Solution Sets of Abstract Volterra Equations

Abstract

In this chapter we examine the solution set of the abstract Volterra equation

$$ \left\{ {\begin{array}{*{20}c} {x'(t) = x(t){\rm{ a}}{\rm{.e t}} \in {\rm{[0,T]}}} \\ {x(0) = x_{0;} } \\\end{array}} \right. $$

(3.1.1)

here V is the abstract Volterra operator. By placing mild conditions on the operator V we will show that the set of solutions of (3.1.1) is a R _δ set [6] (so in particular nonempty, compact and connected). Aronszajn in 1942 was the first to discuss the structure of the solution set of differential equations. However it was only in the late 1970’s that a general theory was established. In particular we refer the reader to the papers of Szufla (see [14]) and his coworkers. In this chapter we present a modern theory (adapted from [4, 13]) on the topological structure of the solution set of abstract Volterra equations. The technique used in this chapter involves applying a well known result of Szufla (see [6 page 161] for an elementary proof) together with a trick involving the Urysohn function. For convenience we recall the result of Szufla here.