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, ∞). 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, ∞). The technique is well suited to integral equations.
Highlights
As all crafts people know, a main part of any task is finding the right tools
In the mid 1950s Krasnoselskii studied the inversion of a perturbed differential operator and arrived at the following working hypothesis
The inversion of a perturbed differential operator yields the sum of a contraction and a compact map
Summary
As all crafts people know, a main part of any task is finding the right tools. In the mid 1950s Krasnoselskii studied the inversion of a perturbed differential operator and arrived at the following working hypothesis. To this set he added his own theorem on the sum of a contraction and compact map. Let (B, · ) be a normed space, P a continuous mapping of B into B which is compact on each bounded subset X of B. Let (B, · ) be a Banach space, C, D : B → B, D a contraction with contraction constant α < 1, and C continuous with C mapping bounded sets into compact sets. We will give a theorem and and example illustrating this This leads us to results on positive solutions, as well as a new idea on uniqueness of solutions. Uniqueness is critical for obtaining a clean statement that there is a solution on [0, ∞) without using a statement that an extension process can be carried out an infinite number of times to get a solution on all of [0, ∞)
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