An eigenvalue result for Hammerstein integral equations with sign changing nonlinearities and functional terms

TL;DR

This paper establishes the existence and location of positive and negative eigenvalues for Hammerstein integral equations with sign-changing nonlinearities, using a Birkhoff-Kellogg approach, and applies the results to boundary value problems.

Contribution

It introduces a novel eigenvalue analysis for nonlinear integral equations with sign-changing terms and applies it to nonlocal boundary value problems.

Findings

Existence of positive and negative eigenvalues proven.

Eigenvalues are located using a norm-based approach.

Application demonstrated on boundary value problems with examples.

Abstract

We discuss, via a version of the Birkhoff-Kellogg theorem, the existence of positive and negative eigenvalues of Hammerstein integral equations with sign-changing nonlinearities and functional terms. The corresponding eigenfunctions have a given norm that, in turn, provides a location for the eigenvalues. As an application, we study the solvability of parameter-dependent boundary value problems for nonlocal ordinary differential equations. Two examples illustrate the applicability of the theory in the case of mixed and Dirichlet boundary conditions.