Dai Qun，Li Hui-laiand Liu Su-li

(1.College of Science，Changchun University of Science and Technology，Changchun，130022)

(2.School of Mathematics，Jilin University，Changchun，130012)

Communicated by Wang Chun-peng

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Existence and Uniqueness of Positive Solutions for a System of Multi-order Fractional Differential Equations

Dai Qun1，Li Hui-lai2and Liu Su-li2

(1.College of Science，Changchun University of Science and Technology，Changchun，130022)

(2.School of Mathematics，Jilin University，Changchun，130012)

Communicated by Wang Chun-peng

In this paper，we prove the existence and uniqueness of positive solutions for a system of multi-order fractional differential equations.The system is used to represent constitutive relation for viscoelastic model of fractional differential equations.Our results are based on the fixed point theorems of increasing operator and the cone theory，some illustrative examples are also presented.

fractional differential equation，Caputo fractional derivative，fixed point theorem，positive solution

2010 MR subject classification:34A08，35B44

Document code:A

Article ID:1674-5647(2016)03-0249-10

1　Introduction

In recent years，fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics，chemistry，aerodynamics，electrodynamics of complex medium，polymer rheology，economics，blood flow phenomena，etc.，see[1]–[5].

Recently，the existence of solutions to fractional differential equations have been studied by many authors，see[6]–[14].In this paper，we consider the following initial value problem for a system of fractional differential equations:

where L1(D)=DSn-an?1DSn?1-···-a1DS1，0＜α＜S1＜···＜Sn?1＜Sn≤1，ai＞0，L2(D)=DKm-bm?1DKm?1-···-b1DK1，0＜β＜K1＜···＜Km?1＜Km≤1，bj＞0，α≤Km，β≤Sn，DSi，DKj(i=1，2，···，n-1；j=1，2，···，m-1)are the standard Caputo fractional derivatives，and f，g:[0，1]×[0，+∞)→[0，+∞)are continuous.

The system(1.1)is used to represent constitutive relation for viscoelastic model of fractional differential equations.Moreover，the system(1.1)describes processes of heat diffusion and combustion in two-component continua with nonlinear heat conduction and volumetric release.When we take?u，?v instead of u，v and let S1=K1=n=m=1，it expresses as macroscopic models II for electrodiffusion of ions in nerve cells when molecular diffusion is anomalous subdiffusion due to binding，crowding or trapping(refer to[9]).

This paper is organized as follows.In Section 2，we present some basic materials needed to prove our main results.In Section 3，we prove the existence and uniqueness of positive solutions for(1.1)by applying the fixed point theorems of increasing operator and the cone theory.

2　Preliminaries

We recall several known definitions and properties from fractional calculus theory(refer to [1]and[15]).

Definition 2.1The Riemann-Liouville fractional integral of a function f:(0，1]→R of order α＞0 is given by

provided that the right-hand side is pointwise defined on(0，1]，and Γ is the gamma function.

Definition 2.2For at least n-times continuously differentiable function f:[0，+∞)→R，the Caputo derivative of order α is defined as

Hereafter Dαdenotesand Iαdenotes.

Definition 2.3A Banach space E endowed with a closed cone K is an ordered Banach space(E，K)with a partial order≤in E as follows:

Definition 2.4For x，y∈E，the order interval?x，y?is defined as

Lemma 2.1Let β＞α＞0，f∈L1[a，b].Then

Lemma 2.2Let(E，K)be an ordered Banach space，x0，y0∈K，x0≤y0，F(xiàn):?x0，y0?→?x0，y0?be an increasing operator，and Fx0≥x0，F(xiàn)y0≤y0.If F is continuous and compact and K is a normal cone，then F has a fixed point which lies in?x0，y0?.

Lemma 2.3Let(E，K)be an ordered Banach space，U1，U2be open subsets of E withbe completely continuous.Suppose either

or

Then F has a fixed point.

Let us introduce the space X={u(t):u(t)∈C1[0，1]}endowed with the norm

Let

Obviously，K is a normal cone.

3　Main Results

Lemma 3.1Assume that g and f:[0，1]×R→[0，+∞)are continuous.Then(1.1)is equivalent to the following system of integral equations:

Proof.Let

Then in view of(1.1)we find that

Applying ISn?αto both sides of(3.3)，by Lemma 2.1，we can get

For u(t)=Iαx(t)and the initial value condition u(0)=0.

By(3.5)，we can get(3.1).Similarly，we also have(3.2).

F，G:K→K can be defined as

Lemma 3.2Let M be a bounded subset of K.Then there exists a positive constant l such that for any uare compact sets.

Proof.We only need to verify that F and G:K→K are completely continuous operators. First，we prove that F(M)is a bounded set.

Let

We have

Similarly，we also have

Thus，F(xiàn)(M)and G(M)are bounded sets.

Second，we prove that the operator F is equicontinuous.

Let u∈M，and for any 0≤t1＜t2≤1，|t1-t2|＜δ.Then

where W1is independent of t1，t2.

Similarly，we also have

where W2is independent of t1，t2.

Thus，F(xiàn) and G are equicontinuous operators.By Arzela-Ascoli Theorem，we know thatandare compact sets.

Theorem 3.1If there exists a constant λ＞0 such thatand there existsuch that

then(1.1)has at least one positive solution.

Proof.We only need to verify that F，G have fixed points.By Lemma 3.2，we know that F and G are completely continuous.For u1＜u2，v1＜v2，u1，u2，v1，v2∈K，we have

Therefore，

In the same way，we can obtain that

Thus，F(xiàn) and G are increasing operators.

we can get

By Lemma 2.2，F(xiàn) and G have fixed points

Theorem 3.2If there exists a constant λ＞0 such thatand for any t∈[0，1]，

then(1.1)has at least one positive solution.

Proof.There exist N＞0 and R＞0 such that for any|Dβv|≥R and|Dαu|≥R，

Let

Hence，for any v，u＞0，one has

We consider the following system of fractional differential equations:

where

By Lemma 3.1，(3.6)is equivalent to the following integral equations:

We have

There exist x(t)=y(t)=0 such that

By Theorem 3.1，this completes the proof.

Theorem 3.3Let

Assume that there exist two distinct positive constantsandsuch that

Then(1.1)has at least one positive solution.

Proof.Let

where

For v，u∈K∩?U2，we have

Since f(t，Dβv(t))≤，we have

Hence

Similarly，we also have

On the other hand，for u∈K∩?U1，we have

Hence

Similarly，we also have

By Lemma 2.3，we know that(1.1)has a fixed point in K∩

Theorem 3.4Assume that there exists a constant M＞0 such that for any t∈[0，1]，f，g satisfy the Lipschitz conditions:

Let

Proof.By Theorem 3.2，we know that(1.1)has at least one positive solution.For u1，u2，v1，v2∈K，we have

By(3.2)，one has

For h＞0，we have

Therefore，

Similarly，we also have

Thus，F(xiàn) and G are contraction operators and so(1.1)has a unique solution.

Example 3.1Consider the system of fractional equations

In view of Theorem 3.2 there exists a positive solution for this system.

Example 3.2The system

has a positive solution in view of Theorem 3.3.

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10.13447/j.1674-5647.2016.03.07

date:Sept.22，2015.

Jilin province education department“twelfth five-year”science and technology research plan project([2015]No.58)，and the science and technology innovation fund(No.XJJLG-2014-02)of Changchun University of Science and Technology.

E-mail address:daiqun1130@163.com(Dai Q).