On a nonlocal fractional thermostat eigenvalue problem

TL;DR

This paper investigates positive solutions for a nonlocal fractional boundary value problem involving Caputo derivatives, extending classical thermostat models by analyzing sign-changing Green's functions and providing explicit eigenvalue intervals.

Contribution

It introduces a novel approach using Birkhoff-Kellogg theorem in cones to handle sign-changing Green's functions in fractional thermostat eigenvalue problems.

Findings

Existence of positive eigenvalues with specified norms established.

Explicit intervals for localizing positive eigenvalues provided.

Application demonstrated through illustrative examples.

Abstract

We study the existence of positive solutions for a parameter-dependent nonlocal boundary value problem involving a Caputo fractional derivative, which generalizes a classic thermostat model. Our approach extends previous work by considering two nonlinear functionals occurring in the boundary conditions and, crucially, by analyzing cases where the associated Green's function is not necessarily positive and is allowed to change sign. We employ a Birkhoff-Kellogg type theorem in cones to establish the existence of positive eigenvalues with associated eigenfunctions with given norms. Furthermore, we provide explicit intervals that localize the corresponding positive eigenvalues. The applicability of our theoretical framework is illustrated with examples.