Existence and uniqueness of solutions for linear Fredholm-Stieltjes integral equations via Henstock-Kurzweil integral M. Federson∗ and R. Bianconi† Abstract We consider the linear Fredholm-Stieltjes integral equation Z b x (t) − α (t, s) x (s) dg(s) = f (t) , t ∈ [a, b] , a in the frame of the Henstock-Kurzweil integral and we prove a Fredholm Alternative type result for this equation. The functions α, x and f are real-valued with x and f being Henstock-Kurzweil integrable. Keywords and phrases: Linear integral equations, Fredholm integral equations, Fredholm Alternative, Henstock-Kurzweil integral. 2000 Mathematical Subject Classification: 45A05, 45B05, 45C05, 26A39. 1 Introduction The purpose of this paper is to prove a Fredholm Alternative type result for the following linear Fredholm-Stieltjes integral equation Z b x (t) − α (t, s) x (s) dg(s) = f (t) , t ∈ [a, b] , (1) a in the frame of the Henstock-Kurzweil integral. Let g :[a, b] → R be an element of a certain subspace of the space of continuous functions from [a, b] to R. Let Kg([a, b], R) denote the space of all functions f :[a, b] → R such that ∗Instituto de CiˆenciasMatem´aticase de Computa¸c˜ao,Universidade de S˜aoPaulo, CP 688, S˜aoCarlos, SP 13560-970, Brazil. E-mail: [email protected] †Instituto de Matem´atica e Estatstica, Universidade de S˜aoPaulo, CP 66281, S˜aoPaulo, SP 05315-970, Brazil. E-mail: [email protected] 1 R b the integral a f(s)dg(s) exists in the Henstock-Kurzweil sense. It is known that even when g(s) = s, an element of Kg([a, b], R) can have not only many points of discontinuity, but it can also be of unbounded variation. The Henstock-Kurzweil integral encompasses the Lebesgue, Riemann and Newton integrals as well as its improper integrals. When (1) is considered in the sense of Henstock-Kurzweil, f can have singularities, for instance, and yet it may be possible to find a solution xf in Kg([a, b], R). In [4], the authors considered the linear Fredholm integral equation Z b x (t) − α (t, s) x (s) ds = f (t) , t ∈ [a, b] , a and established a Fredholm Alternative for such equation in the frame of the Henstock- Kurzweil integral (see [4], Theorem 3.10). On that occasion, the functions α, x and f were considered as being Banach-space valued and x and f were Kurzweil integrable. The latter could also be considered in any of the spaces of Henstock, Bochner or improper Riemann integrable functions (see [4], Remarks 3.2 and 3.3). Also, an example was given showing that, under the hypothesis we considered, the existing results were not enough. Suppose the function α satisfies certain conditions. In the present paper, we prove that: • if for every f ∈ Kg([a, b], R), equation (1) has a unique solution xf ∈ Kg([a, b], R), then the application f 7→ xf is bicontinuous and equation (1) has a resolvent with similar integral representation; if besides α(t, s) = 0 for m-almost every s > t, then f 7→ xf is causal and the resolvent of (1) is given by the Neumann series (see Theorem 5.1 in the sequel). We also state the following Fredholm Alternative for equation (1): • for every f ∈ Kg([a, b], R), either equation (1) has a unique solution xf ∈ Kg([a, b], R) and (1) has a resolvent with similar integral representation, or the corresponding ho- mogeneous equation admits non-trivial solutions in Kg([a, b], R) (see Theorem 5.2 in the sequel). Corollary 5.1 in the sequel mixes the two results. Although the above results are proved in the case where α, x and f are real-valued, the auxiliary theory developed throughout the paper is placed in a general abstract context. As in [6], the reason for considering functions taking values in a finite dimensional space is because some results of the Kurzweil-Henstock integration theory either hold for one integral or for the other. For instance, in a Banach-space valued context, the spaces of Henstock and of Kurzweil integrals may not coincide; a Riemann integrable function may not be Henstock integrable; the Fundamental Theorem of Calculus may fail for the Kurzweil integral. See [7], [8] and [3] for examples. See also [18]. Another obstacle we encountered is the fact that the normed space of Henstock-Kurzweil integrable functions is not complete (it is ultrabornological though). Therefore one can not 2 apply usual fixed point theorems in order to get existence results, for instance. This difficulty was also faced by the authors in [4] and in [6] and by H¨onig in [11], but it can be overcome when some ideas due to H¨onigare applied. Such ideas concern applying representation theorems, an integration by parts formula and the fundamental theorem of calculus in order to transform integral equations in the sense of the Henstock-Kurzweil integral into integral equations in the sense of the Riemann-Stieltjes integral. As we did in [6] for a Volterra-Stieltjes type integral equation, we adapt, in the present paper, the ideas mentioned above to the Stieltjes case, that is, by means of representation theorems, integration by parts and substitution formulas and the fundamental theorem of calculus for Henstock-Kurzweil vector integrals, we transform the Fredholm-Stieltjes integral equation in the sense of Henstock-Kurzweil integral into an ordinary Fredholm-Stieltjes in- tegral equation. In particular, we apply the Fredholm Alternative for the Riemann-Stieltjes integral (proved by the authors in [4], Theorems 2.4 and 2.5) to obtain a Fredholm Alter- native for our equation (1). This result is presented in Section 5. The other sections are organized as follows. Section 2 is devoted to the fundamental theory of the Riemann-Stieltjes integral in Banach spaces, where we present basic results, representation theorems and the Fredholm Alternative. In Section 3, we give some basic definitions of the Henstock-Kurzweil integration theory. In Section 4, we present auxiliary results for the Henstock-Kurzweil integral such as the fundamental theorem of calculus and a substitution formula. 2 The Riemann-Stieltjes integral in Banach spaces In this section we introduce the definitions and terminology that we are going to use through- out the paper for Riemann-Stieltjes integrals and we mention some properties of these inte- grals. We start by defining functions of bounded B-variation, of bounded semi-variation and of bounded variation and giving some results concerning these spaces. 2.1 Functions of bounded B-variation, of bounded semi-variation and of bounded variation A bilinear triple (we write BT ) is a set of three vector spaces E, F and G, where F and G are normed spaces with a bilinear mapping B : E × F → G. For x ∈ E and y ∈ F , we write xy = B (x, y) and we denote the BT by (E, F, G)B or simply by (E, F, G). A topological BT is a BT (E, F, G) where E is also a normed space and B is continuous. We suppose that kBk ≤ 1. We denote by L (E, F ) the space of all linear continuous functions from E to F , where E and F are normed spaces. We write L (E) = L (E, E) and E0 = L (E, R), where R denotes the real line. Throughout this paper, X, Y and Z will always denote Banach spaces. 3 Example 2.1. As an example of a BT we can consider E = L (X, Y ), F = L (Z, X), G = L (Z, Y ) and B (v, u) = v ◦ u. In particular, when Z = R, we have E = L (X, Y ), F = X, G = Y and B (u, x) = u (x); when X = R, we have E = Y , F = Y 0, G = R and B (y, y0) = hy, y0i; when X = Z = R, we have E = G = Y , F = R and B (y, λ) = λy. Let [a, b] be a compact interval of R. Any finite set of closed non-overlapping subintervals [ti−1, ti] of [a, b] such that the union of all intervals [ti−1, ti] equals [a, b] is called a division of [a, b]. In this case we write d = (ti) ∈ D[a,b], where D[a,b] denotes the set of all divisions of [a, b]. By |d| we mean the number of subintervals in which [a, b] is divided through a given d ∈ D[a,b]. Given a BT (E, F, G)B and a function α :[a, b] → E, for every division d = (ti) ∈ D[a,b] we define |d| X SBd (α) = SB[a,b],d (α) = sup [α (ti) − α (ti−1)] yi ; yi ∈ F, kyik ≤ 1 i=1 and SB (α) = SB[a,b] (α) = sup SBd (α); d ∈ D[a,b] . Then SB (α) is the B-variation of α on [a, b]. We say that α is a function of bounded B-variation whenever SB (α) < ∞. When this is the case, we write α ∈ SB ([a, b] ,E). The following properties are not difficult to prove: (SB1) SB ([a, b] ,E) is a vector space and the mapping α ∈ SB ([a, b] ,E) 7→ SB (α) ∈ R+ is a seminorm; (SB2) for α ∈ SB ([a, b] ,E), the function t ∈ [a, b] 7→ SB[a,t] (α) ∈ R+ is monotonically increasing; (SB3) for α ∈ SB ([a, b] ,E) and c ∈ [a, b], SB[a,b] (α) ≤ SB[a,c] (α) + SB[c,b] (α) Consider the BT (L (X, Y ) , X, Y ). In this case, we write SV ([a, b] ,L (X, Y )) and SV (α) respectively by SB ([a, b] ,L (X, Y )) and SB (α), and any element of SV ([a, b] ,L (X, Y )) is called a function of bounded semi-variation.