) s∈(t0,t] Proof. By Lemma 10.2.2, we have

Z t Z t X ∆+Q(s) dP (s) − ∆−P (s) dQ(s) = (∆+Q(s)∆P (s) − ∆−P (s)∆Q(s)) t0 t0 s∈[t0,t] X = (∆+Q(s)(∆+P (s) + ∆−P (s)) − ∆−P (s)(∆+Q(s) + ∆−Q(s)))

s∈[t0,t] X = (∆+Q(s)∆+P (s) − ∆−P (s)∆−Q(s)).

s∈[t0,t] (For t > t, all sums of the form P h(s) should be interpreted as − P h(s).) According to our 0 s∈[t0,t] s∈[t,t0] + + − − convention, ∆ Q(t) = ∆ P (t) = 0 and ∆ Q(t0) = ∆ P (t0) = 0. Hence, the two deﬁnitions of P ⊕ Q are equivalent. For t ∈ [a, b], let X X R(t) = ∆−Q(s)∆−P (s) and T (t) = ∆+Q(s)∆+P (s).

s∈(t0,t] s∈[t0,t) These functions have bounded variation and

∆−R(t) = ∆−Q(t)∆−P (t), ∆+R(t) = 0,

∆−T (t) = 0, ∆+T (t) = ∆+Q(t)∆+P (t). In view of this, it is clear that P ⊕ Q has bounded variation on [a, b]. Moreover,

1 − ∆−(P ⊕ Q)(t) = 1 − ∆−P (t) − ∆−Q(t) − ∆−T (t) + ∆−R(t) = 1 − ∆−P (t) − ∆−Q(t) + ∆−Q(t)∆−P (t) (10.3.1) = 1 − ∆−P (t)1 − ∆−Q(t) 6= 0,

167 for t ∈ (t0, b]. Proceeding in a similar way, we can show that

+ + + 1 + ∆ (P ⊕ Q)(t) = 1 + ∆ P (t) 1 + ∆ Q(t) 6= 0, t ∈ [a, t0). (10.3.2)

Therefore, the exponential function t 7→ ed(P ⊕Q)(t, t0) is well deﬁned.

For t ∈ [a, b], integration by parts gives

Z t Z t edP (t, t0)edQ(t, t0) = edP (t0, t0)edQ(t0, t0) + edP (s, t0) d[edQ(s, t0)] + edQ(s, t0) d[edP (s, t0)] t0 t0

X + − + (∆edP (s, t0)∆ edQ(s, t0) − ∆ edP (s, t0)∆edQ(s, t0)).

s∈[t0,t] Using the substitution theorem, we have

Z t Z t Z s Z t edP (s, t0) d[edQ(s, t0)] = edP (s, t0) d 1 + edQ(u, t0) dQ(u) = edP (s, t0)edQ(s, t0) dQ(s), t0 t0 t0 t0

Z t Z t Z s Z t edQ(s, t0) d[edP (s, t0)] = edQ(s, t0) d 1 + edP (u, t0) dP (u) = edQ(s, t0)edP (s, t0) dP (s). t0 t0 t0 t0 Also, Lemma 10.2.2, Theorem 10.3.2 and the substitution theorem imply

X + − (∆edP (s, t0)∆ edQ(s, t0) − ∆ edP (s, t0)∆edQ(s, t0))

s∈[t0,t]

Z t Z t + − = ∆ edQ(s, t0) d[edP (s, t0)] − ∆ edP (s, t0) d[edQ(s, t0)] t0 t0

Z t Z t + − = ∆ Q(s)edQ(s, t0)edP (s, t0) dP (s) − ∆ P (s)edP (s, t0)edQ(s, t0) dQ(s) t0 t0

Z t Z s Z s + − = edQ(s, t0)edP (s, t0) d ∆ Q(u) dP (u) − ∆ P (u) dQ(u) . t0 t0 t0 By combining the previous results, we obtain

Z t Z s Z s + − edP (t, t0)edQ(t, t0) = 1+ edP (s, t0)edQ(s, t0) d P (s) + Q(s) + ∆ Q(u) dP (u) − ∆ P (u) dQ(u) , t0 t0 t0 which proves the relation edP (t, t0)edQ(t, t0) = ed(P ⊕Q)(t, t0).

Similarly, the reciprocal value of an exponential function is again an exponential function.

Theorem 10.3.4. Assume that P :[a, b] → C has bounded variation, 1 + ∆+P (t) 6= 0 for every t ∈ [a, b), and 1 − ∆−P (t) 6= 0 for every t ∈ (a, b]. Then

−1 (edP (t, t0)) = ed( P )(t, t0), t ∈ [a, b], where X (∆+P (s))2 X (∆−P (s))2 ( P )(t) = −P (t) + − . 1 + ∆+P (s) 1 − ∆−P (s) s∈[t0,t) s∈(t0,t]

168 Proof. For t ∈ [a, b], let

X (∆+P (s))2 X (∆−P (s))2 R (t) = ,R (t) = . 1 1 + ∆+P (s) 2 1 − ∆−P (s) s∈[t0,t) s∈(t0,t]

These functions have bounded variation on [a, b] and satisfy

(∆+P (t))2 (∆−P (t))2 ∆−R (t) = 0, ∆+R (t) = , ∆+R (t) = 0, ∆−R (t) = . (10.3.3) 1 1 1 + ∆+P (t) 2 2 1 − ∆−P (t) Thus, P has bounded variation on [a, b] and

(∆+P (t))2 1 1 + ∆+( P )(t) = 1 − ∆+P (t) + = 6= 0, t ∈ [a, t ), (10.3.4) 1 + ∆+P (t) 1 + ∆+P (t) 0

(∆−P (t))2 1 1 − ∆−( P )(t) = 1 + ∆−P (t) + = 6= 0, t ∈ (t , b], (10.3.5) 1 − ∆−P (t) 1 − ∆−P (t) 0 which implies that the exponential function ed( P ) is well deﬁned.

Using the relations (10.3.3) together with the deﬁnition of ⊕ given in Theorem 10.3.3, we obtain

X + + (P ⊕ ( P ))(t) = P (t) − P (t) + R1(t) − R2(t) + ∆ (−P + R1 − R2)(s)∆ P (s)

s∈[t0,t)

X − − − ∆ (−P + R1 − R2)(s)∆ P (s) = R1(t) − R2(t)

s∈(t0,t] X (∆+P (s))3 X (∆−P (s))3 + −(∆+P (s))2 + − −(∆−P (s))2 − 1 + ∆+P (s) 1 − ∆−P (s) s∈[t0,t) s∈(t0,t] X (∆+P (s))2 X (∆−P (s))2 = R (t) − R (t) − + = 0. 1 2 1 + ∆+P (s) 1 − ∆−P (s) s∈[t0,t) s∈(t0,t]

Now, it follows from Theorems 10.3.3 and 10.3.2 that edP (t, t0)ed( P )(t, t0) = ed(P ⊕( P ))(t, t0) = 1.

The preceding two theorems have the following algebraic interpretation: Let BV∗([a, b], R) be the class consisting of all functions P :[a, b] → R that have bounded variation on [a, b] and satisfy 1 + ∆+P (t) 6= 0 − for every t ∈ [a, b), and 1 − ∆ P (t) 6= 0 for every t ∈ (a, b]. Given a pair of functions P,Q ∈ BV∗([a, b], R), write P ∼ Q if and only if P − Q is a constant function. Clearly, the relation ∼ is an equivalence. Now, the quotient set BV∗([a, b], R)/ ∼ is a commutative group equipped with the binary operation ⊕. The zero element of this group is the class of all constant functions, and inverse elements are obtained using the operation. Moreover, it is not diﬃcult to verify that

((P ⊕ Q) ⊕ R)(t) = (P ⊕ (Q ⊕ R))(t) = P (t) + Q(t) + R(t) X X + ∆+Q(s)∆+P (s) + ∆+R(s)∆+P (s) + ∆+Q(s)∆+R(s) + ∆+P (s)∆+Q(s)∆+R(s)

s∈[t0,t) s∈[t0,t) X X − ∆−Q(s)∆−P (s) + ∆−R(s)∆−P (s) + ∆−Q(s)∆−R(s) + ∆−P (s)∆−Q(s)∆−R(s),

s∈(t0,t] s∈(t0,t] i.e., the operation ⊕ is associative. The next theorem provides some information about the sign of the exponential function.

Theorem 10.3.5. Consider a function P :[a, b] → R, which has bounded variation on [a, b] and satisﬁes + − 1 + ∆ P (t) 6= 0 for every t ∈ [a, b), and 1 − ∆ P (t) 6= 0 for every t ∈ (a, b]. Then, for every t0 ∈ [a, b], the following statements hold:

169 1. edP (t, t0) 6= 0 for all t ∈ [a, b].

+ 2. If 1 + ∆ P (t) < 0, then edP (t, t0)edP (t+, t0) < 0.

− 3. If 1 − ∆ P (t) < 0, then edP (t, t0)edP (t−, t0) < 0.

+ − 4. If 1 + ∆ P (t) > 0 and 1 − ∆ P (t) > 0, then t 7→ edP (t, t0) does not change sign in the neighborhood of t.

Proof. If edP (t, t0) = 0 for a certain t ∈ [a, b], we can use Theorem 10.3.2 to obtain 1 = edP (t0, t0) = edP (t0, t)edP (t, t0) = 0, which is a contradiction. The second and third statement follow immediately from + − the third part of Theorem 10.3.2. Finally, if 1 + ∆ P (t) > 0 and 1 − ∆ P (t) > 0, then edP (t+, t0) and edP (t−, t0) have the same sign as edP (t, t0), which proves the fourth statement. According to the previous theorem, the exponential function changes sign at all points t such that 1 + ∆+P (t) < 0 or 1 − ∆−P (t) < 0. Since P has bounded variation, we conclude that every ﬁnite interval can contain only a ﬁnite number of points where the exponential function changes its sign. In view of Theorem 10.3.5, it makes sense to introduce the class BV+([a, b], R) consisting of all functions P :[a, b] → R that have bounded variation on [a, b] and satisfy 1 + ∆+P (t) > 0 for every t ∈ [a, b), and 1 − ∆−P (t) > 0 for every t ∈ (a, b].

Theorem 10.3.6. The elements of BV+([a, b], R) have the following properties:

1. If P ∈ BV+([a, b], R), then edP (t, t0) > 0 for all t, t0 ∈ [a, b].

2. If P,Q ∈ BV+([a, b], R), then P ⊕ Q ∈ BV+([a, b], R).

3. If P ∈ BV+([a, b], R), then P ∈ BV+([a, b], R).

Proof. The ﬁrst statement follows from the fourth part of Theorem 10.3.5 and the fact that edP (t0, t0) is positive. The second statement is a consequence of the formulas (10.3.1) and (10.3.2), which were obtained in the proof of Theorem 10.3.3:

1 − ∆−(P ⊕ Q)(t) = 1 − ∆−P (t)1 − ∆−Q(t),

1 + ∆+(P ⊕ Q)(t) = 1 + ∆+P (t)1 + ∆+Q(t), Similarly, the third statement is a consequence of the formulas (10.3.4) and (10.3.5) from the proof of Theorem 10.3.4: 1 1 1 + ∆+( P )(t) = , 1 − ∆−( P )(t) = . 1 + ∆+P (t) 1 − ∆−P (t)

According to the previous theorem, the set BV+([a, b], R)/ ∼ is a subgroup of BV∗([a, b], R)/ ∼.

10.4 Hyperbolic and trigonometric functions

Deﬁnition 10.4.1. Consider a function P :[a, b] → C, which has bounded variation on [a, b]. Let t0 ∈ [a, b] + 2 − 2 and assume that 1 − (∆ P (t)) 6= 0 for every t ∈ [a, t0), and 1 − (∆ P (t)) 6= 0 for every t ∈ (t0, b]. Then we deﬁne the hyperbolic functions t 7→ coshdP (t, t0) and t 7→ sinhdP (t, t0), t ∈ [a, b], by the formulas

edP (t, t0) + e (t, t0) edP (t, t0) − e (t, t0) cosh (t, t ) = d(−P ) , sinh (t, t ) = d(−P ) . dP 0 2 dP 0 2 Note that the condition 1 − (∆+P (t))2 6= 0 is equivalent to (1 + ∆+P (t))(1 + ∆+(−P )(t)) 6= 0, and − 2 − − 1 − (∆ P (t)) 6= 0 is equivalent to (1 − ∆ P (t))(1 − ∆ (−P )(t)) 6= 0. Therefore, edP and ed(−P ) are well deﬁned. Obviously, the two hyperbolic functions are real if P is real, and for P (s) = s, we obtain the classical hyperbolic functions: coshdP (t, t0) = cosh(t − t0), sinhdP (t, t0) = sinh(t − t0). More generally, if P is continuous, then coshdP (t, t0) = cosh(P (t) − P (t0)) and sinhdP (t, t0) = sinh(P (t) − P (t0)).

170 Theorem 10.4.2. Let P :[a, b] → C be a bounded variation function satisfying 1 − (∆+P (t))2 6= 0 for − 2 every t ∈ [a, t0), and 1 − (∆ P (t)) 6= 0 for every t ∈ (t0, b]. The generalized hyperbolic functions have the following properties:

1. coshdP (t0, t0) = 1, sinhdP (t0, t0) = 0. R t 2. coshdP (t, t0) = 1 + sinhdP (s, t0) dP (s), t ∈ [a, b]. t0 R t 3. sinhdP (t, t0) = coshdP (s, t0) dP (s), t ∈ [a, b]. t0

2 2 4. coshdP (t, t0) − sinhdP (t, t0) = edQ(t, t0), t ∈ [a, b], where

Z t X X Q(t) = (P ⊕ (−P ))(t) = (∆−P (s) − ∆+P (s)) dP (s) = (∆−P (s))2 − (∆+P (s))2. t0 s∈(t0,t] s∈[t0,t)

Proof. The ﬁrst statement is obvious. Using the deﬁnition of the exponential function, we obtain

1 Z t Z t coshdP (t, t0) = 1 + edP (s, t0) dP (s) + 1 + ed(−P )(s, t0) d(−P )(s) 2 t0 t0

1 Z t Z t = 1 + (edP (s, t0) − ed(−P )(s, t0)) dP (s) = 1 + sinhdP (s, t0) dP (s). 2 t0 t0 The third identity can be proved similarly. To verify the fourth one, observe that

 2 2 edP (t, t0) + e (t, t0) edP (t, t0) − e (t, t0) cosh2 (t, t ) − sinh2 (t, t ) = d(−P ) − d(−P ) dP 0 dP 0 2 2

= edP (t, t0)ed(−P )(t, t0) = ed(P ⊕(−P ))(t, t0). From Theorem 10.3.3, we have

Z t Z t X X (P ⊕ (−P ))(t) = − ∆+P (s) dP (s) + ∆−P (s) dP (s) = − (∆+P (s))2 + (∆−P (s))2. t0 t0 s∈[t0,t) s∈(t0,t]

Deﬁnition 10.4.3. Consider a function P :[a, b] → C, which has bounded variation on [a, b]. Let t0 ∈ [a, b] + 2 − 2 and assume that 1 + (∆ P (t)) 6= 0 for every t ∈ [a, t0), and 1 + (∆ P (t)) 6= 0 for every t ∈ (t0, b]. Then we deﬁne the generalized trigonometric functions t 7→ cosdP (t, t0) and t 7→ sindP (t, t0), t ∈ [a, b], by the formulas

e (t, t0) + e (t, t0) cos (t, t ) = d(iP ) d(−iP ) = cosh (t, t ), dP 0 2 d(iP ) 0 e (t, t0) − e (t, t0) sin (t, t ) = d(iP ) d(−iP ) = −i sinh (t, t ). dP 0 2i d(iP ) 0 Note that the condition 1 + (∆+P (t))2 6= 0 is equivalent to (1 + ∆+(iP )(t))(1 + ∆+(−iP )(t)) 6= 0, and − 2 − − 1 + (∆ P (t)) 6= 0 is equivalent to (1 − ∆ (iP )(t))(1 − ∆ (−iP )(t)) 6= 0. Therefore, ed(iP ) and ed(−iP ) are well defined. When P is a real function, both conditions are always satisfied. Again, it is easy to see that for P (s) = s, our definitions coincide with the classical trigonometric functions: cosdP (t, t0) = cos(t − t0), sindP (t, t0) = sin(t − t0). Also, if P is continuous, then cosdP (t, t0) = cos(P (t) − P (t0)) and sindP (t, t0) = sin(P (t) − P (t0)). If P is real, the trigonometric functions are real as well: By the seventh statement of Theorem 10.3.2, ed(iP ) + ed(−iP ) = ed(iP ) + ed(iP ), which is purely real. Similarly, ed(iP ) − ed(−iP ) = ed(iP ) − ed(iP ), which is purely imaginary.

171 Theorem 10.4.4. Let P :[a, b] → C be a bounded variation function satisfying 1 + (∆+P (t))2 6= 0 for − 2 every t ∈ [a, t0), and 1 + (∆ P (t)) 6= 0 for every t ∈ (t0, b]. The generalized trigonometric functions have the following properties:

1. cosdP (t0, t0) = 1, sindP (t0, t0) = 0. R t 2. cosdP (t, t0) = 1 − sindP (s, t0) dP (s), t ∈ [a, b]. t0 R t 3. sindP (t, t0) = cosdP (s, t0) dP (s), t ∈ [a, b]. t0 2 2 4. cosdP (t, t0) + sindP (t, t0) = edQ(t, t0), t ∈ [a, b], where Z t X X Q(t) = (iP ⊕ (−iP ))(t) = (∆+P (s) − ∆−P (s)) dP (s) = (∆+P (s))2 − (∆−P (s))2. t0 s∈[t0,t) s∈(t0,t]

Proof. The ﬁrst statement is obvious. Using the deﬁnition of the exponential function, we get 1 Z t Z t cosdP (t, t0) = 1 + ed(iP )(s, t0) d[iP (s)] + 1 + ed(−iP )(s, t0) d[−iP (s)] 2 t0 t0 i Z t = 1 + (ed(iP )(s, t0) − ed(−iP )(s, t0)) dP (s) 2 t0 i Z t Z t = 1 + (2i sindP (s, t0)) dP (s) = 1 − sindP (s, t0) dP (s). 2 t0 t0 Similarly, we can obtain the third identity. To prove the last identity, we observe that 2 2 e (t, t0) + e (t, t0) e (t, t0) − e (t, t0) cos2 (t, t ) + sin2 (t, t ) = d(iP ) d(−iP ) + d(iP ) d(−iP ) dP 0 dP 0 2 2i

= ed(iP )(t, t0) ed(−iP )(t, t0) = ed(iP ⊕(−iP ))(t, t0). From Theorem 10.3.3, we have

Z t Z t X X (iP ⊕ (−iP ))(t) = ∆+P (s) dP (s) − ∆−P (s) dP (s) = (∆+P (s))2 − (∆−P (s))2. t0 t0 s∈[t0,t) s∈(t0,t]

10.5 Time scale elementary functions

In this section, we demonstrate that the deﬁnitions and properties of the elementary functions on time scales correspond to special cases of our earlier results. We assume some basic familiarity with the time scale calculus; in particular, we need the concepts of the ∆-derivative and ∆-integral (in the sense of Lebesgue and Henstock-Kurzweil), as well as the deﬁnitions of the elementary functions; see [7, 9, 38, 63].

Let T be a time scale. We use the symbol [a, b]T to denote the interval [a, b] ∩ T. Consider a point t0 ∈ [a, b]T and an rd-continuous function p :[a, b]T → R such that 1 + µ(t)p(t) 6= 0 for all t ∈ [a, t0)T. The time scale exponential function t 7→ ep(t, t0) is usually deﬁned as the unique solution of the initial-value problem

∆ x (t) = p(t)x(t), t ∈ [a, b]T, x(t0) = 1. (There exist alternative deﬁnitions of the exponential function; see [12] for a nice overview.) For our purposes, it is more convenient to work with the equivalent integral form Z t x(t) = 1 + p(s)x(s)∆s, t ∈ [a, b]T. t0

172 At this point, we need the relationship between the ∆-integral and Kurzweil-Stieltjes integral, which was described in Chapter 5. Given a real number t ≤ sup T, let t∗ = inf{s ∈ T; s ≥ t}. Since T is a closed ∗ ∗ set, we have t ∈ T. Further, given a function f :[a, b]T → R, we consider its extension f :[a, b] → R given by f ∗(t) = f(t∗). The next statement combines Theorems 5.4.2 and 5.4.5.

∗ Theorem 10.5.1. Let f :[a, b]T → R be an arbitrary function. Deﬁne g(s) = s for every s ∈ [a, b]. Then R b R b ∗ the Kurzweil-Henstock ∆-integral a f(t)∆t exists if and only if the Kurzweil-Stieltjes integral a f (t) dg(t) exists; in this case, both integrals have the same value. Moreover, if Z t F1(t) = f(s)∆s, t ∈ [a, b]T, a Z t ∗ F2(t) = f (s) dg(s), t ∈ [a, b], a ∗ then F2(t) = F1 (t) for every t ∈ [a, b].

R b ∗ ∗ It is useful to note that the value of a f dg does not change if we replace f by a diﬀerent function which coincides with f on [a, b]T. This is the content of Theorem 4.4.2, which is repeated here for reader’s convenience.

∗ Theorem 10.5.2. Let g(s) = s for every s ∈ [a, b] and consider a pair of functions f1, f2 :[a, b] → R R b R b such that f1(t) = f2(t) for every t ∈ [a, b]T. If a f1 dg exists, then a f2 dg exists as well and both integrals have the same value.

Lemma 10.5.3. Let p :[a, b]T → R be a Lebesgue ∆-integrable function satisfying 1 + µ(t)p(t) 6= 0 for every t ∈ [a, t0)T. Then, the function Z t P (t) = p∗(s) dg(s), t ∈ [a, b], t0 where g(s) = s∗ for every s ∈ [a, b], has the following properties: 1. P has bounded variation on [a, b]. 2. P is left-continuous at all points t ∈ (a, b].

+ 3. P is right-continuous at all points t ∈ [a, b) \ T, ∆ P (t) = p(t)µ(t) for every t ∈ [a, b)T, and + 1 + ∆ P (t) 6= 0 for every t ∈ [a, t0). Proof. The Lebesgue ∆-integrability of p implies that both p and |p| are Henstock-Kurzweil ∆-integrable R b ∗ [63, Theorem 2.19]. By Theorem 10.5.1, the Kurzweil-Stieltjes integral a |p (s)| dg(s) exists. If a = τ0 < τ1 < ··· < τm = b is an arbitrary partition of [a, b], then

m m Z τi Z b X X ∗ ∗ |P (τi) − P (τi−1)| ≤ |p (s)| dg(s) = |p (s)| dg(s), i=1 i=1 τi−1 a which proves the ﬁrst statement. The second and third statement follow from Theorem 10.2.1, because g is left-continuous on (a, b], right-continuous on [a, b) \ T, and ∆+P (t) = p(t)∆+g(t) = p(t)µ(t) for t ∈ [a, b)T. We now return back to the exponential functions.

Theorem 10.5.4. Let p :[a, b]T → R be a Lebesgue ∆-integrable function satisfying 1 + µ(t)p(t) 6= 0 for every t ∈ [a, t0)T. Then, there exists a unique solution of the equation Z t x(t) = 1 + p(s)x(s)∆s, t ∈ [a, b]T. (10.5.1) t0

R t ∗ ∗ The solution is given by x(t) = edP (t, t0), t ∈ [a, b] , where P (t) = p (s) dg(s) and g(s) = s . T t0

173 Proof. As a consequence of Theorem 10.5.1, we see that Eq. (10.5.1) is equivalent to

Z t x∗(t) = 1 + p∗(s)x∗(s) dg(s), t ∈ [a, b]. t0 By the substitution theorem, the last equation is equivalent to the generalized linear diﬀerential equation

Z t x∗(t) = 1 + x∗(s) dP (s), t ∈ [a, b]. t0 According to Lemma 10.5.3, the function P satisﬁes the assumptions of the existence and uniqueness ∗ theorem for generalized linear diﬀerential equations, and we conclude that x (t) = edP (t, t0), t ∈ [a, b]. By the previous theorem, we have the following relationship between the two exponential functions:

ep(t, t0) = edP (t, t0), t ∈ [a, b]T. Moreover, the theorem shows that the time scale exponential function, deﬁned as the unique solution of Eq. (10.5.1), makes a perfect sense for functions p which are not rd-continuous but merely Lebesgue ∆-integrable. The next theorem lists some well-known properties of the time scale exponential function; the proof for the case when p is rd-continuous can be found in [8, Theorem 2.36]. Our goal is to show that these statements follow from the results in Section 3, and hence the properties of the exponential function are preserved for Lebesgue ∆-integrable functions p. In the statements below, we use the convention that σ(t) = t and µ(t) = 0 at the right endpoint t = b. Also, we assume that all exponentials appearing in the identities are well deﬁned.

Theorem 10.5.5. Let p :[a, b]T → R be a Lebesgue ∆-integrable function. The time scale exponential function has the following properties:

1. e0(t, t0) = 1 for every t ∈ [a, b]T.

2. ep(t, t) = 1 for every t ∈ [a, b]T.

3. ep(σ(t), t0) = (1 + µ(t)p(t))ep(t, t0) for every t ∈ [a, b]T.

4. ep(t, t0)eq(t, t0) = er(t, t0) for every t ∈ [a, b]T, where r(t) = p(t) + q(t) + µ(t)p(t)q(t). −1 −1 5. ep(t, t0) = eu(t, t0) for every t ∈ [a, b]T, where u(t) = −p(t)(1 + p(t)µ(t)) .

Proof. The ﬁrst two statements are obvious. To prove the third statement, observe that, for t ∈ [a, b]T,

+ ep(σ(t), t0) = edP (σ(t), t0) = edP (t+, t0) = (1 + ∆ P (t))edP (t, t0) = (1 + µ(t)p(t))ep(t, t0).

In the fourth statement, we have

ep(t, t0)eq(t, t0) = edP (t, t0)edQ(t, t0) = ed(P ⊕Q)(t, t0), where P (t) = R t p∗(s) dg(s) and Q(t) = R t q∗(s) dg(s). Hence, the formula for P ⊕ Q reduces to t0 t0 Z t Z t (P ⊕ Q)(t) = P (t) + Q(t) + ∆+Q(s) dP (s) = P (t) + Q(t) + ∆+Q(s)p∗(s) dg(s) t0 t0 Z t Z t Z t = (p∗(s)+q∗(s)+q∗(s)∆+g(s)p∗(s)) dg(s) = (p∗(s)+q∗(s)+q∗(s)p∗(s)µ∗(s)) dg(s) = r∗(s) dg(s). t0 t0 t0 In the next-to-last integral, we have replaced by ∆+g(s) by ∆+g(s∗) = µ∗(s), which is correct thanks to Theorem 10.5.2. Now, it follows that ed(P ⊕Q)(t, t0) = er(t, t0).

174 Finally, we have −1 −1 ep(t, t0) = edP (t, t0) = ed( P )(t, t0), Using the formula for P together with Lemma 10.2.2, we get

X (∆+P (s))2 Z t ∆+P (s) ( P )(t) = −P (t) + + = −P (t) + + dP (s) 1 + ∆ P (s) t0 1 + ∆ P (s) s∈[t0,t)

Z t Z t ∗ ∗ Z t Z t ∗ 2 ∗ ∗ p (s)µ (s) ∗ p (s) µ (s) = − p (s) dg(s) + ∗ ∗ dP (s) = − p (s) dg(s) + ∗ ∗ dg(s) t0 t0 1 + p (s)µ (s) t0 t0 1 + p (s)µ (s) Z t ∗ ∗ Z t ∗ Z t ∗ p (s)µ (s) p (s) ∗ = − p (s) 1 − ∗ ∗ dg(s) = − ∗ ∗ dg(s) = u dg(s), t0 1 + p (s)µ (s) t0 1 + p (s)µ (s) t0 and hence ed( P )(t, t0) = eu(t, t0), t ∈ [a, b]T. Remark 10.5.6. In the time scale literature, the functions r and u appearing in the fourth and ﬁfth statement of the previous theorem are usually denoted by p ⊕ q and p. To avoid confusion, we emphasize that in our chapter, the symbols ⊕ and have a diﬀerent meaning.

Before we finish our discussion of the time scale exponential function, we remark that Theorems 10.3.5 and 10.3.6 might be used to derive some information about the sign of ep(t, t0); we leave it up to the reader to check that these results are in agreement with Theorems 2.44, 2.48 and Lemma 2.47 from [8]. We now proceed to the time scale hyperbolic functions, which are defined in a natural way using the exponential function (cf. [8, Definition 3.17]). Hence, we immediately see their relation to our generalized hyperbolic functions: e + e edP + e cosh = p −p = d(−P ) = cosh , p 2 2 dP

e − e edP − e sinh = p −p = d(−P ) = sinh , p 2 2 dP where P (t) = R t p∗(s) dg(s). We already know that the right-hand sides are defined only if P satisfies the t0 − 2 + 2 conditions 1 − (∆ P (t)) 6= 0, t ∈ (t0, b], and 1 − (∆ P (t)) 6= 0, t ∈ [a, t0). For our function P , the first 2 2 condition is always satisfied, while the second reduces to 1 − p(t) µ(t) 6= 0, t ∈ [a, t0)T; this coincides with the condition in [8, Definition 3.17]. When defining the time scale hyperbolic functions, it is usually assumed that p is a rd-continuous function. However, it follows from our discussion concerning the exponential function that it is enough if p is Lebesgue ∆-integrable. The next theorem summarizes some basic facts about the hyperbolic functions; in a slightly different form, the statement can be found in [8, Lemma 3.18]. We show that these identities are consequences of the results from Section 3, and hence the properties of the hyperbolic functions are still valid for Lebesgue ∆-integrable functions p.

2 2 Theorem 10.5.7. Let p :[a, b]T → R be a Lebesgue ∆-integrable function satisfying 1 − p(t) µ(t) 6= 0, t ∈ [a, t0)T. The time scale hyperbolic functions have the following properties: R t 1. coshp(t, t0) = 1 + p(s) sinhp(s, t0) ∆s, t ∈ [a, b] . t0 T R t 2. sinhp(t, t0) = p(s) coshp(s, t0) ∆s, t ∈ [a, b] . t0 T

2 2 3. coshp(t, t0) − sinhp(t, t0) = e−p2µ(t, t0), t ∈ [a, b]T. Proof. The ﬁrst and second statement follow from Theorems 10.4.2 and 10.5.1. The third identity can be obtained as follows:

2 2 2 2 coshp(t, t0) − sinhp(t, t0) = coshdP (t, t0) − sinhdP (t, t0) = edQ(t, t0),

175 where Z t Z t Q(t) = − ∆+P (s) dP (s) = − p∗(s)∆+g(s)p∗(s) dg(s) t0 t0 Z t Z t = − p∗(s)2∆+g(s∗) dg(s) = − p∗(s)2µ∗(s) dg(s). t0 t0

It follows that edQ(t, t0) = e−p2µ(t, t0), t ∈ [a, b]T. Finally, we have the following relation between the time scale trigonometric functions (see [8, Deﬁni- tion 3.25]) and our generalized trigonometric functions:

e + e e + e cos = ip −ip = d(iP ) d(−iP ) = cos , p 2 2 dP e − e e − e sin = ip −ip = d(iP ) d(−iP ) = sin , p 2i 2i dP R t ∗ + 2 − 2 where P (t) = p (s) dg(s). The conditions 1 + (∆ P (t)) 6= 0, t ∈ [a, t0), and 1 + (∆ P (t)) 6= 0, t0 t ∈ (t0, b] from the deﬁnition of the generalized trigonometric functions reduce to the single condition 2 2 1 + p(t) µ(t) 6= 0, t ∈ [a, t0)T, which agrees with [8, Deﬁnition 3.25]. The following theorem summarizes the basic properties of these functions (cf. [8, Lemma 3.26]). We leave it up to the reader to verify that the identities are immediate consequences of Theorems 10.4.4 and 10.5.1. Again, these formulas hold in the general case when p is Lebesgue ∆-integrable.

2 2 Theorem 10.5.8. Let p :[a, b]T → R be a Lebesgue ∆-integrable function satisfying 1 + p(t) µ(t) 6= 0, t ∈ [a, t0)T. The time scale trigonometric functions have the following properties: R t 1. cosp(t, t0) = 1 + p(s) sinp(s, t0) ∆s, t ∈ [a, b] . t0 T R t 2. sinp(t, t0) = p(s) cosp(s, t0) ∆s, t ∈ [a, b] . t0 T

2 2 3. cosp(t, t0) − sinp(t, t0) = ep2µ(t, t0), t ∈ [a, b]T.

10.6 Conclusion

Our definition of the exponential function is a non-constructive one as it relies on the existence-uniqueness theorem for generalized linear differential equations. A different approach could be based on product integration theory (see [67, Chapter 7]), which expresses the solution of a generalized linear equation as a limit of certain products. Various authors have extended the definitions of the time scale elementary functions to matrix-valued arguments (see e.g. [8, Chapter 5], [44], [80]). Similarly, given a matrix-valued function P :[a, b] → Rn×n, it is natural to define the generalized exponential function t 7→ edP (t, t0) as the unique solution of the generalized linear differential equation

Z t Z(t) = I + d[P ] Z, t ∈ [a, b], t0 where I is the identity matrix. According to Theorem 10.2.6, it is enough to assume that A has bounded + − variation on [a, b], I + ∆ A(t) is invertible for every t ∈ [a, t0), and I − ∆ A(t) is invertible for every t ∈ (t0, b]. In fact, this deﬁnition of the exponential function coincides with the notion of a fundamental matrix; some of its properties can be found in [67, Theorem 6.15]. Unfortunately, the behavior of this matrix-valued exponential function is not as nice as in the scalar case. For example, to obtain an analogue of Theorem 10.3.3, it is necessary to impose certain commutativity conditions on P and Q. With this in mind, we have restricted our attention to scalar functions only.

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