AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation76892010.1155/2009/768920768920Research ArticleExistence of Positive Solutions for Multiterm Fractional Differential Equations of Finite Delay with Polynomial CoefficientsBabakhaniA.1EnteghamiE.2AticiFerhan1Department of Basic ScienceBabol University of TechnologyBobol 47148-71167Irannit.ac.ir2Department of MathematicsUniversity of MazandaranBabolsar 47418-1468Iranumz.ac.ir2009279200920090902200907062009010920092009Copyright © 2009This 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.

Existence of positive solutions has been studied by A. Babakhani and V. Daftardar-Gejji (2003) in case of multiterm nonautonomous fractional differential equations with constant coefficients. In the present paper we discuss existence of positive solutions in case of multiterm fractional differential equations of finite delay with polynomial coefficients.

1. Introduction

In last 30 years, the theory of ordinary differential equations of fractional order has become a new important branch (see, e.g.,  and the references therein). Numerous applications of such equations have been presented . Existence of positive solution of fractional ordinary differential equations has been well investigated for fractional functional differential equations [1, 6, 1114]. Ye et al.  have addressed the question of existence of positive solutions for the nonlinear fractional functional differential equation Dα[x(t)-x(0)]=x(t)f(t,xt),t(0,T],x(t)=ϕ(t)0,t[-w,0], by using the sub- and supersolution method, where 0<α<1, Dα is the standard Riemann-Liouville fractional derivative, ϕC and f:[0,T]×C+ is continuous, as usual, C=C([-w,0],+) is the space of continuous function from [-w,0] to +, w>0, equipped with the sup norm: ϕ=max-wΘ0|ϕ(t)|, and xt denotes the function in C defined by xt(θ)=x(t+θ),-wθ0. They require that the nonlinearity f(t,xt) is increasing in xt for each t[0,T].

As a pursuit of this in the present paper, we deal with the existence of positive solutions in the case of multiterm differential equations with polynomial coefficients of the fractional type: (D)[x(t)-x(0)]=f(t,xt),t(0,T],x(t)=ϕ(t)0,t[-w,0], where (D)=Dαn-j=1n-1pj(t)Dαn-j,0<α1<<αn<1,pj(t)=k=0Njajktk,pj(2m)(t)0,pj(2m+1)(t)0,m=0,1,,[Nj2],j=1,2,,n-1, and Dαj is the standard Riemann-Liouville fractional derivative, T>0, w>0, ϕC=C([-w,0],+) and f:I×C+ is a given continuous function, I=[0,T].

2. Preliminaties

Let E be a real Banach space with a cone K. K introduces a partial order in E in the following manner : xyify-xK.

Definition 2.1 (see [<xref ref-type="bibr" rid="B12">15</xref>]).

For x,yE the order interval x,y is defined as x,y={zE:xzy}.

Definition 2.2 (see [<xref ref-type="bibr" rid="B12">15</xref>]).

A cone K is called normal, if there exists a positive constant r such that f,gK and ϑfg implies frg, where ϑ denotes the zero element of K.

Definition 2.3 (see [<xref ref-type="bibr" rid="B13">16</xref>, <xref ref-type="bibr" rid="B14">17</xref>]).

Let f:[a,b], and fL1[a,b]. The left-sided Riemann-Liouville fractional integral of f of order α is defined as Iaαf(x)=1Γ(α)ax(x-t)α-1f(t)dt,α>0,x[a,b].

Definition 2.4 (see [<xref ref-type="bibr" rid="B13">16</xref>, <xref ref-type="bibr" rid="B14">17</xref>]).

The left-sided Riemann-Liouville fractional derivative of a function f:[a,b] is defined as Daαf(x)=Dm[Iam-αf(x)],x[a,b], where m=[α]+1, Dm=dm/dtm. We denote D0α by Dα and I0α by Iα. If the fractional derivative Daαf(x) is integrable, then [16, page 71] Iaα(Daβf(x))=Iaα-βf(x)-[Ia1-βf(x)]x=axα-1Γ(α),0<βα<1.

If f is continuous on [a,b], then [Ia1-βf(x)]x=a=0 and (2.5) reduces to Iaα(Daβf(x))=Iaα-βf(x),0<βα<1.

Proposition 2.5.

Let y be continuous on [0,T], T>0 and let n be a nonnegative integer, then Iα(tnx(t))=k=0n(-αk)[Dktn][Iα+kx(t)]=k=0n(-αk)n!tn-k(n-k)!Iα+kx(t), where (-αk)=(-1)kΓ(α+1)k!Γ(α)=(-1)k(αk)=Γ(1-α)Γ(k+1)Γ(1-α-k).

The proof of the above proposition can be found in [17, page 53].

Corollary 2.6.

Let xC[0,T], T>0 and pj(t)=k=0Njajktk, Nj{0}, j=1,2,,n, then Iα(j=1npj(t)x(t))=j=1nk=0Njr=0kajk(-αr)k!tk-r(k-r)![Iα+rx(t)].

Theorem 2.7 (see [<xref ref-type="bibr" rid="B19">10</xref>]).

Let E be a Banach space with CE closed and convex. Assume that U is a relatively open subset of C with 0U and F:U¯C is a continuous and compact map. Then either

F has a fixed point in U¯, or

there exists uU and λ(0,1) with u=λF(u).

3. Existence of Positive Solution

In this section, we discuss the existence of positive solutions for (1.4). Using (2.5), (2.6), and Corollary 2.6, (1.4) is equivalent to the integral equation x(t)={x(0)+[x(t)-x(0)]+Iαnf(t,xt),t(0,T],ϕ(t),t[-w,0], where =j=1n-1k=0Njr=0kajk(-αnr)k!tk-r(k-r)!Iαn-αn-j+r. Let y(·):[-w,T][0,+) be the function defined by y(t)={ϕ(0),tI,ϕ(t)0,t[-w,0], then y0=ϕ, for each zC(I,) with z(0)=0, we denote by z̅ the function define by z̅(t)={z(t),tI,0,t[-w,0]. We can decompose x(·) as x(t)=z̅(t)+y(t), t[-w,T], which implies xt=z̅t+yt, for tI. Therefore, (3.1) is equivalent to the integral equation z(t)=z(t)+Iαnf(t,z̅t+yt),tI, where is defined (3.2). Set A0={zC(I,):z0=0} and let zT be the seminorm in A0 defined by zT=z0+z=z=:sup{|z(t)|:tI},zA0, and A0 is a Banach space with norm ·T. Let K be a cone of A0, K={zA0;z(t)0,tI} and K*={xC([-w,T],+);x(t)=ϕ(t)0,t[-w,0]}. Define the operator F:KK by Fz(t)=z(t)+Iαnf(t,z̅t+yt),tI.

Theorem 3.1.

Suppose that the following conditions hold:

there exist p,qC(I,+) such that f(t,xt)p(t)+q(t)xt, for tI, xtC, and Iαnp=supt[0,T]Iαnp(t)<, Iαnq=supt[0,T]Iαnq(t)<,

1-(T)-Iαnq>0, where (T)=j=1n-1k=0Njr=0k|ajk(-αnr)|k!Tαn-αn-j+k(k-r)!Γ(αn-αn-j+r).

Then (1.4) has at least a positive solution x*K*, satisfying x*max{ϕ,h}, where h=λϕIαnq+λIαnp1-λ(T)-λIαnq+1.

Proof.

We will show that the operator F:KK is continuous and completely continuous.Step 1.

The operator F:KK is continuous in view of the continuity of f.

Step 2.

F maps bounded sets into bounded sets in K.

Let GK be bounded; that is, there exists a positive constant l such that zTl, for all zG. For each zG, we have |Fz(t)|j=1n-1k=0Njr=0k|ajk(-αnr)|k!tk-r(k-r)!Iαn-αn-j+r|z(t)|+Iαnf(t,z̅t+yt)j=1n-1k=0Njr=0k|ajk(-αnr)|lk!tαn-αn-j+k(k-r)!Γ(αn-αn-j+r)+Iαnf(t,z̅t+yt)l(T)+Iαn{p(t)+q(t)z̅t+yt}, where (T) is defined in (3.9). It follows that FzTl(T)+Iαnp+lIαnq+ϕ(t)Iαnq. Hence FG is bounded.

Step 3.

F maps bounded sets into equicontinuous sets of K.

We will show that FG is equicontinuous. For each zG, t1,t2I and t1<t2, then for given ϵ>0, choose δ=min{[ϵC(j,k,r)4]1/(αn-αn-j+r),[ϵΓ(αn+1)4(p+q(l+ϕ))]1/αn}, where j=1,2,,n-1, k=0,1,,Nj, r=0,1,,k, C(j,k,r)=(k-r)!i=1n-1(Ni+1)(Ni+2)×Γ(αn-αn-j+r+1)|ajk(-αnr)|lηk! and η=max{1,TNj,j=1,2,,n-1}. If |t1-t2|<δ, |Fz(t1)-Fz(t2)|=|j=1n-1k=0Njr=0k|ajk(-αnr)|k!t1k-r(k-r)!Γ(αn-αn-j+r)0t1(t1-s)αn-αn-j+r-1z(s)ds-j=1n-1k=0Njr=0k|ajk(-αnr)|k!t2k-r(k-r)!Γ(αn-αn-j+r)0t2(t2-s)αn-αn-j+r-1z(s)ds|+1Γ(αn)|0t1(t1-s)αn-1f(s,z̅s+ys)ds-0t2(t2-s)αn-1f(s,z̅s+ys)ds||j=1n-1k=0Njr=0k|ajk(-αnr)|k!t2k-r(k-r)!Γ(αn-αn-j+r)0t1(t2-s)αn-αn-j+r-1z(s)ds-j=1n-1k=0Njr=0k|ajk(-αnr)|k!t1k-r(k-r)!Γ(αn-αn-j+r)0t1(t1-s)αn-αn-j+r-1z(s)ds|+j=1n-1k=0Njr=0k|ajk(-αnr)|k!t2k-r(k-r)!Γ(αn-αn-j+r)t1t2(t2-s)αn-αn-j+r-1|z(s)|ds+fΓ(αn){0t1[(t2-s)αn-1-(t1-s)αn-1]ds+t1t2(t2-s)αn-1ds}j=1n-1k=0Njr=0k|ajk(-αnr)|lk!Tk-r(k-r)!Γ(αn-αn-j+r)×0t2{(t2-s)αn-αn-j+r-1-(t1-s)αn-αn-j+r-1}ds+p+q(l+ϕ)Γ(αn){0t1[(t2-s)αn-1-(t1-s)αn-1]ds+t1t2(t2-s)αn-1ds}j=1n-1k=0Njr=0k|ajk(-αnr)|2lk!η(t2-t1)αn-αn-j+r(k-r)!Γ(αn-αn-j+r+1)+2(p+q(l+ϕ))(t2-t1)αnΓ(αn+1)=j=1n-1k=0Njr=0k|ajk(-αnr)|2lk!ηδαn-αn-j+r(k-r)!Γ(αn-αn-j+r+1)+2(p+q(l+ϕ))δαnΓ(αn+1)ϵ2+ϵ2=ϵ. Hence FG is equicontiuous. The Arzela-Ascoli theorem implies that F(G)¯ is compact and F:KK is continuous and completely continuous.

Step 4.

We now show that there exists an open set UK with zλF(z) for λ(0,1) and zU. Let zK be any solution of z=λFz, λ(0,1), where F is given by (3.8); since F:KK is continuous and completely continuous, we have z(t)=λFz(t)λj=1n-1k=0Njr=0k|ajk(-αnr)|k!tk-r(k-r)!Γ(αn-αn-j+r)×0t(t-s)αn-αn-j+r-1z(s)ds+λΓ(αn)0t(t-s)αn-1f(s,z̅s+ys)dsλj=1n-1k=0Njr=0k|ajk(-αnr)|zk!tk-r(k-r)!Γ(αn-αn-j+r)0t(t-s)αn-αn-j+r-1ds+λΓ(αn)0t(t-s)αn-1[p(s)+q(s)z̅s+ys]dsλ{z(T)+zIαnq(t)+ϕIαnq(t)+Iαnp(t)}. So z(1-λ(T)-λIαnp)λϕIαnq+λIαnp. Now, by (3.10) and (3.17), we know that any solution z of (3.8) satisfies zh; let U={zK;z<h}. Therefore, Theorem 2.7 guarantees that (3.1) has at least a positive solution zU̅. Hence, (1.4) has at least a positive solution x*K*, satisfying x*max{ϕ,h} and the proof is complete.

Note that we can complete the above mentioned procedure by using only the continuity of f without condition (1), but with our procedure and details of condition (1) in Theorem 3.1 answers all the questions exist in the following remark.

Remark 3.2.

When f is continuous on (0,T]×C, limt0+f(t,·)=+, (i.e., f is singular at t=0) in (1.4). Suppose σ(0,αn], such that tσf(t,xt) is a continuous function on [0,T]×C, then Iαnf(t,xt)=Iαnt-σtσf(t,xt) is continuous on I×C by Lemma 2.1 in [12, page 613]. We also obtain results about the existence to (1.4) by using a nonlinear alternative of Leray-Schauder type. The proof is similar to that of Theorem 3.1 as long as we let

tσf(t,xt)p(t)+q(t)xt, for tI, xtC, and Iαnt-σp<, Iαnt-σq<,

1-(T)-Iαnt-σq>0, then (1.4) has at least a positive solution x*K*, satisfying x*max{ϕ,h}, where h=λIαnt-σp+λϕIαnt-σq1-λ(T)-λIαnt-σq+1.

4. Unique Existence of Solution

In this section, we will give uniqueness of positive solution to (1.4).

Theorem 4.1.

Let f:I×C+ be continuous and λL1([0,T],+) with Iαnλ<. Further assume

|f(t,u̅t+yt)-f(t,v̅t+yt)|λ(t)u̅t-v̅t, for all u,vK, t[0,T],

(T)+Iαnλ<1.

Then (1.4) has unique solution which is positive, where (T) is given in (3.9).

Proof.

Let u,vK. Then we obtain |Fu(t)-Fv(t)|j=1n-1k=0Njr=0k|ajk(-αnr)|k!tk-r(k-r)!Iαn-αn-j+r|u(t)-v(t)|+Iαn|f(t,u̅t+y+t)-f(t,v̅t+yt)|u-vT{j=1n-1k=0Njr=0k|ajk(-αnr)|k!tαn-αn-j+k(k-r)!Γ(αn-αn-j+r+1)+Iαnλ(t)}u-vT{j=1n-1k=0Njr=0k|ajk(-αnr)|k!Tαn-αn-j+k(k-r)!Γ(αn-αn-j+r+1)+Iαnλ(t)}, where F is given in (3.8). Hence, Fu-FvT((T)+Iαnλ)u-vT. In view of Banach fixed point theorem F has unique fixed point in K, which is the unique positive solution of (2.7) and (1.4) has a unique positive solution in K*.

Remark 4.2.

When λ(t)=L>0, then condition (i) reduces to the Lipschitz condition.

Example 4.3.

Let λ(t)=L>0 and f(t,xt)=Lxt+et=Lx(t-ω)+et, ω>0. Consider the equation (D1/2-at2D1/4-btD1/6-cD1/8)x=Lx(t-w)+et,t(0,64],x(t)=0,t[-w,0]. Then (4.3) is equivalent to the integral equation, x(t)=j=13k=0Njr=0kajk(-12r)k!tk-r(k-r)!I1/2-α3-j+rx(t)+I1/2(Lx(t-ω)+et). Here α3=1/2, p1(t)=k=02a1ktk=at2, then N1=2, a10=a11=0, a12=a, p2(t)=k=01a2ktk=bt, then N2=1, a20=0,a21=b and p3(t)=k=00a3ktk=c, then N3=0, a30=c. Hence x(t)=a10(-120)I1/2-1/4x+a11[(-120)tI1/2-1/4x+(-121)I1/2-1/4+1]+a12[(-120)t2I1/2-1/4x+2(-121)tI1/2-1/4+1x+2(-122)I1/2-1/4+2x]+a20(-120)I1/2-1/6x+a21[(-120)tI1/2-1/6x+(-121)I1/2-1/6+1x]+a30(-120)I1/2-1/8x+LI1/2x(t-ω)+I1/2et. In view of (2.8) and that Γ(1/2)=π, Γ(-1/2)=-2π and Γ(-3/4)=4π/3 we obtain x(t)=a[t2I1/4x(t)-tI5/4x(t)+34I9/4x(t)]+b[(1+t)I1/3x(t)-12I4/3x(t)]+cI3/8+LI1/2x(t-ω)+I1/2et. If |a|3/5, |b|2/5, |c|1/5, 0<L<4/5 in the above equation satisfy the conditions required in Theorem 4.1, the iterated sequence is x1(t)=I1/2et=t1/2E1,3/2(t),x2(t)=[a(t2I1/4-tI5/4+34I9/4)+b((1+t)I1/3-12I4/3)+cI3/8+LI1/2]x1(t)+x1(t),xn+1(t)=k=0n[a(t2I1/4-tI5/4+34I9/4)+b((1+t)I1/3-12I4/3)+cI3/8+LI1/2]n-kx1(t), for n=1,2,3,, where Iαx1=tα+1/2E1,α+3/2(t), α>0, x(t)=limnxn(t) is the unique solution, which may not be positive, where Eα,β(t)=k=0(tk/Γ(αk+β)) is Mittag-Leffler function.

Acknowledgment

The authors thank the referee’s efforts for their remarks and Professor Ferhan Merdivenci Atici, Western Kentucky University, USA, for her regular contact.

BabakhaniA.Daftardar-GejjiV.Existence of positive solutions of nonlinear fractional differential equationsJournal of Mathematical Analysis and Applications20032782434442MR197401710.1016/S0022-247X(02)00716-3ZBL1027.34003BabakhaniA.Daftardar-GejjiV.Existence of positive solutions for N-term non-autonomous fractional differential equationsPositivity200592193206MR2189743El-SayedA. M. A.Nonlinear differential equations of orbitrary orderNonlinear Analysis: Theory, Methods & Applications199832181186KilbasA. A.SrivastavaH. M.TrujilloJ. J.Theory and Applications of Fractional Differential Equations2006204Amsterdam, The NetherlandsElsevier Sciencexvi+523North-Holland Mathematics StudiesMR2218073SamkoS. G.KilbasA. A.MarichevO. I.Fractional Integrals and Derivatives1993Yverdon, SwitzerlandGordon and Breach Sciencexxxvi+976MR1347689YeH.DingY.GaoJ.The existence of a positive solution of Dα[x(t)x(0)]=x(t)f(t,xt)Positivity2007112341350MR2321625ZBL1121.34064ZhangX.Some results of linear fractional order time-delay systemApplied Mathematics and Computation20081971407411MR239632310.1016/j.amc.2007.07.069ZBL1138.34328BenchohraM.HendersonJ.NtouyasS. K.OuahabA.Existence results for fractional order functional differential equations with infinite delayJournal of Mathematical Analysis and Applications2008338213401350MR238650110.1016/j.jmaa.2007.06.021LakshmikanthamV.Theory of fractional functional differential equationsNonlinear Analysis: Theory, Methods & Applications2008691033373343MR2450543ZBL1162.34344GranasA.GuentherR. B.LeeJ. W.Some general existence principles in the Carathéodory theory of nonlinear differential systemsJournal de Mathématiques Pures et Appliquées1991702153196MR1103033ZBL0687.34009ZhangS.The existence of a positive solution for a nonlinear fractional differential equationJournal of Mathematical Analysis and Applications20002522804812MR180018010.1006/jmaa.2000.7123ZBL0972.34004BaiC.FangJ.The existence of a positive solution for a singular coupled system of nonlinear fractional differential equationsApplied Mathematics and Computation20041503611621MR203966210.1016/S0096-3003(03)00294-7ZBL1061.34001Daftardar-GejjiV.Positive solutions of a system of non-autonomous fractional differential equationsJournal of Mathematical Analysis and Applications200530215664MR210654610.1016/j.jmaa.2004.08.007ZBL1064.34004BaiC.Positive solutions for nonlinear fractional differential equations with coefficient that changes signNonlinear Analysis: Theory, Methods & Applications2006644677685MR2197088ZBL1152.34304ZhongC.FanX.ChenW.Nonlinear Functional Analysis and Its Application1998Lanzhou, ChinaLanzhou University PressPodlubnyI.Fractional Differential Equations1999198London, UKAcademic Pressxxiv+340Mathematics in Science and EngineeringMR1658022MillerK. S.RossB.An Introduction to the Fractional Calculus and Fractional Differential Equations1993New York, NY, USAJohn Wiley & Sonsxvi+366A Wiley-Interscience PublicationMR1219954