Hilfer-type fractional differential switched inclusions with noninstantaneous impulsive and nonlocal conditions

Jinrong Wang,Donal O’Regan,Ahmed Gamal Ibrahim

doi:10.15388/na.2018.6.7

Abstract

In this paper, we study a new class of nonlocal problems for noninstantaneous impulsive Hilfer-type fractional differential switched inclusions in Banach spaces. First, we introduce a mild solution formula for this noninstantaneous impulsive inclusion problem. Second, we show the existence of mild solutions using the Hausdorff measure of noncompactness on the space of piecewise weighted continuous functions. Finally, an example is provided to illustrate the theory.

Highlights

Fractional differential equations and fractional differential inclusions are important because of their applications in physics, mechanics, and engineering [6, 25]
The Hilfer fractional derivative of order 0 < α < 1 and type 0 β 1 and with lower limit at a for a function f : [a, b] → E is defined by Daα+,βf (t) = (Iaβ+(1−α)D(Ia(1+−β)(1−α)f ))(t), t ∈ [a, b], provided that the right side is point-wise defined on [a, b]; here D = d/dt
Since Daα+,βx = (Iaβ+(1−α)Daγ+ x))(t), it follows from Lemma 2(i) that C1γ−γ (J, E) ⊆ C1α−,βγ (J, E)

Summary

Introduction

Fractional differential equations and fractional differential inclusions are important because of their applications in physics, mechanics, and engineering [6, 25]. Abstract nonlocal semilinear initial-value problems was initiated in [11, 12], where existence and uniqueness of mild solutions for nonlocal differential equations without impulsive was discussed (Lipschitz-type conditions were considered). The authors in [3] discuss existence of integral solutions for nonlocal differential inclusions when X is separable, the operator semigroup is compact, and the nonlinearity F is closed valued and lower semicontinuous in its second variable, and, using the measure of noncompactness, the authors in [39] obtain existence results for mild solutions for nonlocal problems when the evolution system is not compact. The aim of the paper is to study the following nonlocal problem for Hilfer-type fractional noninstantaneous impulsive differential inclusions: Dsα+ i,βx(t) ∈ F t, x(t) , a.e. t ∈

Preliminaries

Main results

An example