A general class of noninstantaneous impulsive fractional differential inclusions in Banach spaces

Jinrong Wang,D O’Regan,Yong Zhou,Ag Ibrahim

doi:10.1186/s13662-017-1342-8

Abstract

In this paper we introduce the concept of a PC-mild solution to a general new class of noninstantaneous impulsive fractional differential inclusions involving the generalized Caputo derivative with the lower bound at zero in infinite dimensional Banach spaces. Using the formula of a PC-mild solution, we give two classes of sufficient conditions to guarantee the existence of PC-mild solutions via fixed point theorems for multivalued functions. Also we characterize the compactness of the solution set. We introduce the concept of generalized Ulam-Hyers stability and present a generalized Ulam-Hyers stability result using multivalued weakly Picard operator theory. Examples are given to illustrate the theoretical results.

Highlights

1 Introduction Fractional inequalities, equations and inclusions arise in various fields, such as physics, mechanics and engineering [ – ] and in particular fractional differential inclusions arise in mathematical modelling of problems in game theory, stability and optimal control
We study existence and stability of noninstantaneous impulsive fractional differential inclusions of the form
(iii) We introduce a concept of generalized Ulam-Hyers stability for noninstantaneous impulsive fractional differential inclusions and we present a generalized-Ulam-Hyers stability result using multivalued weakly Picard operator theory

Summary

Introduction

Fractional inequalities, equations and inclusions arise in various fields, such as physics, mechanics and engineering [ – ] and in particular fractional differential inclusions arise in mathematical modelling of problems in game theory, stability and optimal control. For developments in the study of mild solutions to instantaneous impulsive differential equations we refer the reader to [ – ] and the references therein. The introduction of the drugs in the bloodstream and the consequent absorption for the body are gradual and continuous process In this situation the impulsive action starts at any arbitrary fixed point and stays active on a finite time interval. We study existence and stability of noninstantaneous impulsive fractional differential inclusions of the form. (ii) Using fixed point theorems for multivalued functions we obtain two classes of sufficient conditions to guarantee existence of PC-mild solutions in piecewise continuous spaces endowed with the Chebyshev norm and the Bielecki norm. (iii) We introduce a concept of generalized Ulam-Hyers stability for noninstantaneous impulsive fractional differential inclusions and we present a generalized-Ulam-Hyers stability result using multivalued weakly Picard operator theory

Preliminaries and notations

Methods