Abstract
In this research study, we are concerned with the existence and stability of solutions of a boundary value problem (BVP) of the fractional thermostat control model with ψ-Hilfer fractional operator. We verify the uniqueness criterion via the Banach fixed-point principle and establish the existence by using the Schaefer and Krasnoselskii fixed-point results. Moreover, we apply the arguments related to the nonlinear functional analysis to discuss various types of stability in the format of Ulam. Finally, by several examples we demonstrate applications of the main findings.
Highlights
Fractional differential and integral equations have demonstrated high visibility and capability in applications of various topics related to physics, signal processing, mechanics, electromagnetics, economics, biology, and many more [1, 2]
Speaking, it has been recognized that fractional integro-differential equations, whose kernels allow much freedom to describe various processes involving memory and hereditary properties, often appear in different fractional models caused by many real-life processes such as phenomena related to electromagnetic waves and heat transfer
By Lemma 3.3 we find at least one fixed-point for Q, which is the corresponding solution of the suggested ψ-Hilfer fractional boundary value problems (FBVPs) describing the thermostat control model (3)
Summary
Fractional differential and integral equations have demonstrated high visibility and capability in applications of various topics related to physics, signal processing, mechanics, electromagnetics, economics, biology, and many more [1, 2]. Where (·, ·) is defined in (15), the ψ-Hilfer FBVP describing the thermostat control model (3) has a unique solution x in E. By Lemma 3.1 we get that the solution x ∈ E is unique on J for the supposed ψ-Hilfer FBVP describing the thermostat control model (3).
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