Existence results for hybrid Caputo fractional models for a thermostat with hybrid boundary conditions

In this paper, we study a generalized hybrid fractional thermostat model involving the Caputo fractional derivative and the Riemann–Liouville fractional integral under nonlocal hybrid boundary conditions with feedback control. The problem describes a class of heat regulation systems in which the controller's response depends on both the process's thermal memory and sensor measurements taken at interior points of the domain. Working in the Banach space of integral-type Hölder functions ${\mathcal{J}}_{\alpha ,\beta }$, we establish sufficient conditions for the existence and uniqueness of solutions to the associated boundary value problem. This functional framework is more suitable than the classical spaces $\mathcal{C}(E_0)$ and ${\mathcal{C}}^{\alpha }(E_0)$, since it provides stronger regularity, better stability under fractional integral operators, and improved continuity properties for the nonlinear superposition terms involved in the hybrid structure. Our analysis is based on fractional calculus techniques, an appropriate measure of noncompactness, and Darbo-type fixed-point theorem. The obtained results extend several existing thermostat and hybrid fractional models in the literature. Finally, a numerical example is presented to illustrate and verify the theoretical findings for different values of $\alpha$ and $\beta$.