Abstract
In this manuscript, we establish the existence of results of fractional impulsive differential equations involving ψ-Hilfer fractional derivative and almost sectorial operators using Schauder fixed-point theorem. We discuss two cases, if the associated semigroup is compact and noncompact, respectively. We consider here the higher-dimensional system of integral equations. We present herewith new theoretical results, structural investigations, and new models and approaches. Some special cases of the results are discussed as well. Due to the nature of measurement of noncompactness theory, there exists a strong relationship between the sectorial operator and symmetry. When working on either of the concepts, it can be applied to the other one as well. Finally, a case study is presented to demonstrate the major theory.
Highlights
Academic Editors: WłodzimierzFechner and Jacek ChudziakReceived: 17 July 2021Accepted: 2 October 2021Published: 8 October 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.Licensee MDPI, Basel, Switzerland.This article is an open access articleHilfer [1] started the Hilfer fractional derivative, an extended Riemann–Liouville fractional derivative that interpolates Caputo fractional derivatives and Riemann–Liouville fractional derivatives
In the conventional equation for exponential relaxation, the infinitesimal generator of time evolution is substituted by the infinitesimal generator of composite fractional translations [2]
The ψ-Hilfer fractional derivative differential equations with boundary value problems were discussed by Mali et al in [7]
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Solved is the fractional differential equation for composite fractional relaxation. At high frequencies in the imaginary part, the composite fractional susceptibility function might show an asymmetric relaxation peak and an excess wing. In [6], a monotone iterative technique was used to solve initial value problems for nonlinear fractional differential equations with ψ-Caputo derivative. The ψ-Hilfer fractional derivative differential equations with boundary value problems were discussed by Mali et al in [7]. In [8], Kucche et al discussed the nonlinear ψ-Hilfer fractional derivative differential equations with initial value problems of the form. The work is organized as follows: in Section 2, we discuss the Hilfer derivative, almost sectorial operators, measure of noncompactness, and mild solutions of Equations (1)–(3), as well as some basic definitions and lemmas. The following section describes the supporting results of the given problem which generalizes the results in [12]:
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