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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 Tvrd´y,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, M´arciaFederson, 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 Beˇcv´aˇr;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 2 [ti−1; ti], i 2 f1; : : : ; kg, is called δ-fine if [ti−1; ti] ⊂ (τi − δ(τi); τi + δ(τi)); i 2 f1; : : : ; kg: 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 2 X such that for every " > 0, there is a function δ :[a; b] ! R+ such that k X (U(τi; ti) − U(τi; ti−1)) − I < " 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 2 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 ! : [0; 1) ! R is a continuous increasing function with !(0) = 0. The class F(G; h1; h2;!) consists of all functions F : G ! X satisfying the following conditions: kF (x; t2) − F (x; t1)k ≤ h1(t2) − h1(t1); x 2 B; [t1; t2] ⊂ I; kF (x; t2) − F (x; t1) − F (y; t2) + F (y; t1)k ≤ !(kx − yk)(h2(t2) − h2(t1)); x; y 2 B; [t1; t2] ⊂ I: For the rest of the thesis, let us make the following agreement: Whenever we write F 2 F(G; h1; h2;!), we assume that h1, h2, ! 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; !) instead of F(G; h1; h2;!). 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 2 F(G; h1; h2;!), while the particular values of h1 and h2 play no role.
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