New Study of the Existence and Dimension of the Set of Solutions for Nonlocal Impulsive Differential Inclusions with a Sectorial Operator

Nawal Alsarori,Sami Alabiad,Salvatore Sessa,Hayel Saleh,Kirtiwant Ghadle

doi:10.3390/sym13030491

Abstract

In this article, we are interested in a new generic class of nonlocal fractional impulsive differential inclusions with linear sectorial operator and Lipschitz multivalued function in the setting of finite dimensional Banach spaces. By modifying the definition of PC-mild solutions initiated by Shu, we succeeded to determine new conditions that sufficiently guarantee the existence of the solutions. The results are obtained by combining techniques of fractional calculus and the fixed point theorem for contraction maps. We also characterize the topological structure of the set of solutions. Finally, we provide a demonstration to address the applicability of our theoretical results.

Highlights

Fractional calculus deals with integrations and derivatives in case of a non-integer order, which is a generalized shape of classical integrals and derivatives
The significance of fractional calculus is due to its numerous applications in many significant scientific fields such as physics, control theory, fluid dynamics, image processing, computer networking, sign processing, biology and others
There are numerous examples in biotechnology, physics, population dynamics and processes economics which are characterized by the reality that the model parameters are subjected to short-time period perturbations

Summary

Introduction

Fractional calculus deals with integrations and derivatives in case of a non-integer order, which is a generalized shape of classical integrals and derivatives. The significance of fractional calculus is due to its numerous applications in many significant scientific fields such as physics, control theory, fluid dynamics, image processing, computer networking, sign processing, biology and others. The first ones had been extensively researched in finite and Banach spaces; see, for instance, the studies [9,10,11,12,13]. The existence of mild solutions for the impulsive differential equations and inclusions in Banach spaces have been examined through many researchers, we refer the reader to [14,15,16,17,18,19,20,21,22,23,24]

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