The method of upper and lower solutions and impulsive fractional differential inclusions

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Abstract

In this paper the concept of lower and upper solutions combined with the fixed point theorem of Bohnnenblust–Karlin is used to investigate the existence of solutions for a class of the initial value problem for impulsive differential inclusions involving the Caputo fractional derivative.

Introduction

This paper is concerned the existence of solutions for the initial value problems (IVP for short), for impulsive fractional order 0<α1 differential inclusions cDαy(t)F(t,y),for each ,tJ=[0,T],ttk,k=1,,m,Δy|t=tk=Ik(y(tk)),k=1,,m,y(0)=y0, where cDα is the Caputo fractional derivative, F:J×RP(R) is a multivalued map (P(R) is the family of all nonempty subsets of R), Ik:RR,k=1,,m and y0R,0=t0<t1<<tm<tm+1=T,Δy|t=tk=y(tk+)y(tk),y(tk+)=limh0+y(tk+h) and y(tk)=limh0y(tk+h) represent the right and left limits of y(t) at t=tk,k=1,,m.

Differential equations of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, control, porous media, electromagnetic, etc. (see [1], [2], [3], [4], [5], [6], [7]). There has been a significant development in fractional differential equations in recent years; see the monographs of Lakshmikantham et al. [8], Kilbas et al. [9], Kiryakova [10], Miller and Ross [11], Samko et al. [12] and the papers of Agarwal et al. [13], [14], Belarbi et al. [15], [16], Benchohra et al. [17], [18], [19], [20], Delbosco and Rodino [21], Diethelm et al. [1], [22], [23], Kilbas and Marzan [24], Mainardi [5], Podlubny et al. [25] and Zhang [26] and the references therein.

Applied problems require definitions of fractional derivatives allowing the utilization of physically interpretable initial conditions, which contain y(0),y(0), etc. the same requirements of boundary conditions. Caputo’s fractional derivative satisfies these demands. For more details on the geometric and physical interpretation for fractional derivatives of both the Riemann–Liouville and Caputo types see [27], [28]. The web site http://people.tuke.sk/igor.podlubny/, authored by Igor Podlubny contains more information on fractional calculus and its applications, and hence it is very useful for those that are interested in this field.

Integer order impulsive differential equations have become important in recent years as mathematical models of phenomena in both the physical and social sciences. There has a significant development in impulsive theory especially in the area of impulsive differential equations with fixed moments; see for instance the monographs by Benchohra et al. [29], Lakshmikantham et al. [30], and Samoilenko and Perestyuk [31] and the references therein. In [32], Benchohra and Slimani have initiated the study of fractional differential equations with impulses.

The method of upper and lower solutions plays an important role in the investigation of solutions for differential equations and inclusions. See the monographs by Benchohra et al. [29], Heikkila and Lakshmikantham [33], Ladde et al. [34] and the references therein.

By means of the concept of upper and lower solutions combined with fixed point theorem of Bohnnenblust–Karlin, we present an existence result for the problem (1), (2), (3). This paper initiates the application of the upper and lower solution method for impulsive fractional differential inclusions at fixed moments of impulse.

Section snippets

Preliminaries

In this section, we introduce notations, definitions, and preliminary facts that will be used in the remainder of this paper. Let [a,b] be a compact interval. C([a,b],R) be the Banach space of all continuous functions from [a,b] into R with the norm y=sup{|y(t)|:atb}, and we let L1([a,b],R) denote the Banach space of functions y:[a,b]R that are Lebesgue integrable with norm yL1=ab|y(t)|dt.AC(J,R) is the space of functions y:JR, which are absolutely continuous. Let (X,) be a Banach

Main result

Consider the following space PC(J,R)={y:JR:yC((tk,tk+1],R),k=0,,m+1 and there exist y(tk) and y(tk+),k=1,,m with y(tk)=y(tk)}.PC(J,R) is a Banach space with norm yPC=sup{|y(t)|:0tT}. Set J[0,T]{t1,,tm}.

Definition 3.1

A function yPC(J,R)k=0mAC((tk,tk+1),R) is said to be a solution of (1), (2), (3) if there exists a function vL1(J,R) with v(t)F(t,y(t)), for a.e. tJ, such that the differential equation cDαy(t)=v(t) on J, and conditions Δy|t=tk=Ik(y(tk)),k=1,,m, and y(0)=y0 are satisfied.

Definition 3.2

A

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