Integral equations, transformations, and a Krasnoselskii-Schaefer type fixed point theorem

Abstract

In this paper we extend the work begun in 1998 by the author and Kirk for integral equations in which we combined Krasnoselskii's fixed point theorem on the sum of two operators with Schaefer's fixed point theorem. Schaefer's theorem eliminates a difficult hypothesis in Krasnoselskii's theorem, but requires an a priori bound on solutions. Here, we simplify the work by means of a transformation which often reduces the a priori bound to a triviality. Our work is focused on an integral equation in which the goal is to prove that there is a unique continuous positive solution on $[0,\infty)$. In addition to the transformation, there are two techniques which we would emphasize. A technique is introduced yielding a lower bound on the solutions which enables us to deal with problems threatening non-uniqueness. The technique offers a solution to a classical problem and it seems entirely new. We show that when the equation defines the sum of a contraction and a Lipschitz operator, then we first get existence on arbitrary intervals $[0,E]$ and then introduce a technique which we call a progressive contraction which allows us to prove uniqueness and then parlay the solution to $[0,\infty)$. The technique is well suited to integral equations.

Keywords

• fixed points
• krasnoselskii-schaefer fixed point theorem
• integral equations
• positive solutions
• progressive contractions