GUO Yu-chen, SHU Xiao-bao

(College of Mathematics and Econometrics, Hunan University, Changsha 410082, China)

Abstract: In this paper, we investigate the existence and Ulam stability of solution for impulsive Riemann-Liouville fractional neutral function differential equation with infinite delay of order 1<β <2.Firstly, the solution for the equation is proved.By using the fixed point theorem as well as Hausdorff measure of noncompactness, the existence results are obtained and the Ulam stability of the solution is proved.

Keywords: impulsive Riemann-Liouville fractional differential equation;the fixed point theorem; Hausdorff measure of noncompactness; Ulam stability

1 Introduction

Fractional differential equation, as an excellent tool for describing memory and hereditary properties of various materials and processes in natural sciences and engineering, received a great deal of attention in the literature [1–4]and there were some works on the investigation of the solution of fractional differential equation [5,6].

On the other hand,Riemann-Liouville fractional derivatives or integrals are strong tools for resolving some fractional differential problems in the real world.It is possible to attribute physical meaning to initial conditions expressed in terms of Riemann-Liouville fractional derivatives or integrals which were verified by Heymans and Podlubny[7], and such initial conditions are more appropriate than physically interpretable initial conditions.For another,they considered the impulse response with Riemann-Liouville fractional derivatives as widely used in the fields of physics, such as viscoelasticity.

In recent years, many authors investigated the existence and stability of solutions to fractional differential equations with Caputo fractional derivative, and there were a lot of interesting and excellent results on this fields.However, there is still little literature on the existence and stability of solutions to Riemann-Liouville fractional differential equations.Three years ago,Weera Yukunthorn et al.[8]studied the existence and uniqueness of solutions to impulsive multiorders Riemann-Liouville fractional differential equations

whereβ∈R,0=t0

and ??x(tk) is defined by

Motivated by this work,we use Mnch’s fixed point theorem via measure of noncompactness as well as the basic theory of Ulam stability to investigate the existence and stability of solution to the following impulsive Riemann-Liouville fractional neutral function differential equation with infinite delay in a Banach spaceX.

wherek=1,2,··· ,mandis the Riemann-Liouville fractional derivative of order 1 <β< 2.0=t0

The rest of the paper is organized as follows: in section 2, some basic definitions,notations and preliminary facts that are used throughout the paper are presented.In Section 3, we prove the solution of the equation and present the main results for problem (1.1).

2 Preliminaries

In this section, we mention some notations, definitions, lemmas and preliminary facts needed to establish our main results.

LetXbe a complex Banach space, whose norm is denoted by[0,t1],Jk=(tk,tk+1]fork=1,2,··· ,m.Let

PC(J,X):={x:J→X, is continuous everywhere except for sometkat which

We introduce the spaceC2?β,k(Jk,X) :={x:Jk→X:t2?βx(t) ∈C(Jk,X)} with the normfor each t ∈Jkandwith the norm

ClearlyPC2?βis a Banach space.We useBr(x,X) to denote the closed ball inXwith center atxand radiusr.

Before introducing the fractional-order functional differential equation with infinite delay, we define the abstract phase spaceBv.Letv:(∞,0]→(0,∞) be a continuous function that satisfiesthe Banach spaceinduced byvis then given by

endowed with the norm

Define the following space

where?kis the restriction of?toJk,J0=[0,t1],Jk=(tk,tk+1],k=1,2,··· ,m.

where

Now we consider some definitions about fractional differential equations.

Definition 2.1The Riemann-Liouville fractional derivative of orderα> 0 of a continuous functionf;(a,b)→Xis defined by

wheren=[α]+1, [α]denotes the integer part of numberα, provided the right-hand side is pointwise defined on (a,b), Γ is the gamma function.

Definition 2.2The Riemann-Liouville fractional integral of orderα> 0 of a continuous functionf:(a,b)→Xis defined by

provided the right-hand side is pointwise defined on (a,b).

Lemma 2.1(see [9]) Letα>0.Then forx∈C(a,b)∩L(a,b), it holds

wheren?1<α

Lemma 2.2(see [9]) Ifα≥0 andβ>0, then

Before investigating the solutions to equation (1.1), we consider a simplified version of(1.1), given by

Theorem 2.1Let 1 <β< 2 andf:J→Xbe continuous.Ifx∈PC2?β(J,X) is a solution of (2.1) if and only ifxis a solution of the following the fractional integral equation

wherek=1,2,··· ,m.

ProofFor allt∈(tk,tk+1]wherek=0,1,··· ,mby Lemma 2.1 and 2.2, we obtain

Thus, expression (2.2) satisfies the first equation of problem (2.1).Fork=1,2,··· ,m, it follows from (2.1) that

Therefore, we have

Consequently, all the conditions of problem (2.1) are satisfied.Hence, (2.2) is a solution of problem (2.1)

Next, based on Theorem, we consider the solutions of the Cauchy problem(1.1)

Definition 2.3Suppose functionx: (?∝,T]→X.The solution of the fractional differential equation, given by

will be called a fundamental solution of problem (1.1).

Lemma 2.3(see [10]) Assumethen fort∈J,xt∈Bv.Moreover

Next, we consider some definitions and properties of the measures of noncompactness.

The Hausdorff measure of noncompactnessβ(·) defined on each bounded subsetBof Banach spaceXis given by

Ah! how good milk is! What a pity it is so ruinously expensive! So they made a little shelter of branches for the beautiful creature which was quite gentle, and followed Celandine about like a dog when she took it out every day to graze

Some basic properties ofβ(·) are given in the following lemma.

Lemma 2.4(see[11–13]) IfXis a real Banach space andB,D?Xare bounded,then the following properties are satisfied

(1) monotone: if for all bounded subsetsB,DofX,B?Dimpliesβ(B)≤β(D);

(2) nonsingular:β({x}∪B)=β(B) for everyx∈Xand every nonempty subsetB?X;

(3) regular:Bis precompact if and only ifβ(B)=0;

(4)β(B+D)≤β(B)+β(D), whereB+D={x+y;x∈B,y∈D};

(5)β(B∪D)≤max{β(B),β(D)};

(6)β(λB)≤|λ|β(B);

where

(9) ifWis bounded, then for eachε>0, there is a sequencesuch that

The following lemmas about the Hausdorff measure of noncompactness will be used in proving our main results.

Lemma 2.5(see[14]) LetDbe a closed convex subset of a Banach SpaceXand 0 ∈D.Assume thatF:D→Xis a continuous map which satisfies the Mnch’s condition, that is,M?Dis countable,is compact.ThenFhas a fixed point inD.

Next, we consider the Ulam stability for the equation.

Consider the following inequality

Definition 2.4Equation (1.1) is Hyers-Ulam stable if, for anyε> 0, there exists a solutiony(t) which satisfies the above inequality and has the same initial value asx(t),wherex(t) is a solution to (1.1).Theny(t) satisfiesin whichKis a constant.

3 Existence

To prove our main results, we list the following basic assumptions of this paper.

(H1) The functionf:J×Bv→Xsatisfies the following conditions.

(i)f(·,φ) is measurable for allφ∈Bvandf(t,·) is continuous for a.e.t∈J.

(ii) There exist a constantand a positive integrable function ?:R+→R+such thatfor all (t,φ)∈J×Bv, where ?satisfies

(iii) There exist a constantα2∈(0,α) and a functionsuch that, for any bounded subsetF1?Bv,

for a.e.t∈J, whereF1(θ)={v(θ):v∈F1} andβis the Hausdorff MNC.

(H2) The functiong:J×Bv→Xsatisfies the following conditions.

(i)gis continuous and there exist a constantH1>0 and

(ii) There exist a constantα3∈(0,α) andsuch that, for any bounded subsetF2?Bv,

(H3)Ik,Jk:X→X,k=1,2,··· ,mare continuous functions and satisfy

whereT?=max{1,T,T2}, Γ?=min{Γ(β+1),Γ(β),Γ(β?1)}.

Theorem 3.1Suppose conditions (H1)?(H4) are satisfied.Then system(1.1) has at least one solution onJ.

ProofWe define the operator Γ:

The operator Γ has a fixed point if and only if system (1.1) has a solution.Forφ∈Bv,denote

It is easy to see thatysatisfiesy0=0,t∈(?∝,0]and

if and only ifx(t) satisfiesx(t)=φ(t),t∈(?∝,0]and

with the norm

Step 1We prove that there exists somer> 0 such thatN(Br)?Br.If this is not true, then, for each positive integerr, there existyr∈Brandtr∈(?∝,T]such thatOn the other hand, it follows from the assumption that

So we have

Dividing both sides byrand takingr→+∝from

and

yields

This contradicts (H4).Thus, for some numberr,N(Br)?Br.

Step 2Nis continuous onBr.Letwithyn→yinBrasn→+∝.Then, by using hypotheses (H1),(H2) and (H3), we have

(i)

(ii)

(iii)

Now, for everyt∈[0,t1], we have

Moreover, for allt∈(tk,tk+1],k=1,2,··· ,m, we have

We thus obtain

implying thatNis continuous onBr.

Step 3The mapN(Br) is equicontinuous onJ.The functions {Ny:y∈Br} are equicontinuous att=0.Fort1,t2∈Jk,t1

where there existC1(t1) > 0.The right side is independent ofy∈Brand tend to zero ast1→t2sincet2?βNy(t) ∈C(Jk,X)andast1→t2.ast1→t2.Hence,N(Br) is equicontinuous onJ.

Step 4Mnch’s condition holds.

LetN=N1+N2+N3, where

AssumeW?Bris countable andwe show thatβ(W)=0, whereβis the Hausdorff MNC.Without loss of generality, we may suppose thatSinceN(W)is equicontinuous onis equicontinuous onJkas well.Using Lemma 2.4, (H1)(iii), (H2)(ii),(H3), we have

We thus obtain

whereM?is defined in assumption (H4).SinceWandN(W) are equicountinuous on everyJk, it follows from Lemma 2.4 that the inequality impliesβ(NW)≤M?β(W).Thus, from Mnch’s condition, we have

SinceM?<1, we getβ(W)=0.It follows thatWis relatively compact.Using Lemma 2.5,we know thatNhas a fixed pointyinW.So the theorem is proved.

4 Ulam Stability

(H5) The functiong(t,x) satisfies the condition thatLis a constant and 0

Theorem 3.2Suppose conditions (H1)(H3)(H4)(H5) are satisfied.Then system(1.1)has at least one solution onJand this solution is Ulam stable.

ProofIt is easy to see that the solution satisfies condition (H2) when the solution satisfies condition (H5).By using Theorem 3.1, we can prove the existence of this solution.Then we consider the inequality

Suppose there exists a functionf1(t,yt) satisfiesThen for the equation

We have the fundamental solution of this equation as

It is obvious to see that the solution is Ulam stable in the interval (?∝,0], so, first, let’s have a look at the intervalt∈(0,t1],

So, we have

Second, consider the intervalt∈(t1,t2],

As we have had the conclusion that in the intervalt∈(0,t1]that |y(t)?x(t)|

due toIk,Jkare continuous functions.

So

So

So, in the intervalt∈(t1,t2],

In this way, whentis in the intervalt∈(ti?1,ti]can be proved.