On a nonlocal fractional thermostat eigenvalue problem

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.