# Fractional hybrid systems involving $\varphi$-Caputo derivative.

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## Keywords:

Hybrid differential equation, $\varphi$-Caputo derivative, measure of noncompactness, Banach algebra.## Abstract

This paper delves into the intricate dynamics of a coupled hybrid thermostat system, a sophisticated framework utilized in temperature regulation applications. Our focus lies in elucidating the behavior of this system, specifically characterized by the $\varphi$-Caputo fractional derivative, operating within the confines of a Banach algebra. To address the existence of solutions within this complex framework, we employ a refined application of Darbo’s fixed-point theorem. Additionally, we harness the measure of noncompactness technique, a powerful tool in functional analysis, to further bolster our analysis and establish crucial existence results. Moreover, to offer practical insights into the implications of our theoretical findings, we present an illustrative example highlighting our approach's real-world applicability. Through this example, we demonstrate how our theoretical framework can be effectively applied to analyze and understand the behavior of hybrid thermostat systems.

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*7*(3), 163–176. Retrieved from https://nonlinear-analysis.com/index.php/pub/article/view/453

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