Solvability of Hammerstein integral equations with applications to boundary value problems

TL;DR

This paper establishes new solvability results for nonlinear Hammerstein integral equations using fixed point theorems, with applications to boundary value problems and integral boundary conditions.

Contribution

It introduces novel solvability criteria for Hammerstein equations in specific function cones and applies these to boundary value problems with integral conditions.

Findings

Proved existence of solutions for nonlinear Hammerstein equations.

Applied fixed point theorems to boundary value problems.

Provided examples illustrating the theoretical results.

Abstract

In this paper we present some new results regarding the solvability of nonlinear Hammerstein integral equations in a special cone of continuous functions. The proofs are based on a certain fixed point theorem of Leggett and Williams type. We give an application of the abstract result to prove the existence of nontrivial solutions of a periodic boundary value problem. We also investigate, via a version of Krasnosel{\cprime}slki{\u\i}'s theorem for the sum of two operators, the solvability of perturbed Hammerstein integral equations in the space of continuous functions of bounded variation in the sense of Jordan. As an application of these results, we study the solvability of a boundary value problem subject to integral boundary conditions of Riemann--Stieltjes type. Some examples are presented in order to illustrate the obtained results.