An eigenvalue result for Neumann BVPs with functional terms

TL;DR

This paper investigates eigenvalues and eigenfunctions for parameter-dependent Neumann boundary value problems with functional terms, using integral reformulation and numerical methods for validation.

Contribution

It introduces a new approach combining integral reformulation and fixed-point iteration to analyze eigenvalues for Neumann BVPs with functional terms.

Findings

Established existence and localization of eigenvalues.

Provided numerical algorithms and MATLAB code for eigenvalue approximation.

Validated theoretical bounds through numerical experiments.

Abstract

We study the existence and localization of eigenvalue-eigenfunction pairs for parameter-dependent Neumann BVPs with a functional term. By reformulating the problems as a Hammerstein integral equation, we apply an existence and localization result and propose a convergent fixed-point iteration scheme. Finally, two pseudocodes and a MATLAB implementation are provided to numerically approximate the eigenvalues and validate the theoretical localization bounds. We also illustrate an approximation of the eigenfunctions for a fixed norm.