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Eigenvalues of a third order BVP subject to functional BCs

Gennaro Infante, Paolo Lucisano

TL;DR

This work addresses eigenvalues of a third-order boundary value problem with nonlocal functional boundary conditions and nonlinearities depending on higher derivatives. It reframes the problem as a perturbed Hammerstein integral equation in $C^2[0,1]$ with a compact operator $T = F + \Gamma$, where $F u(t) = \int_{0}^{1} k(t,s) f(s,u,u',u'') ds$ and $\Gamma u(t) = \gamma_1(t) H_1[u] + \gamma_2(t) H_2[u]$. Using a variant of the Birkhoff-Kellogg invariant-direction theorem, the authors establish the existence of eigenvalues of opposite signs with corresponding eigenfunctions localized by the $C^2$-norm, and they provide conditions under which uncountably many eigenpairs arise as the radius $\rho$ varies. An explicit example demonstrates the applicability and computability of the constants involved, highlighting the results’ relevance for nonlocal third-order problems and related applications.

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.

Eigenvalues of a third order BVP subject to functional BCs

TL;DR

This work addresses eigenvalues of a third-order boundary value problem with nonlocal functional boundary conditions and nonlinearities depending on higher derivatives. It reframes the problem as a perturbed Hammerstein integral equation in with a compact operator , where and . Using a variant of the Birkhoff-Kellogg invariant-direction theorem, the authors establish the existence of eigenvalues of opposite signs with corresponding eigenfunctions localized by the -norm, and they provide conditions under which uncountably many eigenpairs arise as the radius varies. An explicit example demonstrates the applicability and computability of the constants involved, highlighting the results’ relevance for nonlocal third-order problems and related applications.

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.
Paper Structure (2 sections, 3 theorems, 28 equations)

This paper contains 2 sections, 3 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Eigenvalues existence

Key Result

Theorem 2.3

Let $X$ be an infinite dimensional real Banach space, and let $\hat{F}\colon \overline{B_\rho(X)}\to X$ be a compact operator such that: Then there exist $\lambda_+,\lambda_-\in{\Lambda_\rho(\hat{F})}$, with $\lambda_+>0,\lambda_-<0.$

Theorems & Definitions (6)

  • Definition 2.1
  • Theorem 2.3: applbook, Theorem 10.2
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • Example 2.6