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.