Existence and Uniqueness of Mild Solutions for Nonlinear Fractional Integro-Differential Evolution Equations

HOU Mimi(侯咪咪),XI Xuanxuan(西宣宣),ZHOU Xianfeng(周先锋)

(School of Mathematical Science,Anhui University,Hefei 230601,China)

Abstract: In this paper,we will study a class of nonlinear fractional integro-differential evolution equations in a Banach space X. We use the fractional power of operators and the theory of analytic semigroups to prove the existence and uniqueness of the solution for the given problem.Furthermore,we give the Hölder continuity of the obtained mild solution.

Key words: Analytic semigroup; Mild solution; Existence and uniqueness; Hölder continuity

1.Introduction

Fractional differential equations have been applied in many fields,including electrochemistry,control,quantum mechanics,biological systems,porous media and electrodynamics.There has been a significant development in fractional differential equations in recent years.[1−5]

Evolution equations are abstract differential equations in Banach spaces and are generalizations of concrete equations.Theory of evolution equations have lots of applications in partial differential equations.Theory of semigroups play an important role in evolution equations.Recently,there has been a rapid development in fractional evolution equations.[6−12]

In this paper,we consider the following nonlinear fractional integro-differential equation

Regarding earlier works on the problems of existence and uniqueness for nonlinear fractional differential equations.[13−17]Dhage et al.[17]has studied the equation with form as follows:

wherex(t)∈C(J,R),f ∈C(J ×R,R),g:J ×R→R,which is a Lebesgue integrable function.It is worthy to mention that the equation is not abstract.Pandey et al.[18]has considered the existence and uniqueness of the following fractional integro-differential equation

in a Banach spaceE,whereAgenerates an analytic semigroup.The authors used the theories of the semigroup and fixed point theorem to prove the existence and uniqueness of the solution.

Our investigation is motivated by the work of Dhage et al.[17]and Pandey et al.[18]The authors in [18]have discussed the existence and uniqueness of Eq.(1.3).In the present work,we use some of the ideas of Kumar et al.[19]and Musilm[20]to obtain the results of existence and uniqueness.Our calculation is different from [19-20].

In this paper,we will study the existence and uniqueness of mild solution of Eq.(1.1)by using the fixed point theorem and theories of analytic semigroup.Furthermore we prove the Hölder continuity of the obtained mild solution for Eq.(1.1).The rest of paper is organized as follows.In Section 2,we give some useful preliminaries about the fractional differential equations.In Section 3,we give our main theorem and its proof.In Section 4,we give the Hölder continuity of the obtained mild solution for Eq.(1.1).

2.Preliminaries

In this section,we introduce some definitions and preliminary facts which we will used throughout this paper.We observe that if−Ais the infinitesimal generator of an analytic semigroupS(t),t ≥0,then−(A+ϱI)is invertible and generates a bounded analytic semigroup forϱ>0 large enough.Thus we can reduce the general case in which−Ais the infinitesimal generator of an analytic semigroup to the case in which the semigroup is bounded and the generator is invertible.Hence without loss of generality,we suppose that

and

whereρ(−A)is the resolvent of−A.The fractional powerA−qis defined by

Aqis the invertible operator ofA−q.It follows that for 0< q ≤1,Aqis a closed linear invertible operator.Its domainD(Aq)is dense inXandD(Aq)⊃D(A).The closeness ofAqmeans thatAq,endowed with the graph norm ofAq,|||u|||=||u||+||Aqu||,u ∈D(Aq),is a Banach space.SinceAqis invertible,its graph norm|||·|||is equivalent to the norm||u||q=||Aqu||.Thus,D(Aq)equipped with the norm||·||qis a Banach space which we denote byXq.Whenq=1,we denoteD(A)byX1with norm||Au||=||u||1.About more details,one can refer to [21].

Definition 2.1[2,4−5]The Rieman-Liouville fractional integral operator of orderα ≥0,of a integrable functionfwith the lower limit 0,is defined as follows:

The Riemann-Liouville derivative has certain disadvantages when trying to model real world phenomena with fractional differential equations.Therefore,we will introduce a modified fractional differential operatorcDαproposed by M.Caputo.

Definition 2.2[2,4−5]The M.Caputo derivative of orderαwith the lower limit 0 for a functionfcan be written as

where the functionf(t)has absolutely continuous derivatives up to ordern−1.

Remark 2.1Iffis an abstract function with values inX,the integral which appear in Definitions (2.1)and (2.2)are taken in Bocher’s sense.Integrating both sides of Eq.(1.1),we have

For more details,one can refer to [8].

By using the method of Lemma 3.1 of [9],we obtain the following lemma.

Lemma 2.1If Eq.(2.3)holds,then we have

whereζα(θ)is a probability density function defined on (0,+∞),that is

and

Its Laplace transform is given by

Lemma 2.2[19,21]Let−Abe the infinitesimal generator of an analytic semigroupS(t).If 0∈ρ(A),then

(a)S(t):X→D(Aq)for everyt>0 andq ≥0;

(b)S(t)Aqu=AqS(t)ufor everyu ∈D(Aq);

(c)The operatorAqS(t)is bounded and||AqS(t)||≤Cqt−q,t>0.

Lemma 2.3[21]Let−Abe the infinitesimal generator of an analytic semigroup,then

(a)Aαis a close operator with domainD(Aα)=R(A−α)=the range ofA−α;

(b)0<β ≤αimpliesD(Aα)⊂D(Aβ);

(d)Ifα,βare real thenAα+βx=AαAβx.

We use the similar calculation presented in Theorem 3.2 of [22]to find the upper bound of integral,thus we get the following lemma.

Lemma 2.4For any 0≤t′

where 0<δ <1,c=1−

ProofIt is easy to see that

SettingR(t)=we can write

wheret′

3.Local Existence of Mild Solutions

Before we give our main result,we first state the following definition.

Definition 3.1The local mild solution of Eq.(1.1)onJmeans that a continuous functionudefined fromJintoXwhich satisfies the following integral equation:

Remark 3.1Ais the infinitesimal generator of an analytic semigroup,andA−qis defined by(2.1),hence||Aq−1||is bounded for 0max {1,||Aq−1||}.

Theorem 3.1Suppose that the operator−Agenerates the analytic semigroupS(t)with||S(t)||≤M,t ≥0,0∈ρ(A),and the following assumptions hold:

(H1)LetDbe an open subset ofJ ×X1,f:D→X1satisfy the conditions:for every(t,u)∈D,there is a neighborhoodVof (t,u)(V ⊂D)and the constantsL,vandHsatisfyingand 0<ν ≤1,such that

for all (ti,ui)∈V,i=1,2,where we denoteD(A)byX1.

(H2)For an open subsetEofJ ×Xq,h:E→X,g:E→Xsatisfy the conditions:For every (t,u)∈E,there is a neighborhoodUof (t,u)(U ⊂E)and the constantsL1,L2andνsatisfyingL1>0,L2>0 and 0<ν ≤1,such that

for all (ti,ui)∈U,i=1,2,whereXis a Banach space,and we denoteD(Aq)byXq.

Suppose further that the real valued mapais integrable onJ.Then Eq.(1.1)has a unique local mild solution for everyu0∈X1.

ProofChooset∗>0 andr >0 such that the estimates (3.2),(3.3)and (3.4)hold on the sets

By the inequalities (3.2),(3.3)and (3.4),there exist positive constantsB,B1andB2such that

ChooseTsuch that for 0≤t ≤T,

for a givenθ ∈[0,+∞),and

whereCqis a positive constant depended onqsatisfying

ForT >0,denote L

etY=C([0,T];X)with usual supremum normWe define a map onYbyFywith the following form

for everyy ∈Y,t>0,whereFy(0)=Aq[u0−f(0,u0)]+Aqf(0,u0)=Aqu0.

LetGbe the nonempty closed and bounded subset ofYdefined by

We first prove that the mapFis fromGintoG.For anyy ∈G,we have by (2.5)that

where

By (3.7),we have

By (2.6),(3.9)and the assumption (H1),we have that

We have by (2.6),(3.3)and (3.9)that

We get by (2.6),(3.4),(3.9)and (3.10)that

By (3.8),(3.14),(3.15),(3.16)and (3.17),we get that

Therefore,FmapsGinto itself.Moreover,for anyy1,y2∈G,we have

where

According to the assumption (H1)and Lemma 2.3 (d),we get

We have by (2.6),(3.9)and the assumption (H1)that

One can derive by (2.6),(3.9)and the assumption (H2)that

By (2.6),(3.9),(3.10)and the assumption (H2),we get

Therefore,we have by (3.20),(3.21),(3.22)and (3.23)that

Letu=A−qy(t),then fort ∈[0,T],we obtain by (3.25)that

Thusuis a unique local mild solution to Eq.(1.1).The proof is complete.

4.Hölder Continuous

In this section,we give the proof of the Hölder continuity of the solution of Eq.(1.1).

Theorem 4.1Assume that the assumptions(H1)and(H2)hold.Then the mild solutionu(t)of Eq.(1.1)is Hölder continuous.

ProofBy the previous arguments in Theorem 3.1,there exists a positive numberT0such that Eq.(1.1)has a unique mild solution on the intervalJ0=[0,T0]given by

Letv(t)=Aqu(t),we have by Lemma 2.2(b)that

For convenience,set

Then (4.2)can be rewritten as

Since the mapsf,gandhsatisfy the assumptions (H1)and (H2),u(t)is continuous onJ0,we can let

Next,we have

where

Now,we calculateK1.For eachx ∈X1,we can write

Hence,we get that

Furthermore,we get by (2.6),(3.9),Lemma 2.2(b)and Lemma 2.3(d)that

By Lemma 2.3 (d)and the assumption (H1),we have

We get by (2.6),(3.9)and (4.4)that

We obtain by (2.6),(3.9)and (4.4)that

whereλ1=1−αq,µ1=andαq≠1.By Lemma 2.4,we have

Furthermore,we have

By the boundness ofA−qand (4.14),u(t)is Hölder continuous.The proof is complete.