Existence and Uniqueness Results for Caputo Fractional Differential Equations with Integral Boundary Value Conditions

XUE Yimin,DAI Zhenxiang,LIU Jie and SU Ying

School of Mathematics and Physics,Xuzhou University of Technology,Xuzhou221018,China.

?Corresponding author.Email addresses:xueym@xzit.edu.cn(Y.M.Xue),402898530@qq.com(Z.X.Dai),1046550060@qq.com(J.Liu),suyingxzit@163.com(Y.Su)

1 Introduction

Fractional differential equations can be used to describe many phenomena in a number of fi elds.For examples in physics,polymer rheology,chemistry,electrodynamics of complex medium,regular variation in thermodynamics,control theory,signal and image processing,biophysics,and so forth.There are many papers dealing with the existence and uniqueness results of boundary value problems for nonlinear fractional differential equations[1-5].Meanwhile,boundary value problems with integral boundary value conditions of nonlinear fractional differential equations have aroused considerable attention.Boundary value problems with integral boundary value conditions have various applications in population dynamics,chemical engineering,etc.For some recent development on the integral boundary conditions,see the texts[6-9]and the references cited therein.

Recently,in[10]Bai and L¨u used classical fixed point theorems to prove the multiple positive solutions for the following nonlinear fractional differential equation

where 1＜α≤2,Dαis the Riemann-Liouville fractional derivative of orderαand

In[11],Xu et al.investigated the existence of positive solutions for the following fractional boundary value problem

where 2＜α≤3,Dαis a fractional derivative in the sense of Riemann-Liouville andf∈C([0,1]×[0,∞),[0,∞)).The existence,multiplicity,uniqueness of positive solutions are established by using some fixed point theorems.

In[12],Cabada and Wang used Guo-Krasnoselskii fixed point theorem to show the existence of positive solutions for a class of nonlinear boundary value problem with integral boundary conditions as

where 2＜α＜3,0＜λ＜2,cDαis the Caputo fractional derivative of orderα,andf∈C([0,1]×[0,∞),[0,∞)).

Motivated by the results of[10-12],we consider the existence and uniqueness results for the following Caputo fractional differential equations with integral boundary value condition

wheren≥2(n∈N),cDαis the standard Caputo fractional derivative of orderα,andf∈C([0,1]×R,R)is a given function.The contraction mapping principle,Krasnoselskii’s fixed point theorem and Leray-Schauder degree theory, which partly improves and extends the associated results of fractional differentialequations. Four examples illustrating our main results are included.

2 Preliminaries

For the convenience of the reader,we will demonstrate and study some necessary definitions and theorems which can been founded in[13-18].

Definition 2.1.([13,14])For a function g:[0,∞)→R,the Caputo fractional derivative of order α＞0is given as

provided the previous integral exists.where n=[α]+1,[α]denotes the integer part of α.

Definition2.2.([15,16])Forafunctiong:(0,∞)→R,the Riemann-Liouville fractional integral of order α＞0is the integral

provided that the integral of the right-hand side of the previous equation exists.

Lemma 2.1.([15])Let α＞0,then the fractional differential equationcDαu(t)=0has a unique solution given as

Lemma 2.2.([15])Let α＞0,then the integral and derivative of the previous have the composite property as

To obtain the solution of the boundary problem(1.1),we establish the following fractional differential equation with integral boundary value condition

from which,we have the following lemma：

Lemma 2.3.For any y(t)∈C[0,1],n≥2(n∈N),0＜λ＜n,then problem(2.1)has a unique solution given by：

where

Proof.We may apply Lemma 2.2 to reduce problem(2.1)to an equivalent integral equation：

Sinceu(0)=u′(0)=···=u(n-2)(0)=u(n)(0)=0,we deduce that

Using the integral boundary conditionwe find that

In view of(2.3)and(2.4),yields

Now we integrate the previous equality from 0 to 1 on both sides ont,yields

Therefore,we have

Substituting(2.6)into(2.5),we have

The proof of Lemma 2.3 is completed.

Now,we present the following lemma,which will play major in the proofs of our main results.

Lemma 2.4.(Arzel`a-Ascoli[17])Let D?X be a compact set with a sequence{xn}?D being uniformly bounded and equicontinuous,then the sequence has a uniformly convergent subsequence.

Lemma 2.5.([18])Let X be a Banach space with??X being open and bounded,0∈?.let T→X be a completely continuous operator such that

Then T has a fixed point in

Lemma 2.6.(Krasnoselskii’s fixed point theorem[18])Let X be a Banach space withM ?Xbeing closed convex and nonempty.If the operator A and B satisfy the following conditions：

(a)Ax+By∈M,wherever x,y∈M;

(b)A is compact and continuous;

(c)B is a contraction mapping.

Then there exists z∈Msuch that z=Az+Bz.

3 Main results

In this section,we investigate the existence and uniqueness results of positive solutions for boundary value problem(1.1).First,we renew some notions.LetE=C([0,1],R)denote the set of all continuous functions from[0,1]into R.ThenEis a Banach space endowed with the norm defined by

define the operatorF：E→Eas

It follows from Lemma 2.3 that the fixed points of the operatorFcoincide with the solutions of fractional differential equation(1.1).Next,we will prove that the operatorF：E→Eis completely continuous.

Lemma 3.1.The operator F:E→E defined by(3.1)is completely continuous.

Proof.By continuity of functionsfandG(t,s),the operatorFis continuous.Let ??Ebe bounded.Then for anyt∈[0,1]andu∈?,there exists a positive constantK1＞0 such that|f(t,u)|≤K1.From which,we can deduce that

which implies that‖(Fu)(t)‖≤K2.Analogously,for the derivative,we obtain that

Therefore,for all 0≤t1＜t2≤1,one can deduce that

which implies that the operatorFis equicontinuous on[0,1].Thus,by Lemma 2.4,the operatorF：E→Edefined by(3.1)is completely continuous.

Next we consider the following existence results.

Theorem 3.1.Assume f∈C([0,1]×R,R)least one solution on[0,1].

Proof.there exist positive constantsε＞0 andδ＞0 such that

whereεsatisfies

define ?δ={u∈E：|u|＜δ},takingu0∈??δ,i.e.,u0∈E,and|u0|=δ.According to(3.1),(3.2)and(3.3),we can take

By virtue of Lemma 3.1,we know thatFis completely continuous.It follows from above that all conditions of Lemma 2.5 hold.Therefore,the boundary value problem(1.1)has at least one solution on[0,1].

Theorem 3.2.Suppose f∈C([0,1]×R,R).Moreover,there exist two positive real constants L,μsuch that the following conditions hold：

Then the boundary problem(1.1)has a unique solution on[0,1].

Proof.First,we show thatF：Br→Br.defineand select

define a closed ball inEasBr={u∈E：‖u‖≤r}.Foru∈Br,we have

which implies thatF：Br→Br.On the other hand,for allu,v∈E,t∈[0,1],we have

which implies thatFis a contraction operator.Therefore,thank to the contraction mapping principle,the boundary value problem(1.1)has a unique solution on[0,1].This completes the proof.

Theorem 3.3.Let f∈C([0,1]×R,R)satis fi es the following conditions：

(I1) ‖f(t,u)-f(t,v)‖≤L‖u-v‖, ?t∈[0,1],u,v∈E;

(I2) ‖f(t,u)‖≤φ(t), ?t∈[0,1],u∈E,and φ∈L1([0,1],R+),

where

Then the boundary problem(1.1)has at least one solution on[0,1].

Proof.Let

and define a closed ball inEasBr={u∈E：‖u‖≤r}.Now,we define the operatorsT1andT2onBras

For everyu,v∈Br,from(3.4)and(3.5),one can obtain that

which impliesT1+T2∈Br.From the proof of Theorem 3.2,we can takeT2is a contraction operator.Sincefis continuous,T1is also continuous.In addition,for arbitraryu∈Br,we find

by using(I2).Thus,T1is uniformly bounded onBr.Next,we prove the compactness of the operatorT1.Let S=[0,1]×Br,and defineFor anyt1,t2∈[0,1],one can get

which is independent ofu.We can obtain thatT1is equicontinuous.Note thatfmaps bounded subsets into relatively compact subsets,we can get thatT1(Ebs)(t)is relatively compact inEfor allt,whereEbs?Eis bounded.Hence,T1(·)is relatively compact onBr.By the means of Arzel`a-Ascoli theorem,T1is compact onBr,Consequently,in view of Lemma 2.6,we conclude that the nonlinear boundary value problem(1.1)has at least one solution on[0,1].

Theorem 3.4.Assume that f∈C([0,1]×R,R).Moreover,there exist real constant η satisfying

Let constant M＞0such that|f(t,u)|≤η‖u‖+M,for each t∈[0,1],u∈R.Then the boundary problem(1.1)has at least one solution on[0,1].

Proof.Let

and define a open ball inEas

Then,we only need to show that mappingFu：Br→Esatis fi es

LetH(ρ,u)=ρFu,u∈E,ρ∈ [0,1].Then,it is easily show thathρ(u)=u-H(ρ,u)=uρFuis completely continuous by Lemma 3.1.If(3.6)holds,according to the homotopy invariance of topological degree in Leray-Schauder degree theory,we deduce that

whereIdenotes the unit operator.We can get that there existsu∈Brsuch ash1(u)=u-ρFu=0 by using the nonzero property of the Leray-Schauder degree.

Next,we prove(3.6).Suppose thatu=ρFufor someρ∈[0,1]and anyt∈[0,1].Then,we have

Straightforward computation gives

whereNote that

So(3.6)holds.This completes the proof.

4 Examples

In this section,we present several examples to illustrate our theoretical results.

Example 4.1.Consider the following problem of fractional differential equations with integral boundary conditions

from(4.1),2＜α＜3 and 0＜λ＜2.By straightforward calculation we get thatThen,in virtue of Theorem 3.1,the boundary value problem(4.1)has at least one solution on[0,1].

Example 4.2.Consider the following problem of fractional differential equations with integral boundary conditions

whereα=7/2,λ=2,

asn=3,3＜α＜4,0＜λ＜3.One can easily obtain that

When we chooseL=1/2,a simple calculation gives

Obviously,the conditions(H1)and(H2)of Theorem 3.2 are satis fi ed,the boundary value problem(4.2)has a unique solution on[0,1].

Example 4.3.Consider the following problem of fractional differential equations with integral boundary conditions

in this case,α=19/4,λ=10/3,

asn=4,4＜α＜5,0＜λ＜4.We can obtain that

When we chooseby standard calculation,we have

Moreover

Clearly,all the assumptions of Theorem 3.3 are satis fi ed,the boundary problem(4.3)has at least one solution on[0,1].

Example 4.4.Consider the following problem of fractional differential equations with integral boundary conditions

according to(4.4),we haveα=15/4,λ=2,asn=3,3＜α＜4,0＜λ＜3,

We can obtain that

by straightforward calculation.Then,it is easy to see that there exists in fi nitely many positive constantM,such that

Thus,it follows from Theorem 3.4 that the boundary value problem(4.4)has at least one solution on[0,1].

Acknowledgement

Our work was supported by National Natural Science Foundation of China(11301454),the TianYuan Special Funds of the National Natural Science Foundation of China(115261-77),Natural Science Foundation of Jiangsu Province(BK20151160),the Natural Science Foundation for Colleges and Universities in Jiangsu Province(14KJB110025)and the Project of Xuzhou University of Technology(XKY2017113).

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