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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 691651, 13 pages
http://dx.doi.org/10.1155/2012/691651
Research Article

Positive Mild Solutions of Periodic Boundary Value Problems for Fractional Evolution Equations

1School of Mathematics and Computer Science Institute, Northwest University for Nationalities, Gansu, Lanzhou 730000, China
2Department of Mathematics, Northwest Normal University, Gansu, Lanzhou 730000, China

Received 29 October 2011; Accepted 14 February 2012

Academic Editor: Shiping Lu

Copyright © 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.

Abstract

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),(1.1) where 𝐷𝛼 is the Caputo fractional derivative of order 0<𝛼<1,𝐼=[0,𝜔],𝐴𝐷(𝐴)𝑋𝑋 is the infinitesimal generator of an analytic semigroup {𝑇(𝑡)}𝑡0 of uniformly bounded linear operators on 𝑋, and 𝑓𝐼×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 [3].

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,𝜕𝛼𝑡𝑢(𝑦,𝑡)=𝐴𝑢(𝑦,𝑡),𝑡0,𝑦𝑅,(1.2) we can take 𝐴=𝜕𝛽1𝑦, for 𝛽1(0,1], or 𝐴=𝜕𝑦+𝜕𝛽2𝑦 for 𝛽2(1,2], where 𝜕𝛼𝑡, 𝜕𝛽1𝑦, 𝜕𝛽2𝑦 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. [10] 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 [1318]).

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 [19]). 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 [2023].

Definition 2.1. The fractional integral of order 𝛼 with the lower limit zero for a function 𝑓 is defined as: 𝐼𝛼1𝑓(𝑡)=Γ(𝛼)𝑡0𝑓(𝑠)(𝑡𝑠)1𝛼𝑑𝑠,𝑡>0,𝛼>0,(2.1) 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: 𝐿𝐷𝛼1𝑓(𝑡)=𝑑Γ(𝑛𝛼)𝑛𝑑𝑡𝑛𝑡0𝑓(𝑠)(𝑡𝑠)𝛼+1𝑛𝑑𝑠,𝑡>0,𝑛1<𝛼<𝑛.(2.2)

Definition 2.3. The Caputo fractional derivative of order 𝛼 for a function 𝑓 can be written as: 𝐷𝛼𝑓(𝑡)=𝐿𝐷𝛼𝑓(𝑡)𝑛1𝑘=0𝑡𝑘𝑓𝑘!(𝑘)(0),𝑡>0,𝑛1<𝛼<𝑛.(2.3)

Remark 2.4. (i) If 𝑓𝐶𝑛[0,), then 𝐷𝛼1𝑓(𝑡)=Γ(𝑛𝛼)𝑡0𝑓𝑛(𝑠)(𝑡𝑠)𝛼+1𝑛𝑑𝑠,𝑡>0,𝑛1<𝛼<𝑛.(2.4)
(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.(2.5)

Lemma 2.5 (see [4]). If 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)=(𝑡),𝑡𝐼,𝑢(0)=𝑥0𝑋,(2.6) is given by 𝑢(𝑡)=𝑈(𝑡)𝑥0+𝑡0(𝑡𝑠)𝛼1𝑉(𝑡𝑠)(𝑠)𝑑𝑠,(2.7) where 𝑈(𝑡)=0𝜁𝛼(𝜃)𝑇(𝑡𝛼𝜃)𝑑𝜃,𝑉(𝑡)=𝛼0𝜃𝜁𝛼(𝜃)𝑇(𝑡𝛼𝜃)𝑑𝜃,(2.8)𝜁𝛼(𝜃) is a probability density function defined on (0,).

Remark 2.6. (i) See [6, 7] 𝜁𝛼1(𝜃)=𝛼𝜃11/𝛼𝜌𝛼𝜃1/𝛼,𝜌𝛼1(𝜃)=𝜋𝑛=0(1)𝑛1𝜃𝛼𝑛1Γ(𝑛𝛼+1)𝑛!sin(𝑛𝜋𝛼),𝜃(0,),(2.9)
(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 0𝑒𝑝𝜃𝜁𝛼(𝜃)𝑑𝜃=𝑛=0(𝑝)𝑛Γ(1+𝑛𝛼)=𝐸𝛼(𝑝),(2.10) where 𝐸𝛼() is the Mittag-Leffler function (see [20]),
(iv) see [24] by (i) and (ii), we can obtain that for 𝑝00𝑒𝑝𝜃𝜃𝜁𝛼(1𝜃)𝑑𝜃=𝛼𝑛=0(𝑝)𝑛=1Γ(𝛼(𝑛+1))𝛼𝐸𝛼,𝛼(𝑝),(2.11) where 𝐸𝛼(), 𝐸𝛼,𝛼() are the Mittag-Leffler functions.
(v) see [25] for 𝑝<0,0<𝐸𝛼(𝑝)<𝐸𝛼(0)=1,
(vi) see [10] if 𝛿>0 and 𝑡>0, then (1/𝛿)(𝐸𝛼(𝛿𝑡𝛼))=𝑡𝛼1𝐸𝛼,𝛼(𝛿𝑡𝛼).

Remark 2.7. See [6, 8], the operators 𝑈 and 𝑉, given by (2.8), have the following properties:
(i) 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,(2.12)(ii){𝑈(𝑡)}𝑡0 and {𝑉(𝑡)}𝑡0 are strongly continuous.

Definition 2.8. If 𝐶(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 [26])𝑅(𝑡)𝑀1𝑒𝛿𝑡,𝑡0.(2.13) Then𝜈0=inf𝛿thereexist𝑀1>0suchthat𝑅(𝑡)𝑀1𝑒𝛿𝑡,𝑡0(2.14) 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.(2.15)

Definition 2.9 (see [26]). 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 [2731].

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)=(𝑡),𝑡𝐼,𝑢(0)=𝑢(𝜔),(2.16) has a unique mild solution 𝑢(𝑡)=(𝑄)(𝑡)=𝑈(𝑡)𝐵()+𝑡0(𝑡𝑠)𝛼1𝑉(𝑡𝑠)(𝑠)𝑑𝑠,(2.17) where 𝑈(𝑡) and 𝑉(𝑡) are given by (2.8), 𝐵()=(𝐼𝑈(𝜔))1𝜔0(𝜔𝑠)𝛼1𝑉(𝜔𝑠)(𝑠)𝑑𝑠,(2.18)𝑄𝐶(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)𝑀1𝑒𝜈𝑡,𝑡0.(2.19) In 𝑋, give the equivalent norm || by |𝑥|=sup𝑡0𝑒𝜈𝑡𝑇(𝑡)𝑥,(2.20) 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=𝑒𝜈𝑡sup𝑠0𝑒𝜈(𝑠+𝑡)𝑇(𝑠+𝑡)𝑥=𝑒𝜈𝑡sup𝜂𝑡𝑒𝜈𝜂𝑇(𝜂)𝑥𝑒𝜈𝑡|𝑥|.(2.21) Thus, |𝑇(𝑡)|𝑒𝜈𝑡. Then by Remark 2.6, ||𝑈||=||||(𝑡)0𝜁𝛼(𝜃)𝑇(𝑡𝛼||||𝜃)𝑑𝜃0𝜁𝛼(𝜃)𝑒𝜈𝑡𝛼𝜃𝑑𝜃=𝐸𝛼(𝜈𝑡𝛼)<1.(2.22) Therefore, 𝐼𝑈(𝜔) has bounded inverse operator and (𝐼𝑈(𝜔))1=𝑛=0(𝑈(𝜔))𝑛,||(2.23)(𝐼𝑈(𝜔))1||11𝐸𝛼(𝜈𝜔𝛼).(2.24) Set 𝑥0=(𝐼𝑈(𝜔))1𝜔0(𝜔𝑠)𝛼1𝑉(𝜔𝑠)(𝑠)𝑑𝑠,(2.25) then 𝑢(𝑡)=𝑈(𝑡)𝑥0+𝑡0(𝑡𝑠)𝛼1𝑉(𝑡𝑠)(𝑠)𝑑𝑠(2.26) is the unique mild solution of LIVP (2.6) satisfing 𝑢(0)=𝑢(𝜔). So set 𝐵()=(𝐼𝑈(𝜔))1𝜔0(𝜔𝑠)𝛼1𝑉(𝜔𝑠)(𝑠)𝑑𝑠,(𝑄)(𝑡)=𝑈(𝑡)𝐵()+𝑡0(𝑡𝑠)𝛼1𝑉(𝑡𝑠)(𝑠)𝑑𝑠,(2.27) then 𝑢=𝑄 is the unique mild solution of LPBVP (2.16). By Remark 2.7, 𝑄𝐶(𝐼,𝑋)𝐶(𝐼,𝑋) is a bounded linear operator. Furthermore, by Remark 2.6, we obtain that ||𝑉||=||||𝛼(𝑡)0𝜃𝜁𝛼(𝜃)𝑇(𝑡𝛼||||𝜃)𝑑𝜃𝛼0𝜃𝜁𝛼(𝜃)𝑒𝜈𝑡𝛼𝜃𝑑𝜃=𝐸𝛼,𝛼(𝜈𝑡𝛼).(2.28) By (2.24), (2.28) and Remark 2.6, for 𝑡0 we have that ||||||||(𝑄)(𝑡)𝑈(𝑡)(𝐼𝑈(𝜔))1𝜔0(𝜔𝑠)𝛼1𝑉||||+||||(𝜔𝑠)(𝑠)𝑑𝑠𝑡0(𝑡𝑠)𝛼1||||𝐸𝑉(𝑡𝑠)(𝑠)𝑑𝑠𝛼(𝜈𝑡𝛼)1𝐸𝛼(𝜈𝜔𝛼)||||𝜔0(𝜔𝑠)𝛼1𝑉||||||||(𝜔𝑠)𝑑𝑠𝐶+||||𝑡0(𝑡𝑠)𝛼1||||||||𝑉(𝑡𝑠)𝑑𝑠𝐶=𝐸𝛼(𝜈𝑡𝛼)1𝐸𝛼(𝜈𝜔𝛼)1𝜈𝐸𝛼(𝜈(𝜔𝑠)𝛼)||||𝜔0+1𝜈𝐸𝛼(𝜈(𝑡𝑠)𝛼)|||𝑡0||||𝐶=||||𝐶𝜈,(2.29) where ||𝐶=max𝑡𝐼|(𝑡)|. Thus, |𝑄|𝐶||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 [32].

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 𝑄𝐶(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 for𝑡>0, it is well known that 𝜈0 can be obtained by𝜎(𝐴): the spectrum of 𝐴 (see [33]) 𝜈0=inf{Re𝜆𝜆𝜎(𝐴)}.(2.30) We know that 𝑇(𝑡) is continuous in the uniform operator topology for 𝑡>0 if 𝑇(𝑡) is a compact semigroup, see [26]. 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 [31]), 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)},(2.31) 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,(2.32) then LPBVP (2.16) has a unique mild solution 𝑢=𝑄, 𝑄𝐶(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, 𝑄𝐶(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 (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 Results

Theorem 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)𝜃forall𝑡𝐼, and the following conditions are satisfied.(H1) For any 𝑅>0, there exists 𝐶=𝐶(𝑅)>0 such that 𝑓𝑡,𝑥2𝑓𝑡,𝑥1𝑥𝐶2𝑥1,(3.1) for any 𝑡𝐼, 𝜃𝑥1𝑥2, 𝑥1, 𝑥2𝑅.(H2) There exists 𝐿<𝜈0 (𝜈0 is the growth index of {𝑇(𝑡)}𝑡0), such that 𝑓𝑡,𝑥2𝑓𝑡,𝑥1𝑥𝐿2𝑥1,(3.2) for any 𝑡𝐼, 𝜃𝑥1𝑥2. Then PBVP (1.1) has a unique positive mild solution.

Proof. Let 0(𝑡)=𝑓(𝑡,𝜃), then 0𝐶(𝐼,𝑋), 0𝜃. Consider LPBVP 𝐷𝛼𝑢(𝑡)+(𝐴𝐿𝐼)𝑢(𝑡)=0(𝑡),𝑡𝐼,𝑢(0)=𝑢(𝜔).(3.3)(𝐴𝐿𝐼) 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)=(𝑡),𝑡𝐼,𝑢(0)=𝑢(𝜔).(3.4)(𝐴+𝐶𝐼) generates a positive analytic semigroup 𝑇1(𝑡)=𝑒𝐶𝑡𝑇(𝑡), whose growth index is 𝐶+𝜈0<0. By Lemma 2.14 and Remark 2.16, for 𝐶(𝐼,𝑋) LPBVP (3.4) has a unique mild solution 𝑢=𝑄1,𝑄1𝐶(𝐼,𝑋)𝐶(𝐼,𝑋) is a positive bounded linear operator and the spectral radius 𝑟(𝑄1)1/(𝐶𝜈0).
Set 𝐹(𝑢)=𝑓(𝑡,𝑢)+𝐶𝑢, then 𝐹𝐶(𝐼,𝑋)𝐶(𝐼,𝑋) is continuous, 𝐹(𝜃)=0𝜃. By (H1), 𝐹 is an increasing operator on [𝜃,𝑤0]. Set 𝑣0=𝜃, we can define the sequences 𝑣𝑛=𝑄1𝑣𝐹𝑛1,𝑤𝑛=𝑄1𝑤𝐹𝑛1,𝑛=1,2,.(3.5) By (3.4), we have that 𝑤0=𝑄10+𝐿𝑤0+𝐶𝑤0.(3.6) In (H2), we set 𝑥1=𝜃,𝑥2=𝑤0(𝑡), then 𝑓𝑡,𝑤00(𝑡)+𝐿𝑤0𝑤(𝑡),(3.7)𝜃𝐹(𝜃)𝐹00+𝐿𝑤0+𝐶𝑤0.(3.8) By (3.6) and (3.8), the definition and the positivity of 𝑄1, we have that 𝑄1𝜃=𝜃=𝑣0𝑣1𝑤1𝑤0.(3.9) Since 𝑄1𝐹 is an increasing operator on [𝜃,𝑤0], in view of (3.5), we have that 𝜃𝑣1𝑣𝑛𝑤𝑛𝑤1𝑤0.(3.10) Therefore, we obtain that 𝜃𝑤𝑛𝑣𝑛=𝑄1𝐹𝑤𝑛1𝑣𝐹𝑛1=𝑄1𝑓,𝑤𝑛1𝑓,𝑣𝑛1𝑤+𝐶𝑛1𝑣𝑛1(𝐶+𝐿)𝑄1𝑤𝑛1𝑣𝑛1.(3.11) By induction, 𝜃𝑤𝑛𝑣𝑛(𝐶+𝐿)𝑛𝑄𝑛1𝑤0𝑣0=(𝐶+𝐿)𝑛𝑄𝑛1𝑤0.(3.12) In view of the normality of the cone 𝑃, we have that 𝑤𝑛𝑣𝑛𝐶𝑁(𝐶+𝐿)𝑛𝑄𝑛1(𝑤0)𝐶𝑁(𝐶+𝐿)𝑛𝑄𝑛1𝐶𝑤0𝐶.(3.13) On the other hand, since 0<𝐶+𝐿<𝐶𝜈0, for some 𝜀>0, we have that 𝐶+𝐿+𝜀<𝐶𝜈0. By the Gelfand formula, lim𝑛𝑛𝑄𝑛1𝐶=𝑟(𝑄1)1/(𝐶𝜈0). Then there exist 𝑁0, for 𝑛𝑁0, we have that 𝑄𝑛1𝐶1/(𝐶+𝐿+𝜀)𝑛. By (3.13), we have that 𝑤𝑛𝑣𝑛𝐶𝑤𝑁0𝐶𝐶+𝐿𝐶+𝐿+𝜀𝑛0,(𝑛).(3.14) By (3.10) and (3.14), similarly to the nested interval method, we can prove that there exists a unique 𝑢𝑛=1[𝑣𝑛,𝑤𝑛], such that lim𝑛𝑣𝑛=lim𝑛𝑤𝑛=𝑢.(3.15) By the continuity of the operator 𝑄1𝐹 and (3.5), we have that 𝑢=𝑄1𝑢𝐹.(3.16) By the definition of 𝑄1 and (3.10), we know that 𝑢 is a positive mild solution of (3.4) when (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=𝑄1𝐹(𝑢𝑖)=𝑢𝑖 (𝑖=1,2). By (3.14), we have that 𝑢𝑖𝑣𝑛𝐶0,(𝑛,𝑖=1,2).(3.17) 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 forall𝑡𝐼, 𝑓 satisfies (H1) and the following condition:(H2) There exist 𝐿<𝜆1, where 𝜆1 is the first eigenvalue of 𝐴, such that 𝑓𝑡,𝑥2𝑓𝑡,𝑥1𝑥𝐿2𝑥1,(3.18) 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. Examples

Example 4.1. Consider the following periodic boundary value problem for fractional parabolic partial differential equations in 𝑋:𝜕𝛼𝑡𝑢𝑢Δ𝑢=𝑓(𝑡,𝑢(𝑡,𝑥)),(𝑡,𝑥)𝐼×Ω,𝑢𝜕Ω=0,(0,𝑥)=𝑢(𝜔,𝑥),𝑥Ω,(4.1) where 𝜕𝛼𝑡 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, 𝑓𝐼× is continuous.

Let 𝑋=𝐿2(Ω), 𝑃={𝑣𝑣𝐿2(Ω),𝑣(𝑥)0a.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)𝐻10(Ω),𝐴𝑢=Δ𝑢.(4.2) 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 𝑢𝜕Ω=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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