IJDEInternational Journal of Differential Equations1687-96511687-9643Hindawi Publishing Corporation18692810.1155/2010/186928186928Research ArticlePositive Solution to Nonzero Boundary Values Problem for a Coupled System of Nonlinear Fractional Differential EquationsWangJinhuaXiangHongjunLiuZhigangMomaniShaherDepartment of MathematicsXiangnan UniversityChenzhou 423000Chinaxnu.edu.cn201016062009201013042009090620092010Copyright © 2010This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider the existence and uniqueness of positive solution to nonzero boundary values problem for a coupled system of fractional differential equations. The differential operator is taken in the standard Riemann-Liouville sense. By using Banach fixed point theorem and nonlinear differentiation of Leray-Schauder type, the existence and uniqueness of positive solution are obtained. Two examples are given to demonstrate the feasibility of the obtained results.

1. Introduction

Fractional differential equation can describe many phenomena in various fields of science and engineering such as control, porous media, electrochemistry, viscoelasticity, and electromagnetic. There are many papers dealing with the existence and uniqueness of solution for nonlinear fractional differential equation; see, for example, . In , the authors investigated a singular coupled system with initial value problems of fractional order. In , Su discussed a boundary value problem of coupled system with zero boundary values. By means of Schauder fixed point theorem, the existence of the solution is obtained. The nonzero boundary values problem of nonlinear fractional differential equations is more difficult and complicated. No contributions exist, as far as we know, concerning the existence of positive solution for coupled system of nonlinear fractional differential equations with nonzero boundary values.

In this paper, we consider the existence and uniqueness of positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations: Dαu(t)+f(t,v(t))=0,0<t<1,Dβv(t)+g(t,u(t))=0,0<t<1,u(0)=0,u(1)=au(ξ),v(0)=0,v(1)=bv(ξ), where 1<α<2,1<β<2,0a,b1,0<ξ<1, f,g:[0,1]×[0,+)[0,+) are given functions, and D is the standard Riemann-Liouville differentiation. By using Banach fixed point theorem and nonlinear differentiation of Leray-Schauder type, some sufficient conditions for the existence and uniqueness of positive solution to the above coupled boundary values problem are obtained.

The rest of the paper is organized as follows. In Section 2, we introduce some basic definitions and preliminaries used in later. In Section 3, the existence and uniqueness of positive solution for the coupled boundary values problem (1.1) will be discussed, and examples are given to demonstrate the feasibility of the obtained results.

2. Basic Definitions and Preliminaries

In this section, we introduce some basic definitions and lemmas which are used throughout this paper.

Definition 2.1 (see [<xref ref-type="bibr" rid="B6">6</xref>, <xref ref-type="bibr" rid="B7">7</xref>]).

The fractional integral of order α  (α>0) of a function y:(0,)R is given by Iαy(t)=1Γ(α)0t(t-s)α-1y(s)ds, provided that the right side is pointwise defined on (0,).

Definition 2.2 (see [<xref ref-type="bibr" rid="B6">6</xref>, <xref ref-type="bibr" rid="B7">7</xref>]).

The fractional derivative of order α>0 of a continuous function y:(0,)R is given by Dαy(t)=1Γ(n-α)(ddt)n0t(t-s)n-α-1y(s)ds, where n=[α]+1 provided that the right side is pointwise defined on (0,).

Remark 2.3 (see [<xref ref-type="bibr" rid="B2">3</xref>]).

The following properties are useful for our discussion:

IαDαu(t)=u(t)-k=1NCktα-k,Dαu(t)C(0,1)L(0,1), CkR, N=[α]+1,

DαIαu(t)=u(t),

Dαtγ=Γ(γ+1)/Γ(γ+1-α)tγ-α,α>0,γ>-1,γ>α-1,t>0.

Lemma 2.4 (the nonlinear alternative of Leray and Schauder type [<xref ref-type="bibr" rid="B8">8</xref>]).

Let E be a Banach space with CE closed and convex. Let U be a relatively open subset of C with 0U and let T:U¯C be a continuous and compact mapping. Then either

the mapping T has a fixed point in U̅, or

there exist uU and λ(0,1) with u=λTu.

Consider Dαu(t)+y(t)=0,0<t<1,u(0)=0,u(1)=au(ξ), then one has the following lemma.

Lemma 2.5.

Let yC[0,1] and 1<α<2, then u(t) is a solution of BVP (2.3) if and only if u(t) is a solution of the integral equation: u(t)=01G1(t,s)y(s)ds, where G1(t,s)={[t(1-s)]α-1-atα-1(ξ-s)α-1-(t-s)α-1(1-aξα-1)(1-aξα-1)Γ(α),0st1,sξ,[t(1-s)]α-1-(t-s)α-1(1-aξα-1)(1-aξα-1)Γ(α),0<ξst1,[t(1-s)]α-1-atα-1(ξ-s)α-1(1-aξα-1)Γ(α),0tsξ1,[t(1-s)]α-1(1-aξα-1)Γ(α),0ts1,ξs.

Proof.

Assume that u(t) is a solution of BVP (2.3), then by Remark 2.3, we have u(t)=-Iαy(t)+C1tα-1+C2tα-2=-0t(t-s)α-1Γ(α)y(s)ds+C1tα-1+C2tα-2.

By (2.3), we have C2=0,C1=01(1-s)α-1Γ(α)(1-aξα-1)y(s)ds-a0ξ(ξ-s)α-1Γ(α)(1-aξα-1)y(s)ds.

Therefore, we obtain u(t)=-0t(t-s)α-1Γ(α)y(s)ds+01tα-1(1-s)α-1Γ(α)(1-aξα-1)y(s)ds-a0ξtα-1(ξ-s)α-1Γ(α)(1-aξα-1)y(s)ds=01G1(t,s)y(s)ds.

Conversely, if u(t) is a solution of integral equation (2.4), using the relation Dαtα-m=0,m=1,2,,N, where N is the smallest integer greater than or equal to α [3, Remark  2.1], we have Dαu(t)=-Dα(0t(t-s)α-1Γ(α)y(s)ds)+Dαtα-1[01(1-s)α-1Γ(α)(1-aξα-1)y(s)ds-a0ξ(ξ-s)α-1Γ(α)(1-aξα-1)y(s)ds]=-DαIαy(t)=-y(t). A simple computation showed u(0)=0,u(1)=au(ξ). The proof is complete.

Let G2(t,s)={[t(1-s)]β-1-btβ-1(ξ-s)β-1-(t-s)β-1(1-bξβ-1)(1-bξβ-1)Γ(β),0st1,sξ,[t(1-s)]β-1-(t-s)β-1(1-bξβ-1)(1-bξβ-1)Γ(β),0<ξst1,[t(1-s)]β-1-btβ-1(ξ-s)β-1(1-bξβ-1)Γ(β),0tsξ1,[t(1-s)]β-1(1-bξβ-1)Γ(β),0ts1,ξs, we call G(t,s)=(G1(t,s),G2(t,s)) Green's function of the boundary value problem (1.1).

Lemma 2.6.

Let 0a,b1, then the function G(t,s) is continuous and satisfies

G(t,s)>0,for  t,s(0,1),

G(t,s)G(s,s),for  t,s(0,1).

Proof.

It is easy to prove that G(t,s) is continuous on [0,1]×[0,1], here we omit it. Now we prove G1(t,s)>0. Let g1(t,s)=[t(1-s)]α-1-atα-1(ξ-s)α-1-(t-s)α-1(1-aξα-1)(1-aξα-1)Γ(α),0<st1,sξ,g2(t,s)=[t(1-s)]α-1-(t-s)α-1(1-aξα-1)(1-aξα-1)Γ(α),0<ξst1,g3(t,s)=[t(1-s)]α-1-atα-1(ξ-s)α-1(1-aξα-1)Γ(α),0<tsξ1,g4(t,s)=[t(1-s)]α-1(1-aξα-1)Γ(α),0<ts1,ξs. We only need to prove g1(t,s)>0,0<st1,sξ. Since [t(1-s)]α-1-atα-1(ξ-s)α-1-(t-s)α-1(1-aξα-1)=tα-1[(1-s)]α-1-a(ξ-s)α-1-(1-st)α-1(1-aξα-1), set g(t)=(1-s)α-1-a(ξ-s)α-1-(1-s/t)α-1(1-aξα-1), we have g(t)=-(α-1)(1-st)α-2st2(1-aξα-1)0,for  0<s<t1,sξ.

Then g(t) is decreasing on (0,1). Meanwhile, g(1)=(1-s)α-1-a(ξ-s)α-1-(1-s)α-1(1-aξα-1)=aξα-1[(1-s)α-1-(1-sξ)α-1]>0,0<s<t1,sξ.

Therefore, g1(t,s)>0, for 0<s<t1,sξ. Clearly g1(t,s)>0,t=s, so g1(t,s)>0,s,t(0,1). It is easy to show that g2(t,s)>0,g3(t,s)>0, g4(t,s)>0. Hence, G1(t,s)>0,s,t(0,1).

Similarly, G2(t,s)>0,s,t(0,1). The proof of (1) is completed.

Let g2(t)=[t(1-s)]α-1-(t-s)α-1(1-aξα-1)(1-aξα-1)Γ(α),0<ξst1, then, g2(t)=(α-1)tα-2[(1-s)]α-1-(1-s/t)α-2(1-aξα-1)(1-aξα-1)Γ(α),0<ξs<t1,[(1-s)]α-1-(1-s/t)α-2(1-aξα-1)[(1-s)]α-1-(1-s)α-2(1-aξα-1)=[(1-s)]α-2(aξα-1-s)0,0<ξs<t1, therefore, g2(t)0,      0<ξs<t1. So, g2(t,s) is decreasing with respect to t. Similarly, g1(t,s) is decreasing with respect to t. Also g3(t,s) and g4(t,s) are increasing with respect to t. We obtain that G1(t,s) is decreasing with respect to t for st and increasing with respect to t for ts.

With the use of the monotonicity of G1(t,s), we have max0t1G1(t,s)=G1(s,s)={[s(1-s)]α-1-a[s(ξ-s)]α-1Γ(α)(1-aξα-1),s(0,ξ],[s(1-s)]α-1Γ(α)(1-aξα-1),s[ξ,1). Similarly, max0t1G2(t,s)=G2(s,s)={[s(1-s)]β-1-b[s(ξ-s)]β-1Γ(β)(1-bξβ-1),s(0,ξ],[s(1-s)]β-1Γ(β)(1-bξβ-1),s[ξ,1). The proof of (2) is completed.

3. Main Result

In this section, we will discuss the existence and uniqueness of positive solution for boundary value problem (1.1).

We define the space X={u(t)u(t)C[0,1]} endowed with uX=max0t1|u(t)|, Y={v(t)v(t)C[0,1]} endowed with uY=max0t1|v(t)|.

For (u,v)X×Y, let (u,v)X×Y=max{uX,vY}.

Define P={(u,v)X×Yu(t)0,v(t)0}  , then the cone PX×Y.

From Lemma 2.5 in Section 2, we can obtain the following lemma.

Lemma 3.1.

Suppose that f(t,v) and g(t,u) are continuous, then (u,v)X×Y is a solution of BVP (1.1) if and only if (u,v)X×Y is a solution of the integral equations u(t)=01G1(t,s)f(s,v(s))ds,v(t)=01G2(t,s)g(s,u(s))ds. Let T:X×YX×Y be the operator defined as T(u,v)(t)=(01G1(t,s)f(s,v(s))ds,01G2(t,s)g(s,u(s))ds)=:(T1v(t),T2u(t)), then by Lemma 3.1, the fixed point of operator T coincides with the solution of system (1.1).

Lemma 3.2.

Let f(t,v) and g(t,u) be continuous on [0,1]×[0,)[0,), then T:PP defined by (3.2) is completely continuous.

Proof.

Let (u,v)P, in view of nonnegativeness and continuity of functions G(t,s), f, and g, we conclude that T:PP is continuous.

Let ΩP be bounded, that is, there exists a positive constant h>0 such that (u,v)h for all (u,v)Ω.

Let M=max{|f(t,v(t))|+1:0t1,0vh},N=max{|g(t,u(t))|+1:0t1,0uh}, then we have |T1v(t)|=|01G1(t,s)f(s,v(s))ds|M01G1(s,s)ds,|T2u(t)|=|01G2(t,s)g(s,u(s))ds|N01G2(s,s)ds. Hence, T(u,v)max{M01G1(s,s)ds,N01G2(s,s)ds}. T(Ω) is uniformly bounded.

Since G1(t,s) is continuous on [0,1]×[0,1], it is uniformly continuous on [0,1]×[0,1]. Thus, for fixed s[0,1] and for any ε>0, there exists a constant δ>0, such that any t1,t2[0,1] and |t1-t2|<δ,|G1(t1,s)-G1(t2,s)|<ε/M.

Then |T1(v)(t2)-T1(v)(t1)|M01|G1(t2,s)-G1(t1,s)|ds<ε.

Similarly, |T2(u)(t2)-T2(u)(t1)|N01|G2(t2,s)-G2(t1,s)|ds<ε. For the Euclidean distance d on R2, we have that if t1,t2[0,1] are such that |t2-t1|<δ, then d(T(u,v)(t2),T(u,v)(t1))=(T1v(t2)-T1v(t1))2+(T2u(t2)-T2u(t1))2<2ε. That is to say, T(P) is equicontinuous. By the means of the Arzela-Ascoli theorem, we have T:PP is completely continuous. The proof is completed.

Theorem 3.3.

Assume that f(t,v) and g(t,u) are continuous on [0,1]×[0,)[0,), and there exist two positive functions m(t),n(t) that satisfy

|f(t,v2)-f(t,v1)|m(t)|v2-v1|,for  t[0,1],v1,v2[0,),

|g(t,u2)-g(t,u1)|n(t)|u2-u1|,for  t[0,1],u1,u2[0,).

Then system (1.1) has a unique positive solution if ρ=01G1(s,s)m(s)ds<1,θ=01G2(s,s)n(s)ds<1.

Proof.

For all (u,v)P, by the nonnegativeness of G(t,s) and f(t,v),g(t,u), we have T(u,v)(t)0. Hence, T(P)P.T1v2-T1v1=maxt[0,1]|T1v2-T1v1|=maxt[0,1]|01G1(t,s)[f(s,v2(s))-f(s,v1(s))]ds|01G1(s,s)m(s)dsv2-v1ρv2-v1. Similarly, T2u2-T2u1θu2-u1. We have, T(u2,v2)-T(u1,v1)max(ρ,θ)(u2,v2)-(u1,v1). From Lemma 3.2, T is completely continuous, by Banach fixed point theorem, the operator T has a unique fixed point in P, which is the unique positive solution of system (1.1). This completes the proof.

Theorem 3.4.

Assume that f(t,v) and g(t,u) are continuous on [0,1]×[0,)[0,) and satisfy

|f(t,v(t))|a1(t)+a2(t)|v(t)|,

|g(t,u(t))|b1(t)+b2(t)|u(t)|,

A1=01G1(s,s)a2(s)ds<1,0<B1=01G1(s,s)a1(s)ds<,

A2=01G2(s,s)b2(s)ds<1,0<B2=01G2(s,s)b1(s)ds<.

Then the system (1.1) has at least one positive solution (u,v) in C={(u,v)P(u,v)<min(B11-A1,B21-A2)}.

Proof.

Let C={(u,v)X×Y:(u,v)<r} with r=min(B1/(1-A1),B2/(1-A2)), define the operator T:CP as (3.2).

Let (u,v)C, that is, (u,v)<r. Then T1v=maxt[0,1]|01G1(t,s)f(s,v(s))ds|01G1(s,s)(a1(s)+a2(s)|v(s)|)ds01G1(s,s)a1(s)ds+01G1(s,s)a2(s)dsv=B1+A1vr.

Similarly, T2ur, so T(u,v)r, T(u,v)C¯. From Lemma 3.2T:C¯C¯ is completely continuous.

Consider the eigenvalue problem (u,v)=λT(u,v),  λ(0,1).

Under the assumption that (u,v) is a solution of (3.15) for a λ(0,1), one obtains u=λT1v=λmaxt[0,1]|01G1(t,s)f(s,v(s))ds|<01G1(s,s)(a1(s)+a2(s)|v(s)|)ds=01G1(s,s)a1(s)ds+01G1(s,s)a2(s)dsv=B1+A1vr. Similarly, v=λT2u<r, so (u,v)<r, which shows that (u,v)C. By Lemma 2.4, T has a fixed point in C¯. We complete the proof of Theorem 3.4.

Example 3.5.

Consider the problem D7/4u(t)+f(t,v(t))=0,0<t<1,D3/2v(t)+g(t,u(t))=0,0<t<1,u(0)=0,u(1)=12u(12),v(0)=0,v(1)=34v(12), where f(t,v(t))=tv(t)(1+t)(1+v(t)),g(t,u(t))=arctant1+t|sinu(t)|. Set v1(t),v2(t),u1(t),u2(t)[0,) and t[0,1], then we have |f(t,v2(t))-f(t,v1(t))|t1+t|v2(t)-v1(t)|,|g(t,u2(t))-g(t,u1(t))|arctant1+t|u2(t)-u1(t)|. Therefore, ρ=01G1(s,s)m(s)ds01G1(s,s)ds=1Γ(7/4)(1-(1/2)7/4){01/2[s(1-s)]3/4ds-01/212[s(12-s)]3/4ds+1/21[s(1-s)]3/4ds}=2(1+(1/2)7/4)  5·Γ(3/4)Γ(1/2)<45<1,θ=01G2(s,s)n(s)dsπ401G2(s,s)ds=π41Γ(3/2)(1-(3/4)(1/2)1/2){01/2[s(1-s)]1/2ds-01/2  34[s(12-s)]1/2ds+1/21[s(1-s)]1/2ds  }=π4[1-(3/4)(1/2)2][1-(3/4)(1/2)1/2]·Γ(3/2)Γ(3)0.6018<1. With the use of Theorem 3.3, BVP (3.17) has a unique positive solution.

Example 3.6.

Consider the problem D7/4u(t)+f(t,v(t))=0,0<t<1,D3/2v(t)+g(t,u(t))=0,0<t<1,u(0)=0,u(1)=12u(12),v(0)=0,v(1)=34v(12), where f(t,v(t))=t2+t1+tln(1+v(t)),g(t,u(t))=10+t220+u(t). We have |f(t,v(t))|t2+t1+t·|v(t)|,|g(t,u(t))|(10+t220)+|u(t)|. Hence, A1=01G1(s,s)a2(s)ds01G1(s,s)ds=2(1+(1/2)7/4)5·Γ(3/4)Γ(1/2)<1,B1=01G1(s,s)a1(s)ds=01G1(s,s)·s2ds<,A2=01G2(s,s)b2(s)ds=01G2(s,s)ds0.7666<1,B2=01G2(s,s)b1(s)ds=01G2(s,s)(10+s220)ds<. By Theorem 3.4, BVP (3.21) has at least one positive solution in C={(u,v)Pu,v<min(B11-A1,B21-A2)}.

Acknowledgments

This work was jointly supported by the Natural Science Foundation of Hunan Provincial Education Department under Grants 07A066 and 07C700, the Construct Program of the Key Discipline in Hunan Province, Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province, and the Foundation of Xiangnan University.

BaiC.FangJ.The existence of a positive solution for a singular coupled system of nonlinear fractional differential equationsApplied Mathematics and Computation20041503611621MR203966210.1016/S0096-3003(03)00294-7ZBL1061.34001SuX.Boundary value problem for a coupled system of nonlinear fractional differential equationsApplied Mathematics Letters20092216469MR248316310.1016/j.aml.2008.03.001BaiZ.H.Positive solutions for boundary value problem of nonlinear fractional differential equationJournal of Mathematical Analysis and Applications20053112495505MR216841310.1016/j.jmaa.2005.02.052ZBL1079.34048ZhangS.zhangshuqin@tsinghua.org.cnExistence of solution for a boundary value problem of fractional orderActa Mathematica Scientia200626222022810.1016/S0252-9602(06)60044-1ZBL1106.34010ZhangS. Q.Positive of solution for boundary-value problems of nonlinear fractional differential equationsElectronic Journal of Differential Equations200636112PodlubnyI.Fractional Differential Equations1999198San Diego, Calif, USAAcademic Pressxxiv+340Mathematics in Science and EngineeringMR1658022ZBL0924.34008KilbasA. A.SrivastavaH. M.TrujilloJ. J.Theory and Applications of Fractional Differential Equations2006204Amsterdam, The NetherlandsElsevier Sciencexvi+523North-Holland Mathematics StudiesMR2218073ZBL1092.45003ZeidlerE.Nonlinear Functional Analysis and Its Applications—I: Fixed-Point Theorems1986New York, NY, USASpringerxxi+897MR816732ZBL0583.47050