Linear Volterra Integral Equations as the Limit of Discrete Systems

Abstract

We consider the multidimensional abstract linear integral equation of Volterra type 1 $$ x{\left( t \right)} + {\left( * \right)}{\int_{R_{t} } {\alpha {\left( s \right)}x{\left( s \right)}ds = f{\left( t \right)},{\kern 1pt} t \in R} } $$ , as the limit of discrete Stieltjes-type systems and we prove results on the existence of continuous solutions. The functions x, α and f are Banach space-valued defined on a compact interval R of $$\mathbb{R}^{\text n}$$ , R t is a subinterval of R depending on t ∈ R and (⋆) ∫ denotes either the Bochner-Lebesgue integral or the Henstock integral. The results presented here generalize those in [1] and are in the spirit of [3]. As a consequence of our approach, it is possible to study the properties of (1) by transferring the properties of the discrete systems. The Henstock integral setting enables us to consider highly oscillating functions.