Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications

TL;DR

This paper develops a general fixed point index theory to prove the existence of nonzero solutions for perturbed Hammerstein integral equations with deviated arguments, with applications to boundary value and thermostat problems.

Contribution

It introduces a broad theoretical framework for nonzero solutions of perturbed Hammerstein equations with deviated arguments, including optimality discussions and practical examples.

Findings

Established existence of solutions using fixed point index

Applied theory to boundary value problems with reflections

Provided examples illustrating the theory

Abstract

We provide a theory to establish the existence of nonzero solutions of perturbed Hammerstein integral equations with deviated arguments, being our main ingredient the theory of fixed point index. Our approach is fairly general and covers a variety of cases. We apply our results to a periodic boundary value problem with reflections and to a thermostat problem. In the case of reflections we also discuss the optimality of some constants that occur in our theory. Some examples are presented to illustrate the theory.