Shasha Guo,Xiangkui Zhao

(Dept.of Applied Math.,University of Science and Technology Beijing, Beijing 100083,E-mail:guoshasha2014@126.com(S.Guo))

POSITIVE SOLUTIONS TO A COUPLED SYSTEM OF FRACTIONAL INTEGRAL BOUNDARY
VALUE PROBLEM WITH DELAY∗†

Shasha Guo,Xiangkui Zhao

(Dept.of Applied Math.,University of Science and Technology Beijing, Beijing 100083,E-mail:guoshasha2014@126.com(S.Guo))

In this paper,we study the existence of positive solutions to an integral boundary value problem with delay for a coupled system of fractional differential equations.By using the Krasnoselskii fixed point theorem,we obtain sufficient conditions for the existence of at least one or two positive solutions to the problem under some weaker conditions.

positive solution;delay;integral boundary value problem;cone

2000 Mathematics Subject Classification 34A08;34B18

Ann.of Appl.Math.

31:2(2015),148-158

1　Introduction

The aim of this paper is to establish sufficient conditions for the existence of positive solutions to the following integral boundary value problem

where 1＜α,β＜2,a,b∈C((0,1),[0,+∞)),vt(s)=v(t+s),ut(s)=u(t+s)for t∈[0,1], s∈[-r,0],0＜θ＜r＜1.f,g∈C([0,1]×[0,+∞),[0,+∞)).D is the standard Riemann-Liouville fractional derivative.Let Cr(r＞0)be the space of all continuous functions u:[-r,0]→R.

Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modelling of systems and processes in the fields of physics,chemistry,biology, economics,control theory,signal and image processing.From[1-7],we know that a large number of papers have investigated the existence or multiplicity of solutions or positive solutions to initial or boundary value problem for some nonlinear fractional differential equations.

In[7],Feng,Zhang and Ge studied the existence and multiplicity of positive solutions to the following higher-order singular boundary value problem of fractional differential equation:

where D is the standard Riemann-Liouville fractional derivative of order n-1＜α≤n, n≥3,g∈C((0,1),[0,+∞))and g may be singular at t=0 or at t=1,h∈L1[0,1]is nonnegative,and f∈C([0,1]×[0,+∞),[0,+∞)).

Recently,many people have established the existence,nonexistence and uniqueness of solutions to some systems of nonlinear fractional differential equations,see[8-10]and references cited therein.

In[10],Yang studied the existence and nonexistence of positive solutions to boundary values problem for a coupled system of nonlinear fractional differential equations as follows:

where 1＜α,β≤2,a,b∈C((0,1),[0,+∞)),φ,ψ∈L1[0,1]are nonnegative,f,g∈C([0,1]× [0,+∞),[0,+∞)),and D is the standard Riemann-Liouville fractional derivative.

As the studies of positive solution to boundary value problems are increasingly systematic,many authors have studied the delay differential equations.See[11-14]and references therein.In[14],the author used the Krasnoselskii fixed point theorem to study the existence of at least one or two positive solutions to a second-order boundary value problem for a class of nonlinear functional differential equations

where η∈(0,1),yt(s)=y(t+s)for t∈[0,1],s∈[-r,0],0≤r＜1-η.Motivated by the above works,we consider the integral boundary value problem with delay(1.1).

Noting from the assumption that 0＜θ＜r＜1,we conclude that mes(E)≠0,where

Define nonnegative extended real numbersby

with 0＜γβ＜1 being some constant depending on v.

In this paper,we study sufficient conditions for the existence of at least one or two positive solutions to(1.1).The proof of our results is based on the following Krasnoselskii fixed point theorem.

Theorem 1.1[14]Let E be a Banach space,and K ⊂E be a cone.Assume that Ω1and Ω2are open bounded subsets of E with 0∈Ω1,,and let T:be a completely continuous operator such that either

(i)‖Tu‖≤‖u‖,u∈K∩∂Ω1,‖Tu‖≥‖u‖,u∈K∩∂Ω2;or

(ii)‖Tu‖≥‖u‖,u∈K∩∂Ω1,‖Tu‖≤‖u‖,u∈K∩∂Ω2, hold.Then T has at least a fixed point in

2　Preliminaries

In this section,we introduce some definitions and preliminary facts which are used throughout this paper.

Definition 2.1[7]The integral

where α＞0,is called Riemann-Liouville fractional integral of order α.

Definition 2.2[7]For a function f(x)given in the interval[0,1),the expression

where n=[α]+1 and[α]denotes the integer part of number α,is called the Riemann-Liouville fractional derivative of order α.

Lemma 2.1[7]Assume that u∈C(0,1)∩L(0,1)with a fractional derivative of order α＞0 that belongs to u∈C(0,1)∩L(0,1).Then

for some Ci∈R,i=1,2,···,N,where N is the smallest integer greater than or equal to α.

In order to obtain the results conveniently,we make use of the following lemmas.

Proof By Lemma 2.1,we can reduce the equation of problem(2.1)to an equivalent integral equation

By u(0)=0,there is C2=0.Thus,

Differentiating(2.4),we have

Therefore,

From(2.6),we have

It follows that

So,we obtain

The proof is complete.

We call G(t,s)=(G1α(t,s),G1β(t,s))the Green’s function of the boundary value problem(1.1).

(i)G1α(t,s)≥0 is continuous for all t,s∈[0,1],G1α(t,s)＞0 for all t,s∈(0,1);

(ii)G1α(t,s)≤G1α(s)for each t,s∈[0,1],G1α(s)=G2α(s,s)+G3α(1,s).

Lemma 2.4 There exists a γα＞0 such that

where θ∈(0,1/2).

Proof First,we show that

For t∈[θ,1-θ],we divide the proof into the following three cases for s∈[0,1].

Case 1 If s∈[1-θ,1],from the definition of G2α(t,s),we have

On the other hand,

Therefore,G2α(t,s)≥θα-1G2α(s,s).Letting θα-1=γ1,we have G2α(t,s)≥γ1G2α(s,s).

Case 2 If s∈[θ,1-θ],when t≤s this case is similar to Case 1.Then,if t≥s,

So,there exists a γ2=min{θα-1,θα}=θα,such that G2α(t,s)≥γ2G2α(s,s).

Case 3 If s∈[0,θ],from(2.2),it is easy to know that

Then,

And we can have

There exists a constant γ3such that G2α(t,s)≥γ3G2α(s,s).Letting r∗=min{γ1,γ2,γ3}, it follows that(2.7)holds.

Secondly,we show that

From(2.3)we have

Letting θα=γ∗∗,we have G3α(t,s)≥γ∗∗G3α(s,s).From the definitions of G1α(t,s) and G1α(s),let γα=min{γ∗,γ∗∗},then we obtain that

where γ3is defined in Case 3.This completes the proof.

For convenience,we list three assumptions to be used in the paper:

(B1)a,b∈C((0,1),[0,+∞))with a(t)≢0 and b(t)≢0 in any subinterval of(0,1),

(B2)f,g∈C([0,1]×[0,+∞));

(B3)φ,ψ∈C[0,1],φ′＜0,ψ′＜0 and φi∈Cr+(i=1,2).

We define Banach spaces X={u(t)|u(t)∈C[-r,1]}endowed with the norm‖u‖X=endowed with the norm‖v‖Y=

Clearly,(X×Y,‖(u,v)‖X×Y)is a Banach space.First of all,define P={(u,v)∈X× Y|u(t)≥0,v(t)≥0},then the cone P⊂X×Y.And

Lemma 2.5 Suppose that f(t,v)and g(t,u)are continuous,then(u,v)∈X×Y is a solution to the BVP(1.1)if and only if(u,v)∈X×Y is a solution to integral equations

Let T:X×Y→X×Y be an operator defined as

where T1v(t)=u(t),T2u(t)=v(t).Then by Lemma 2.5,the fixed point of operator T coincides with the solution to system(1.1).

Lemma 2.6Let f(t,v)and g(t,u)be continuous on[0,1]×[0,+∞)→ [0,+∞),,then T:P→P and T:K→K defined by(2.8)are completely continuous.

Proof Since the proof of Lemma 2.6 is similar to that of Lemma 3.2 in[22,23],we omit the proof.

3　Existence of at Least One Positive Solution

Set

Theorem 3.1Assume that(B1)-(B3)hold.And supposes that one of the following conditions is satisfied:

Then boundary value problem(1.1)has at least one positive solution.

Proof Assume that condition(i)holds.Considering f0=0,there exists an H1＞‖φ1‖[-r,0]such that f(t,v)≤ ∊1‖v‖[-r,0],for,0≤ ‖v‖[-r,0]≤ H1,where∊1＞0 satisfies∊1Λ2≤1.

Then,for t∈[0,1],(u,v)∈∂KH1,we get

Similarly,we have T2u(t)≤H1,that is(u,v)∈∂KH1implies that

For(u,v)∈∂KH2,we deduce that

Then we have

Hence‖T1v‖[-r,1]≥‖v‖[-r,1].Similarly,we have‖T2u‖[-r,1]≥‖u‖[-r,1],that is(u,v)∈∂KH2implies that

By the first part of Theorem 1.1,T has a fixed point

The proof of condition(ii)is similar to that of(i),so we omit it.

4　The Existence of Multiple Positive Solutions

Now we discuss the multiplicity of positive solutions to the boundary value problem (1.1).We obtain the following existence results.

Theorem 4.1 Assume that(B1)-(B3)hold.If the following two conditions are satisfied:

(ii)There exists a constant p＞0 such that

Then the boundary value problem(1.1)has at least two positive solutions(u1,v1)and(u2,v2) such that 0≤‖(u1,v1)‖[-r,1]＜p＜‖(u2,v2)‖[-r,1].

For(u,v)∈∂KP1,we deduce that,0≤‖vs‖[-r,0]for s∈[0,1],and

Then we have

This implies‖T1v‖[-r,1]≥ ‖v‖[-r,1].Similarly,we have‖T2u‖[-r,1]≥ ‖u‖[-r,1],that is (u,v)∈∂KP1implies that

For(u,v)∈∂Kp2,we haveand

Then we have

So we obtain‖T1v‖[-r,1]≥ ‖(v)‖[-r,1].Similarly,‖T2u‖[-r,1]≥ ‖(u)‖[-r,1],that is (u,v)∈∂Kp2implies that

Finally,we set

From(ii),for any v(t)∈∂Ωp,we have

which implies‖T1v‖[-r,1]≤ ‖v‖[-r,1].Similarly,‖T2u‖[-r,1]≤ ‖u‖[-r,1].Then,we obtain that

for(u,v)∈K and‖(u,v)‖[-r,1]=p.Hence,since p1＜p＜p2and(4.1)-(4.3),it follows from Theorem 1.1 that T has a fixed pointand a fixed point(u2,v2)∈,which are both the positive solutions to the boundary value problem(1.1)and 0≤‖(u1,v1)‖[-r,1]＜p＜‖(u2,v2)‖[-r,1].The proof is complete.

Theorem 4.2 Assume that(B1)-(B3)hold.If the following two conditions are satisfied:

(i)f0=f∞=0,g0=g∞=0.

(ii)There exists a constant q≥max{‖φ1‖[-r,0],‖φ2‖[-r,0]}such that

Then the boundary value problem(1.1)has at least two positive solutions(u1,v1)and(u2,v2) such that 0≤‖(u1,v1)‖[-r,1]＜q＜‖(u2,v2)‖[-r,1].

Proof The proof is similar to that of Theorem 4.1,so we omit it.

5　Examples

We illustrate Theorem 3.1.Consider the following boundary value problem

Then

as‖v‖[-r,0]→+∞,thus we have f∞=0.Similarly,g∞=0.

as‖v‖[-r,0]→0+.Thus,

All the conditions of Theorem 3.1 are satisfied,then the boundary value problem(1.1) has at least one positive solution.

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(edited by Liangwei Huang)

∗Beijing Higher Education Young Elite Teacher Project(No.YETP0388).

†Manuscript December 11,2014;Revised April 6,2015