THE EXISTENCE OF POSITIVE SOLUTIONS FOR IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS WITH BOUNDARY VALUE CONDITIONS

WANG Xian-cun,SHU Xiao-bao

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

THE EXISTENCE OF POSITIVE SOLUTIONS FOR IMPULSIVE FRACTIONAL DIFFERENTIAL EQUATIONS WITH BOUNDARY VALUE CONDITIONS

WANG Xian-cun,SHU Xiao-bao

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

In this paper,we investigate the impulsive fractional diff erential equation with boundary value conditions.By using the theory of Kuratowski measure of noncompactness and Sadovskii’fi xed point theorem,we obtain the existence of positive solution for the impulsive fractional diff erential equations,which generalize the results of previous literatures.

fractional diff erential equations;impulsive fractional diff erential equations; measure of noncompactness; α-contraction

1 Introduction

In the past few decades,fractional diff erential equations arise in many engineering and scientific disciplines,such as the mathematical modeling of systems and processes in the fi elds of physics,chemistry,biology,economics,control theory,signal and image processing, biophysics,blood flow phenomena,aerodynamics,fitting of experimentaldata,etc.Because of this,the investigation of the theory of fractional diff erential equation attracted many researchers attention.

In[4],Ahmad and Sivasundaram studied the solution of a nonlinear impulsive fractional differentialequation with integralboundary conditions given by

wherecDqtis the Caputo fractional derivative of order q ∈ (1,2).The authors investigate the existence ofthe solution for the equation by applying contraction mapping principle and Krasnoselskii’s fixed point theorem.

In[5],Nieto and Pimentelstudied the positive solutions ofa fractionalthermostatmodel of the following

where α ∈ (1,2], β ＞ 0,0 ＜ η ≤ 1 are given numbers.Based on the known Guo-Krasnoselskii fixed point theorem on cones,the authors proved the existence of positive solutons for the fractionalorder thermostat model.

In[6],Zhao etc.investigated the existence of positive solutions for the nonlinear fractionaldifferentialequation with boundary value problem

where 1 ＜ α ≤ 2 is a real number,cDα0+is the Caputo fractional derivative.By using the properties of the Green function and Guo-Krasnoselskii fixed point theorem on cones, the eigenvalue intervals of the nonlinear fractional diff erential equation with boundary value problem are considered,some suffi cient conditions for the nonexistence and existence of at least one or two positive solutions for the boundary value problem are established.

A lot of scholars were engaged in the research about the positive solution of fractional differential equations(see[5–20]).To the best of our knowledge,there is few result about the positive solutions for nonlinear impulsive fractionaldifferentialequations with boundary value conditions so far.

Motivated by the above articles,in this paper,we will consider the positive solution of the following impulsive fractionaldifferential equation with boundary value conditions

2 Preliminaries and Lemmas

Let E be a real Banach space and P be a cone inwhich defined a partial ordering in E by x ≤ y if and only if y − x ∈ P,P is said to be normalif there exists a positive constant N such that θ≤ x ≤ y implies ‖x‖ ≤ N‖y‖,where θdenotes the zero element of E,and the smallest N is called the normalconstant of P,P is called solid ifits interior P is nonempty. If x ≤ y and x/=y,we write x ＜ y.If P is solid and y − x ∈ P˙,we write x ＜＜ y.For details on cone theory,see[1].

A map u ∈ P C1[J,E]is called a nonnegative solution of BVP(1.1)if u ≥ θfor t ∈ J and u(t)satisfi es BVP(1.1).A map u ∈ P C1[J,E]is called a positive solution of BVP(1.1) if it is a nonnegative solution of BVP(1.1)and u(t)/= θ.

Let α, αPC1 be the Kuratowski measure of non-compactness in E and P C1[J,E],respectively.For details on the definition and properties of the measure of non-compactness, the reader is referred to[2].

As the main application of this paper,we fist give the definition of α-contraction and the related lemma to be used to prove our main result.

Defi nition 2.1(see[3])Let X be a Banach space.If there exists a positive constant k ＜ 1 satisfying α(Q(K)) ≤ kα(K)for any bounded closed subset K ⊆ W,then the map Q:W ⊂ X → X is called an α-contraction,where α(·)is the Kuratowski measure of non-compactness.

Lemma 2.1(see[3])If W ⊂ X is bounded closed and convex,the continuous map Q:W → W is an α-contraction,then the map Q has at least one fixed point in W.

Lemma 2.2(see[20])If V ⊂ P C1[J,E]is bounded and the elements of V′are equicontinuous on each(tk,tk+1)(k=1,2,···,m),then

Lemma 2.3(see[20])Let H be a countable set of strongly measurable function x: J → E such that there exists an M ∈ L[J,R+]such that ‖x‖ ≤ M(t)a.e.t ∈ J for all x ∈ H.Then α(H(t)) ∈ L[J,R+]and

Lemma 2.4For a linear function g ∈ C[0,1],a function u is a solution ofthe following impulsive fractional diff erential equation with boundary value conditions

if and only if u satisfies the integralequation

where

ProofA generalsolution u ofequation(2.1)on each interval(tk,tk+1)(k=0,1,2,···,m) can be given by

It is known that

According to impulsive condition of system(2.1),we get (

for k=1,2,···,m,then we can obtain the following relations

which implies that

Thus we get(2.2)considering the above equations.

On the contrary,if u is a solution of(2.2),then a q order fractional differentiation of (2.2)yields

and we can get

Clearly,for k=1,2,···,m,we have

This completes the proof.

3 Main Results

We shall reduce BVP(1.1)to an integral equation in E.To this end,we first consider operator T defined by the following,for t ∈ (tk,tk+1)(k=0,1,···,m),

Hereafter,we write Q={x ∈ KPC1:‖x‖PC1≤ R}.Then Q is a bounded closed and convex subset of P C1[J,E].

We will list the following assumptions,which will stand throughout this paper.

(H1)f ∈ C[J × R+,R+],there exist a,b,c ∈ L[J,R+]and h ∈ C[R+,R+]such that

and

where

and

We write

and

We write

(H4)For any t ∈ J and bounded sets V ⊂ P C1[J,E],there exist positive numbers l, dk,fk(k=1,2,···,m)such that

Theorem 3.1If conditions(H1)–(H3)are satisfi ed,then operator T is a continuous operator form Q into Q.

ProofLet

by(H1),there exist a r ＞ 0 such that

and

where

Hence we get

Let

we see that by(H2)–(H3),for k=1,2,···,m,there exist a r1＞ 0,such that

and

where

Then ∀x ∈ R+,we have

Defi ne

By(H2)–(H3),we have

So

Differentiating(3.1),we get

where

By assumption(H1),we obtain

Thus by(3.2),we also have

Then we can get

So by(3.6),(3.7)and(3.8),we obtain T u ∈ Q.Thus we have proved that T maps Q into Q.

Finally,we show that T is continuous.LetIt is easy to get

It is clear that

and by(3.2),

By(3.10)and(3.11)and the dominated convergence theorem,we obtain that

Obviously,for i=1,2,···,m,

So

Following(3.12),(3.13)and(3.14),we obtain that0 as n → ∞,and the continuity of T is proved.

Theorem 3.2Assumes that conditions(H1)–(H4)are satisfied,if

1,then BVP(1.1)has at least one positive solution on Q.

ProofDefineand.For u ∈ Q,tk＜ t1＜ t2＜tk+1,by(3.2),(3.4)and(3.7),we get

Consequently,

which implies that operator T′is equicontinuous on each(tk,tk+1)(k=1,2,···,m).

By Lemma 2.2,for any bounded and closed subset V ⊂ Q we have

It follows from Lemma 2.3 that

Therefore

Then operator T is a α-contraction as

that operator T has at least one fixed points on Q.Given that T u ≥ 0 for u∈ Q,we learn

By Lemma 2.1,we obtain that problem(1.1)has at least one positive solution.

4 An Example

Consider the following fractionaldifferential equation with boundary value conditions

ConclusionBVP(4.1)has at least one positive solution on[0,1].

ProofLet E=R and P=R+,R+denotes the set of all nonnegative numbers.It is clear,P is a normaland solid cone in E.In this situation,m=1,t1=12,

and

Obviously,f ∈ C([0,1]× R+,R+),I1,J1∈ C(R+,R+).By a direct computation about (4.2),we have

So(H1)is satisfied for a(t)=0,b(t)=c(t)=5+1t,h(x)=2 ln(1+x).

On the other hand,by(4.3),we have that

which imply that condition(H2)and(H3)are satisfied for F1(x)=F2(x)=x and η11= η21= γ11= γ21=15.

where ξ, δ, ζ are all between x1and x2,and clearly l=15,d1=f1=15,which mean that (H4)is satisfied.Then

It is not diffi cult to see that the condition of Theorem 3.2 are satisfied.Hence,boundary value problem(4.1)has at least one positive solution on[0,1].

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带有边界值问题的脉冲分数阶微分方程正解的存在性

王献存,舒小保
(湖南大学数学与计量经济学院,湖南 长沙 410082)

本 文 研 究 了 具 有 边 界 值 条 件 的 脉 冲 分 数 阶 微 分 方 程. 利 用Kuratowski非 紧 性 测 度 理 论和Sadovskii不动点定理, 得到了脉冲分数阶微分方程正解的存在性的结果, 推广了已有文献的结论.

分数阶微分方程;脉冲分数阶微分方程;非紧性测度;α-压缩

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A < class="emphasis_bold">Article ID:0255-7797(2017)02-0271-12

0255-7797(2017)02-0271-12

∗Received date:2014-12-09 Accepted date:2015-04-07

Foundation item:Supported by Doctoral Fund of Ministry of Education of China(200805321017).

Biography:Wang Xiancun(1991–),female,born at Nanyang,Henan,graduate,ma jor in fractional diff erential equation.