Investigation of the neutral fractional differential inclusions of Katugampola-type involving both retarded and advanced arguments via Kuratowski MNC technique

Sina Etemad,Shahram Rezapour,Mohammed K A Kaabar,Benoumran Telli,Mohammed Said Souid

doi:10.1186/s13662-021-03377-x

Abstract

A class of the boundary value problem is investigated in this research work to prove the existence of solutions for the neutral fractional differential inclusions of Katugampola fractional derivative which involves retarded and advanced arguments. New results are obtained in this paper based on the Kuratowski measure of noncompactness for the suggested inclusion neutral system for the first time. On the one hand, this research concerns the set-valued analogue of Mönch fixed point theorem combined with the measure of noncompactness technique in which the right-hand side is convex valued. On the other hand, the nonconvex case is discussed via Covitz and Nadler fixed point theorem. An illustrative example is provided to apply and validate our obtained results.

Highlights

The theory of fractional differential equations, that of fractional boundary value problems, has motivated several mathematicians, physicists, engineers, ecologists, and biologists to study them due to the variety of their applications in multidisciplinary sciences [1,2,3,4,5,6,7,8,9,10,11,12,13,14]
For more information about this interesting research study, some recent research studies have been conducted on fractional differential equations (FrDEqs) in [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]
(2021) 2021:214 as the generalized Caputo fractional derivative [30, 31] which was proposed by Katugampola and Almeida with a purpose to define a fractional derivative that satisfies the property of semigroup, and it is capable of combining other fractional derivatives [32]

Summary

Introduction

The theory of fractional differential equations, that of fractional boundary value problems, has motivated several mathematicians, physicists, engineers, ecologists, and biologists to study them due to the variety of their applications in multidisciplinary sciences [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. Stimulated by the aforesaid research, this research work investigates the existence of solutions for the neutral fractional functional differential inclusions via Katugampola fractional derivative (KaFrD) which involves retarded and advanced arguments as follows: Dξn+ w(t) – q t, wt ∈ K t, wt , t ∈ J := [n, m], 1 < ξ ≤ 2, (3)

Results

Conclusion