JAMJournal of Applied Mathematics1687-00421110-757XHindawi Publishing Corporation69165110.1155/2012/691651691651Research ArticlePositive Mild Solutions of Periodic Boundary Value Problems for Fractional Evolution EquationsMuJia1,2FanHongxia2LuShiping1School of Mathematics and Computer Science InstituteNorthwest University for NationalitiesGansu, Lanzhou 730000Chinanwnu.edu.cn2Department of MathematicsNorthwest Normal UniversityGansu, Lanzhou 730000Chinanwnu.edu.cn20123042012201229102011140220122012Copyright © 2012 Jia Mu and Hongxia Fan.This 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.

The periodic boundary value problem is discussed for a class of fractional evolution equations. The existence and uniqueness results of mild solutions for the associated linear fractional evolution equations are established, and the spectral radius of resolvent operator is accurately estimated. With the aid of the estimation, the existence and uniqueness results of positive mild solutions are obtained by using the monotone iterative technique. As an application that illustrates the abstract results, an example is given.

1. Introduction

In this paper, we investigate the existence and uniqueness of positive mild solutions of the periodic boundary value problem (PBVP) for the fractional evolution equation in an ordered Banach space XDαu(t)+Au(t)=f(t,u(t)),tI,u(0)=u(ω), where Dα is the Caputo fractional derivative of order 0<α<1,  I=[0,ω],  -A:D(A)XX is the infinitesimal generator of an analytic semigroup {T(t)}t0 of uniformly bounded linear operators on X, and f:I×XX is a continuous function.

The origin of fractional calculus goes back to Newton and Leibnitz in the seventieth century. We observe that fractional order can be complex in viewpoint of pure mathematics and there is much interest in developing the theoretical analysis and numerical methods to fractional equations, because they have recently proved to be valuable in various fields of science and engineering. Indeed, we can find numerous applications in viscoelasticity, electrochemistry, electromagnetism, biology, and hydrogeology. For example space-fractional diffusion equations have been used in groundwater hydrology to model the transport of passive tracers carried by fluid flow in a porous medium [1, 2] or to model activator-inhibitor dynamics with anomalous diffusion .

Fractional evolution equations, which is field have abundant contents. Many differential equations can turn to semilinear fractional evolution equations in Banach spaces. For example, fractional diffusion equations are abstract partial differential equations that involve fractional derivatives in space and time. The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α(0,1), namely,tαu(y,t)=Au(y,t),t0,yR, we can take A=yβ1, for β1(0,1], or A=y+yβ2 for β2(1,2], where tα, yβ1, yβ2 are the fractional derivatives of order α, β1, β2, respectively. Recently, fractional evolution equations are attracting increasing interest, see El-Borai [4, 5], Zhou and Jiao [6, 7], Wang et al. [8, 9], Shu et al.  and Mu et al. [11, 12]. They established various criteria on the existence of solutions for some fractional evolution equations by using the Krasnoselskii fixed point theorem, the Leray-Schauder fixed point theorem, the contraction mapping principle, or the monotone iterative technique. However, no papers have studied the periodic boundary value problems for abstract fractional evolution equations (1.1), though the periodic boundary value problems for ordinary differential equations have been widely studied by many authors (see ).

In this paper, without the assumptions of lower and upper solutions, by using the monotone iterative technique, we obtain the existence and uniqueness of positive mild solutions for PBVP (1.1). Because in many practical problems such as the reaction diffusion equations, only the positive solution has the significance, we consider the positive mild solutions in this paper. The characteristics of positive operator semigroup play an important role in obtaining the existence of the positive mild solutions. Positive operator semigroup are widely appearing in heat conduction equations, the reaction diffusion equations, and so on (see ). It is worth noting that our assumptions are very natural and we have tested them in the practical context. In particular to build intuition and throw some light on the power of our results, we examine sufficient conditions for the existence and uniqueness of positive mild solutions for periodic boundary value problem for fractional parabolic partial differential equations (see Example 4.1).

We now turn to a summary of this work. Section 2 provides the definitions and preliminary results to be used in theorems stated and proved in the paper. In particular to facilitate access to the individual topics, the existence and uniqueness results of mild solutions for the associated linear fractional evolution equations are established and the spectral radius of resolvent operator is accurately estimated. In Section 3, we obtain very general results on the existence and uniqueness of positive mild solutions for PBVP (1.1), when the nonlinear term f satisfies some conditions related to the growth index of the operator semigroup {T(t)}t0. The main method is the monotone iterative technique. In Section 4, we give also an example to illustrate the applications of the abstract results.

2. Preliminaries

Let us recall the following known definitions. For more details see .

Definition 2.1.

The fractional integral of order α with the lower limit zero for a function f is defined as: Iαf(t)=1Γ(α)0tf(s)(t-s)1-αds,  t>0,α>0, provided the right side is point-wise defined on [0,), where Γ(·) is the gamma function.

Definition 2.2.

The Riemann-Liouville derivative of order α with the lower limit zero for a function f can be written as:   LDαf(t)=1Γ(n-α)dndtn0tf(s)(t-s)α+1-nds,  t>0,n-1<α<n.

Definition 2.3.

The Caputo fractional derivative of order α for a function f can be written as: Dαf(t)=DLα(f(t)-k=0n-1tkk!f(k)(0)),  t>0,n-1<α<n.

Remark 2.4.

(i) If fCn[0,), then Dαf(t)=1Γ(n-α)0tfn(s)(t-s)α+1-nds,t>0,n-1<α<n.

(ii) The Caputo derivative of a constant is equal to zero.

(iii) If f is an abstract function with values in X, then the integrals and derivatives which appear in Definitions 2.12.3 are taken in Bochner's sense.

Throughout this paper, let X be an ordered Banach space with norm · and partial order ≤, whose positive cone P={yXyθ} (θ is the zero element of X) is normal with normal constant N. Let C(I,X) be the Banach space of all continuous X-value functions on interval I with norm uC=maxtIu(t). Evidently, C(I,X) is also an ordered Banach space with the partial ≤ reduced by the positive function cone PC={uC(I,X)u(t)θ,tI}. PC is also normal with the same constant N. For u,vC(I,X),  uv if u(t)v(t) for all tI. For v,wC(I,X), denote the ordered interval [v,w]={uC(I,X)vuw} in C(I,X), and [v(t),w(t)]={yXv(t)yw(t)}(tI) in X. Set Cα(I,X)={uC(I,X)Dαu exists and DαuC(I,X)}. X1 denotes the Banach space D(A) with the graph norm ·1=·+A·. Suppose that -A is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators {T(t)}t0. This means there exists M1 such thatT(t)M,t0.

Lemma 2.5 (see [<xref ref-type="bibr" rid="B4">4</xref>]).

If h satisfies a uniform Hölder condition, with exponent β(0,1], then the unique solution of the linear initial value problem (LIVP) for the fractional evolution equation, Dαu(t)+Au(t)=h(t),tI,u(0)=x0X, is given by u(t)=U(t)x0+0t(t-s)α-1V(t-s)h(s)ds, where U(t)=0ζα(θ)T(tαθ)dθ,  V(t)=α0θζα(θ)T(tαθ)dθ,ζα(θ) is a probability density function defined on (0,).

Remark 2.6.

(i) See [6, 7] ζα(θ)=1αθ-1-1/αρα(θ-1/α),ρα(θ)=1πn=0(-1)n-1θ-αn-1Γ(nα+1)n!sin(nπα),θ(0,),

(ii) see [6, 24], ζα(θ)0,  θ(0,),  0ζα(θ)dθ=1,  0θζα(θ)dθ=1/Γ(1+α),    0θvζα(θ)dθ=Γ(1+v)/Γ(1+αv) for v(-1,),

(iii) see [4, 5], the Laplace transform of ζα is given by 0e-pθζα(θ)dθ=n=0(-p)nΓ(1+nα)=Eα(-p), where Eα(·) is the Mittag-Leffler function (see ),

(iv) see  by (i) and (ii), we can obtain that for p00e-pθθζα(θ)dθ=1αn=0(-p)nΓ(α(n+1))=1αEα,α(-p), where Eα(·), Eα,α(·) are the Mittag-Leffler functions.

(v) see  for p<0,  0<Eα(p)<Eα(0)=1,

(vi) see  if δ>0 and t>0, then -(1/δ)(Eα(-δtα))=tα-1Eα,α(-δtα).

Remark 2.7.

See [6, 8], the operators U and V, given by (2.8), have the following properties:

For any fixed t0, U(t) and V(t) are linear and bounded operators, that is, for any xX, U(t)xMx,V(t)xαMΓ(1+α)x,

{U(t)}t0 and {V(t)}t0 are strongly continuous.

Definition 2.8.

If hC(I,X), by the mild solution of IVP (2.6), we mean that the function uC(I,X) satisfying the integral (2.7).

We also introduce some basic theories of the operator semigroups. For an analytic semigroup {R(t)}t0, there exist M1>0 and δ such that (see )R(t)M1eδt,t0. Thenν0=inf{δRthere  exist  M1>0  such  that  R(t)M1eδt,t0} is called the growth index of the semigroup {R(t)}t0. Furthermore, ν0 can also be obtained by the following formula:ν0=limsupt+lnR(t)t.

Definition 2.9 (see [<xref ref-type="bibr" rid="B26">26</xref>]).

A C0-semigroup {T(t)}t0 is called a compact semigroup if T(t) is compact for t>0.

Definition 2.10.

An analytic semigroup {T(t)}t0 is called positive if T(t)xθ for all xθ and t0.

Remark 2.11.

For the applications of positive operators semigroup, we can see .

Definition 2.12.

A bounded linear operator K on X is called to be positive if Kxθ for all xθ.

Remark 2.13.

By Remark 2.6(ii), we obtain that U(t) and V(t) are positive for t0 if {T(t)}t0 is a positive semigroup.

Lemma 2.14.

Let X be an ordered Banach space, whose positive cone P is normal. If {T(t)}t0 is an exponentially stable analytic semigroup, that is, ν0=limsupt+(lnT(t)/t)<0. Then the linear periodic boundary value problem (LPBVP), Dαu(t)+Au(t)=h(t),tI,u(0)=u(ω), has a unique mild solution u(t)=(Qh)(t)=U(t)B(h)+0t(t-s)α-1V(t-s)h(s)ds, where U(t) and V(t) are given by (2.8), B(h)=(I-U(ω))-10ω(ω-s)α-1V(ω-s)h(s)ds,Q:C(I,X)C(I,X) is a bounded linear operator, and the spectral radius r(Q)1/|ν0|.

Proof.

For any ν(0,|ν0|), by there exists M1 such that T(t)M1e-νt,t0. In X, give the equivalent norm |·| by |x|=supt0eνtT(t)x, then x|x|M1x. By |T(t)| we denote the norm of T(t) in (X,|·|), then for t0, |T(t)x|=sups0eνsT(s)T(t)x=e-νtsups0eν(s+t)T(s+t)x=e-νtsupηteνηT(η)xe-νt|x|. Thus, |T(t)|e-νt. Then by Remark 2.6, |U(t)|=|0ζα(θ)T(tαθ)dθ|0ζα(θ)e-νtαθdθ=Eα(-νtα)<1. Therefore, I-U(ω) has bounded inverse operator and (I-U(ω))-1=n=0(U(ω))n,|(I-U(ω))-1|11-Eα(-νωα). Set x0=(I-U(ω))-10ω(ω-s)α-1V(ω-s)h(s)ds, then u(t)=U(t)x0+0t(t-s)α-1V(t-s)h(s)ds is the unique mild solution of LIVP (2.6) satisfing u(0)=u(ω). So set B(h)=(I-U(ω))-10ω(ω-s)α-1V(ω-s)h(s)ds,(Qh)(t)=U(t)B(h)+0t(t-s)α-1V(t-s)h(s)ds,   then u=Qh is the unique mild solution of LPBVP (2.16). By Remark 2.7, Q:C(I,X)C(I,X) is a bounded linear operator. Furthermore, by Remark 2.6, we obtain that |V(t)|=|α0θζα(θ)T(tαθ)dθ|α0θζα(θ)e-νtαθdθ=Eα,α(-νtα). By (2.24), (2.28) and Remark 2.6, for t0 we have that |(Qh)(t)||U(t)(I-U(ω))-10ω(ω-s)α-1V(ω-s)h(s)ds|+|0t(t-s)α-1V(t-s)h(s)ds|Eα(-νtα)1-Eα(-νωα)|0ω(ω-s)α-1V(ω-s)ds||h|C+|0t(t-s)α-1V(t-s)ds||h|C=[Eα(-νtα)1-Eα(-νωα)1νEα(-ν(ω-s)α)|0ω+1νEα(-ν(t-s)α)|0t]|h|C=|h|Cν, where |·|C=maxtI|·(t)|. Thus, |Qh|C|h|C/ν. Then |Q|1/ν and the spectral radius r(Q)1/ν. By the randomicity of ν(0,|ν0|), we obtain that r(Q)1/|ν0|.

Remark 2.15.

For sufficient conditions of exponentially stable operator semigroups, one can see .

Remark 2.16.

If {T(t)}t0 is a positive and exponentially stable analytic semigroup generated by -A, by Remark 2.13, then the resolvent operator Q:C(I,X)C(I,X) is also a positive bounded linear operator.

Remark 2.17.

For the applications of Lemma 2.14, it is important to estimate the growth index of {T(t)}t0. If T(t) is continuous in the uniform operator topology for  t>0, it is well known that ν0 can be obtained byσ(A): the spectrum of A (see ) ν0=-inf{Reλλσ(A)}. We know that T(t) is continuous in the uniform operator topology for t>0 if T(t) is a compact semigroup, see . Assume that P is a regeneration cone, {T(t)}t0 is a compact and positive analytic semigroup. Then by the characteristic of positive semigroups (see ), for sufficiently large λ0>-inf{Reλλσ(A)}, we have that λ0I+A has positive bounded inverse operator (λ0I+A)-1. Since σ(A), the spectral radius r((λ0I+A)-1)=1/dist(-λ0,σ(A))>0. By the Krein-Rutmann theorem (see [34, 35]), A has the first eigenvalue λ1, which has a positive eigenfunction x1, and λ1=inf{Reλλσ(A)}, that is, ν0=-λ1.

Corollary 2.18.

Let X be an ordered Banach space, whose positive cone P is a regeneration cone. If {T(t)}t0 is a compact and positive analytic semigroup, and its first eigenvalue of A is λ1=inf{Reλλσ(A)}>0, then LPBVP (2.16) has a unique mild solution u:=Qh, Q:C(I,X)C(I,X) is a bounded linear operator, and the spectral radius r(Q)=1/λ1

Proof.

By (2.32), we know that the growth index of {T(t)}t0 is ν0=-λ1<0, that is, {T(t)}t0 is exponentially stable. By Lemma 2.14, Q:C(I,X)C(I,X) is a bounded linear operator, and the spectral radius r(Q)1/λ1. On the other hand, since λ1 has a positive eigenfunction x1, in LPBVP (3.17) we set h(t)=x1, then x1/λ1 is the corresponding mild solution. By the definition of the operator Q, Q(x1)=x1/λ1, that is, 1/λ1 is an eigenvalue of Q. Then r(Q)1/λ1. Thus, r(Q)=1/λ1.

3. Main ResultsTheorem 3.1.

Let X  be an ordered Banach space, whose positive cone P is normal with normal constant N. If {T(t)}t0 is a positive analytic semigroup, f(t,θ)θ  for  all  tI, and the following conditions are satisfied.

For any R>0, there exists C=C(R)>0 such that f(t,x2)-f(t,x1)-C(x2-x1), for any tI, θx1x2, x1, x2R.

There exists L<-ν0 (ν0 is the growth index of {T(t)}t0), such that f(t,x2)-f(t,x1)L(x2-x1), for any tI, θx1x2.

Then PBVP (1.1) has a unique positive mild solution.

Proof.

Let h0(t)=f(t,θ), then h0C(I,X), h0θ. Consider LPBVP Dαu(t)+(A-LI)u(t)=h0(t),tI,u(0)=u(ω).-(A-LI) generates a positive analytic semigroup eLtT(t), whose growth index is L+ν0<0. By Lemma 2.14 and Remark 2.16, LPBVP (3.3) has a unique mild solution w0C(I,X) and w0θ.

Set R0=Nw0+1,  C=C(R0) is the corresponding constant in (H1). We may suppose C>max{ν0,-L}, otherwise substitute C+|ν0|+|L| for C, (H1) is also satisfied. Then we consider LPBVP Dαu(t)+(A+CI)u(t)=h(t),tI,u(0)=u(ω).-(A+CI) generates a positive analytic semigroup T1(t)=e-CtT(t), whose growth index is -C+ν0<0. By Lemma 2.14 and Remark 2.16, for hC(I,X) LPBVP (3.4) has a unique mild solution u=Q1h,  Q1:C(I,X)C(I,X) is a positive bounded linear operator and the spectral radius r(Q1)1/(C-ν0).

Set F(u)=f(t,u)+Cu, then F:C(I,X)C(I,X) is continuous, F(θ)=h0θ. By (H1), F is an increasing operator on [θ,w0]. Set v0=θ, we can define the sequences vn=Q1F(vn-1),wn=Q1F(wn-1),n=1,2,. By (3.4), we have that w0=Q1(h0+Lw0+Cw0). In (H2), we set x1=θ,  x2=w0(t), then f(t,w0)h0(t)+Lw0(t),θF(θ)F(w0)h0+Lw0+Cw0. By (3.6) and (3.8), the definition and the positivity of Q1, we have that Q1θ=θ=v0v1w1w0. Since Q1·F is an increasing operator on [θ,w0], in view of (3.5), we have that θv1vnwnw1w0. Therefore, we obtain that θwn-vn=Q1(F(wn-1)-F(vn-1))=Q1(f(,wn-1)-f(,vn-1)+C(wn-1-vn-1))(C+L)Q1(wn-1-vn-1). By induction, θwn-vn(C+L)nQ1n(w0-v0)=(C+L)nQ1n(w0). In view of the normality of the cone P, we have that wn-vnCN(C+L)nQ1n(w0)CN(C+L)nQ1nCw0C. On the other hand, since 0<C+L<C-ν0, for some ε>0, we have that C+L+ε<C-ν0. By the Gelfand formula, limnQ1nCn=r(Q1)1/(C-ν0). Then there exist N0, for nN0, we have that Q1nC1/(C+L+ε)n. By (3.13), we have that wn-vnCNw0C(C+LC+L+ε)n0,  (n). By (3.10) and (3.14), similarly to the nested interval method, we can prove that there exists a unique u*n=1[vn,wn], such that limnvn=limnwn=u*. By the continuity of the operator Q1·F and (3.5), we have that u*=Q1F(u*). By the definition of Q1 and (3.10), we know that u* is a positive mild solution of (3.4) when h(t)=f(t,u*(t))+Cu*(t). Then u* is the positive mild solution of PBVP (1.1).

In the following, we prove that the uniqueness. If u1, u2 are the positive mild solutions of PBVP (1.1). Substitute u1 and u2 for w0, respectively, then wn=Q1·F(ui)=ui (i=1,2). By (3.14), we have that ui-vnC0,  (n,i=1,2). Thus, u1=u2=limnvn, PBVP (1.1) has a unique positive mild solution.

Corollary 3.2.

Let X be an ordered Banach space, whose positive cone P is a regeneration cone. If {T(t)}t0 is a compact and positive analytic semigroup, f(t,θ)θ for for  all  tI, f satisfies (H1) and the following condition:

There exist L<λ1, where λ1 is the first eigenvalue of A, such that f(t,x2)-f(t,x1)L(x2-x1), for any tI,  θx1x2.

Then PBVP (1.1) has a unique positive mild solution.

Remark 3.3.

In Corollary 3.2, since λ1 is the first eigenvalue of A, the condition L<λ1in (H2) cannot be extended toL<λ1”. Otherwise, PBVP (1.1) does not always have a mild solution. For example, f(t,x)=λ1x.

4. ExamplesExample 4.1.

Consider the following periodic boundary value problem for fractional parabolic partial differential equations in X:tαu-Δu=f(t,u(t,x)),  (t,x)I×Ω,uΩ=0,u(0,x)=u(ω,x),  xΩ, where tα is the Caputo fractional partial derivative of order 0<α<1,  I=[0,ω],  ΩN is a bounded domain with a sufficiently smooth boundary Ω, Δ is the Laplace operator, f:I× is continuous.

Let X=L2(Ω), P={vvL2(Ω),v(x)0  a.e.  xΩ}. Then X is an Banach space with the partial order “≤” reduced by the normal cone P. Define the operator A as follows:D(A)=H2(Ω)H01(Ω),Au=-Δu. Then -A generates an operator semigroup {T(t)}t0 which is compact, analytic, and uniformly bounded. By the maximum principle, we can find that {T(t)}t0 is a positive semigroup. Denote u(t)(x)=u(t,x),  f(t,u(t))(x)=f(t,u(t,x)), then the system (4.1) can be reformulated as the problem (1.1) in X.

Theorem 4.2.

Assume that f(t,0)0 for tI, the partial derivative fu(t,u) is continuous on any bounded domain and supfu(t,u)<λ1, where λ1 is the first eigenvalue of -Δ under the condition uΩ=0. Then the problem (4.1) has a unique positive mild solution.

Proof.

It is easy to see that (H1) and (H2) are satisfied. By Corollary 3.2, the problem (4.1) has a unique positive mild solution.

Acknowledgments

This research was supported by NNSFs of China (10871160, 11061031) and project of NWNU-KJCXGC-3-47.

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