Eigenvalues of a coupled system of thermostat-type via a Birkhoff-Kellogg type Theorem

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.