Eigenvalues of a third order BVP subject to functional BCs

TL;DR

This paper investigates the eigenvalues of a third order boundary value problem with functional boundary conditions, establishing existence and localization results using the Birkhoff-Kellogg theorem.

Contribution

It introduces new existence and localization results for eigenvalues of third order BVPs with functional BCs, extending classical methods.

Findings

Existence of positive and negative eigenvalues proven.

Eigenfunctions are localized in terms of their norm.

Theoretical results are illustrated with an example.

Abstract

We discuss the existence of eigenvalues for a third order boundary value problem subject to functional boundary conditions and higher order derivative dependence in the nonlinearities. We prove the existence of positive and negative eigenvalues and provide a localization of the corresponding eigenfunctions in terms of their norm. The methodology involves a version of the classical Birkhoff-Kellogg theorem. We illustrate the applicability of the theoretical results in an example.