Eigenvalues of a coupled system of thermostat-type via a Birkhoff-Kellogg type Theorem
Sajid Ullah
TL;DR
This work develops a general eigenvalue existence theory for a parameter-dependent, coupled system of thermostat-type boundary-value problems by reformulating them as a system of Hammerstein integral equations on a cone. Using a Krasnosel'skiĭ–Lyžhenskij-type fixed-point theorem in cones, the authors derive conditions under which a positive eigenpair $(\lambda,(u_1,u_2))$ exists, with $\lambda$ coupling both the differential equations and boundary conditions. The theory is then instantiated to three boundary-condition families (Dirichlet-type, Neumann-type, and mixed), each accompanied by explicit Green's functions and verifiable hypotheses; detailed examples illustrate the applicability and construction of eigenpairs. The results extend existing nonlocal boundary-value problem theory to parameter-dependent coupled systems, providing a robust framework for thermostat-type models and related feedback-control systems in nonlinear settings.
Abstract
In this paper, by means of Birkhoff--Kellogg type Theorem in cones we address the existence of eigenvalues and the corresponding eigenvectors to a family of coupled system of thermostat type. The system is characterized by the presence of a real parameter that influences not only the differential equations but also the boundary conditions. Motivated by models of temperature regulation and feedback-controlled systems, we reformulate the original boundary value problems into systems of Hammerstein integral equations. The theoretical results are applied to three different classes of boundary conditions in $t=0$, which are supported by examples.