An Outlook on Hybrid Fractional Modeling of a Heat Controller with Multi-Valued Feedback Control

Abstract

In this study, we extend the investigations of fractional-order models of thermostats and guarantee the solvability of hybrid Caputo fractional models for heat controllers, satisfying some nonlocal hybrid multi-valued conditions with multi-valued feedback control, which involves the Chandrasekhar kernel, by using hybrid Dhage’s fixed point theorem. A part of this study is dedicated to transforming this problem into an equivalent integral representation and then proving some existence results to achieve our aims. Furthermore, the continuous dependence of the unique solution on the control variable and on the set of selections will be discussed. Moreover, we provide an illustration to support our results.

Qualitative Aspects of a Fractional-Order Integro-Differential Equation with a Quadratic Functional Integro-Differential Constraint

“…It is obvious that all the hypotheses of Theorem 1 are valid. Hence there exists at least one solution x ∈ [0, 1] of ( 15)- (12). Moreover, we have…”

“…In recent years, several scholars have concentrated their efforts on constrained integral equations. Their findings about functional integral equations have been expanded to include a particular set of constrained integral equations on a bounded interval (see [11][12][13]) and unbounded intervals (see [14]). Constrained problems are essential in the mathematical depiction of real-world situations, where such problems are transformed into mathematical models [15][16][17].…”

“…In [30], the author discussed a positive periodic solution to a nonlinear neutral delay population equation with feedback control. In [12], the authors studied fractional-order models of thermostats; they proved the existence of a solution and the continuous dependence of the unique solution on the control variable. In [13], the author investigated the solvability and the asymptotic stability of a class of nonlinear functional-integral equations with feedback control.…”

“…In [13], the author investigated the solvability and the asymptotic stability of a class of nonlinear functional-integral equations with feedback control. For further relevant works, see [12,[31][32][33][34][35]. Fixed-point theorems are a great tool for discussing the solvability of differential equation problems that have been studied in a number of monographs and publications; see [6,31,[36][37][38][39][40].…”

Qualitative Study on ψ−Caputo fractional differential inclusion with non-local conditions and feedback control

“…The demonstration relies on the fixed-point theorem of strictset-contraction operators [26]. El-Sayed et al [5] address a functional integral equation with multi-valued feedback control that meets a constraint functional equation.…”