Fractional q-Difference Inclusions in Banach Spaces

Badr Alqahtani,Mouffak Benchohra,Sara Salem Alzaid,Saïd Abbas

doi:10.3390/math8010091

Abstract

In this paper, we study a class of Caputo fractional q-difference inclusions in Banach spaces. We obtain some existence results by using the set-valued analysis, the measure of noncompactness, and the fixed point theory (Darbo and Mönch’s fixed point theorems). Finally we give an illustrative example in the last section. We initiate the study of fractional q-difference inclusions on infinite dimensional Banach spaces.

Highlights

Fractional differential equations and inclusions have attracted much more interest of mathematicians and physicists which provides an efficiency for the description of many practical dynamical arising in engineering, vulnerability of networks, and other applied sciences [1,2,3,4,5,6,7,8]
Riemann–Liouville and Caputo fractional differential equations with initial and boundary conditions are studied by many authors; [2,9,10,11,12,13,14]
Theorem 3 implies that N has at least one fixed point u ∈ C ( I ) which is a solution of our problem in Equations (1) and (2)

Summary

Introduction

Fractional differential equations and inclusions have attracted much more interest of mathematicians and physicists which provides an efficiency for the description of many practical dynamical arising in engineering, vulnerability of networks (fractional percolation on random graphs), and other applied sciences [1,2,3,4,5,6,7,8]. Riemann–Liouville and Caputo fractional differential equations with initial and boundary conditions are studied by many authors; [2,9,10,11,12,13,14]. Some interesting results about initial and boundary value problems of ordinary and fractional q-difference equations can be found in [22,23,24,25,26,27]. Where ( E, k · k) is a real or complex Banach space, q ∈ (0, 1), α ∈ (0, 1], T > 0, F : I × E → P ( E) is a multivalued map, P ( E) = {Y ⊂ E : y 6= ∅}, and c Dqα is the Caputo fractional q-difference derivative of order α. This paper initiates the study of fractional q-difference inclusions on Banach spaces.

Preliminaries

Existence Results

An Example

Conclusions