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Mathematical Problems in Engineering
Volume 2012, Article ID 746872, 16 pages
http://dx.doi.org/10.1155/2012/746872
Research Article

Periodic Boundary Value Problems for Semilinear Fractional Differential Equations

Department of Mathematics, Northwest Normal University, Lanzhou, Gansu 730000, China

Received 27 September 2011; Accepted 5 December 2011

Academic Editor: Kwok W. Wong

Copyright © 2012 Jia Mu and Yongxiang Li. 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

We study the periodic boundary value problem for semilinear fractional differential equations in an ordered Banach space. The method of upper and lower solutions is then extended. The results on the existence of minimal and maximal mild solutions are obtained by using the characteristics of positive operators semigroup and the monotone iterative scheme. The results are illustrated by means of a fractional parabolic partial differential equations.

1. Introduction

In this paper, we consider the periodic boundary value problem (PBVP) for semilinear fractional differential equation in an ordered Banach space 𝑋,  𝐷𝛼𝑢(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 a 𝐶0-semigroup (i.e., strongly continuous semigroup) {𝑇(𝑡)}𝑡0 of uniformly bounded linear operators on 𝑋, and 𝑓𝐼×𝑋𝑋 is a continuous function.

Fractional calculus is an old mathematical concept dating back to the 17th century and involves integration and differentiation of arbitrary order. In a later dated 30th of September 1695, L’Hospital wrote to Leibniz asking him about the differentiation of order 1/2. Leibniz’ response was “an apparent paradox from which one day useful consequences will be drawn.” In the following centuries, fractional calculus developed significantly within pure mathematics. However, the applications of fractional calculus just emerged in the last few decades. The advantage of fractional calculus becomes apparent in science and engineering. In recent years, fractional calculus attracted engineers’ attention, because it can describe the behavior of real dynamical systems in compact expressions, taking into account nonlocal characteristics like infinite memory [13]. Some instances are thermal diffusion phenomenon [4], botanical electrical impedances [5], model of love between humans [6], the relaxation of water on a porous dyke whose damping ratio is independent of the mass of moving water [7], and so forth. On the other hand, directing the behavior of a process with fractional-order controllers would be an advantage, because the responses are not restricted to a sum of exponential functions; therefore, a wide range of responses neglected by integer-order calculus would be approached [8]. For other advantages of fractional calculus, we can see real materials [913], control engineering [14, 15], electromagnetism [16], biosciences [17], fluid mechanics [18], electrochemistry [19], diffusion processes [20], dynamic of viscoelastic materials [21], viscoelastic systems [22], continuum and statistical mechanics [23], propagation of spherical flames [24], robotic manipulators [25], gear transmissions [26], and vibration systems [27]. It is well known that the fractional-order differential and integral operators are nonlocal operators. This is one reason why fractional differential operators provide an excellent instrument for description of memory and hereditary properties of various physical processes.

In recent years, there have been some works on the existence of solutions (or mild solutions) for semilinear fractional differential equations, see [2836]. They use mainly Krasnoselskii’s fixed-point theorem, Leray-Schauder fixed-point theorem, or contraction mapping principle. They established various criteria on the existence and uniqueness of solutions (or mild solutions) for the semilinear fractional differential equations by considering an integral equation which is given in terms of probability density functions and operator semigroups. Many partial differential equations involving time-variable 𝑡 can turn to semilinear fractional differential equations in Banach spaces; they always generate an unbounded closed operator term 𝐴, such as the time fractional diffusion equation of order 𝛼(0,1), namely, 𝜕𝛼𝑡𝑢(𝑦,𝑡)=𝐴𝑢(𝑦,𝑡),𝑡0,𝑦𝑅,(1.2) where 𝐴 may be linear fractional partial differential operator. So, (1.1) has the extensive application value.

However, to the authors’ knowledge, no studies considered the periodic boundary value problems for the abstract semilinear fractional differential equations involving the operator semigroup generator 𝐴. Our results can be considered as a contribution to this emerging field. We use the method of upper and lower solutions coupled with monotone iterative technique and the characteristics of positive operators semigroup.

The method of upper and lower solutions has been effectively used for proving the existence results for a wide variety of nonlinear problems. When coupled with monotone iterative technique, one obtains the solutions of the nonlinear problems besides enabling the study of the qualitative properties of the solutions. The basic idea of this method is that using the upper and lower solutions as an initial iteration, one can construct monotone sequences, and these sequences converge monotonically to the maximal and minimal solutions. In some papers, some existence results for minimal and maximal solutions are obtained by establishing comparison principles and using the method of upper and lower solutions and the monotone iterative technique. The method requires establishing comparison theorems which play an important role in the proof of existence of minimal and maximal solutions. In abstract semilinear fractional differential equations, positive operators semigroup can play this role, see Li [3741].

In Section 2, we introduce some useful preliminaries. In Section 3, in two cases: 𝑇(𝑡) is compact or noncompact, we establish various criteria on existence of the minimal and maximal mild solutions of PBVP (1.1). The method of upper and lower solutions coupled with monotone iterative technique, and the characteristics of positive operators semigroup are applied effectively. In Section 4, we give also an example to illustrate the applications of the abstract results.

2. Preliminaries

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

If 𝐴 is the infinitesimal generator of a 𝐶0-semigroup in a Banach space, then (𝐴+𝑞𝐼) generates a uniformly bounded 𝐶0-semigroup for 𝑞>0 large enough. This allows us to reduce the general case in which 𝐴 is the infinitesimal generator of a 𝐶0-semigroup to the case in which the semigroup is uniformly bounded. Hence, for convenience, throughout this paper, we suppose that 𝐴 is the infinitesimal generator of a uniformly bounded 𝐶0-semigroup {𝑇(𝑡)}𝑡0. This means that there exists 𝑀1 such that 𝑇(𝑡)𝑀,𝑡0.(2.1)

We need some basic definitions and properties of the fractional calculus theory which are used further in this paper.

Definition 2.1 (see [9, 32]). The fractional integral of order 𝛼 with the lower limit zero for a function  𝑓𝐴𝐶[0,)is defined as 𝐼𝛼1𝑓(𝑡)=Γ(𝛼)𝑡0𝑓(𝑠)(𝑡𝑠)1𝛼𝑑𝑠,𝑡>0,0<𝛼<1,(2.2) provided the right side is pointwise defined on [0,), where Γ() is the gamma function.

Definition 2.2 (see [9, 32]). The Riemann-Liouville derivative of order 𝛼 with the lower limit zero for a function 𝑓𝐴𝐶[0,) can be written as 𝐿𝐷𝛼1𝑓(𝑡)=𝑑Γ(1𝛼)𝑑𝑡𝑡0𝑓(𝑠)(𝑡𝑠)𝛼𝑑𝑠,𝑡>0,0<𝛼<1.(2.3)

Definition 2.3 (see [9, 32]). The Caputo fractional derivative of order 𝛼 for a function 𝑓𝐴𝐶[0,) can be written as 𝐷𝛼𝑓(𝑡)=𝐿𝐷𝛼(𝑓(𝑡)𝑓(0)),𝑡>0,0<𝛼<1.(2.4)

Remark 2.4 (see [32]). (i) If 𝑓𝐶1[0,), then 𝐷𝛼1𝑓(𝑡)=Γ(1𝛼)𝑡0𝑓(𝑠)(𝑡𝑠)𝛼𝑑𝑠,𝑡>0,0<𝛼<1.(2.5)
(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.

For more fractional theories, one can refer to the books [9, 4244].

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. 𝑋1 denotes the Banach space 𝐷(𝐴) with the graph norm 1=+𝐴. Let 𝐶(𝐼,𝑋) be the Banach space of all continuous 𝑋-value functions on interval 𝐼 with norm 𝑢𝐶=max𝑡𝐼𝑢(𝑡). 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} and [𝑣(𝑡),𝑤(𝑡)]={𝑦𝑋𝑣(𝑡)𝑦𝑤(𝑡)}, 𝑡𝐼. Set 𝐶𝛼(𝐼,𝑋)={𝑢𝐶(𝐼,𝑋)𝐷𝛼𝑢 exists and 𝐷𝛼𝑢𝐶(𝐼,𝑋)}.

Definition 2.5. If  𝑣0𝐶𝛼(𝐼,𝑋)𝐶(𝐼,𝑋1)  and satisfies 𝐷𝛼𝑣0(𝑡)+𝐴𝑣0(𝑡)𝑓𝑡,𝑣0𝑣(𝑡),𝑡𝐼,0(0)𝑣(𝜔),(2.6) then 𝑣0 is called a lower solution of PBVP (1.1); if all inequalities of (2.6) are inverse, one calls it an upper solution of PBVP (1.1).

Definition 2.6 (see [29, 45]). If  𝐶(𝐼,𝑋),  by the mild solution of LIVP, 𝐷𝛼𝑢(𝑡)+𝐴𝑢(𝑡)=(𝑡),𝑡𝐼,𝑢(0)=𝑥0𝑋,(2.7) one means that the function 𝑢𝐶(𝐼,𝑋) and satisfies 𝑢(𝑡)=𝑈(𝑡)𝑥0+𝑡0(𝑡𝑠)𝛼1𝑉(𝑡𝑠)(𝑠)𝑑𝑠,(2.8) where 𝑈(𝑡)=0𝜁𝛼(𝜃)𝑇(𝑡𝛼𝜃)𝑑𝜃,𝑉(𝑡)=𝛼0𝜃𝜁𝛼(𝜃)𝑇(𝑡𝛼𝜁𝜃)𝑑𝜃,(2.9)𝛼(1𝜃)=𝛼𝜃11/𝛼𝜌𝛼𝜃1/𝛼,𝜌𝛼1(𝜃)=𝜋𝑛=0(1)𝑛1𝜃𝛼𝑛1Γ(𝑛𝛼+1)𝑛!sin(𝑛𝜋𝛼),𝜃(0,),(2.10) and 𝜁𝛼(𝜃) is a probability density function defined on (0,).

Remark 2.7. (i) [2931] 𝜁𝛼(𝜃)0,𝜃(0,), 0𝜁𝛼(𝜃)𝑑𝜃=1, and 0𝜃𝜁𝛼(𝜃)𝑑𝜃=1/Γ(1+𝛼).
(ii) [33, 34, 46, 47] The Laplace transform of 𝜁𝛼 is given by 0𝑒𝑝𝜃𝜁𝛼(𝜃)𝑑𝜃=𝑛=0(𝑝)𝑛Γ(1+𝑛𝛼)=𝐸𝛼(𝑝),(2.11) where 𝐸𝛼() is Mittag-Leffler function (see [42]).
(iii) [48] For 𝑝<0, 0<𝐸𝛼(𝑝)<𝐸𝛼(0)=1.

Lemma 2.8. If {𝑇(𝑡)}𝑡0 is an exponentially stable 𝐶0-semigroup, there are constants 𝑁1 and 𝛿>0, such that 𝑇(𝑡)𝑁𝑒𝛿𝑡,𝑡0,(2.12) then the linear periodic boundary value problem (LPBVP) 𝐷𝛼𝑢(𝑡)+𝐴𝑢(𝑡)=(𝑡),𝑡𝐼,𝑢(0)=𝑢(𝜔)(2.13) has a unique mild solution (𝑃)(𝑡)=𝑈(𝑡)𝐵()+𝑡0(𝑡𝑠)𝛼1𝑉(𝑡𝑠)(𝑠)𝑑𝑠,(2.14) where 𝑈(𝑡) and 𝑉(𝑡) are given by (2.9) 𝐵()=(𝐼𝑈(𝜔))1𝜔0(𝜔𝑠)𝛼1𝑉(𝜔𝑠)(𝑠)𝑑𝑠.(2.15)

Proof. In 𝑋, give equivalent norm || by |𝑥|=sup𝑡0𝑒𝛿𝑡,𝑇(𝑡)𝑥(2.16) then 𝑥|𝑥|𝑁𝑥. By |𝑇(𝑡)|, we denote the norm of 𝑇(𝑡) in (𝑋,||), then for 𝑡0, ||||𝑇(𝑡)𝑥=sup𝑠0𝑒𝛿𝑠𝑇(𝑠)𝑇(𝑡)𝑥=𝑒𝛿𝑡sup𝑠0𝑒𝛿(𝑠+𝑡)𝑇(𝑠+𝑡)𝑥=𝑒𝛿𝑡sup𝜂𝑡𝑒𝛿𝜂𝑇(𝜂)𝑥𝑒𝛿𝑡|𝑥|.(2.17) Thus, |𝑇(𝑡)|𝑒𝛿𝑡. Then by Remark 2.7, ||𝑈||=||||(𝜔)0𝜁𝛼(𝜃)𝑇(𝜔𝛼||||𝜃)𝑑𝜃0𝜁𝛼(𝜃)𝑒𝛿𝜔𝛼𝜃𝑑𝜃=𝐸𝛼(𝛿𝜔𝛼)<1.(2.18) Therefore, 𝐼𝑈(𝜔) has bounded inverse operator and (𝐼𝑈(𝜔))1=𝑛=0(𝑈(𝜔))𝑛.(2.19) Set 𝑥0=(𝐼𝑈(𝜔))1𝜔0(𝜔𝑠)𝛼1𝑉(𝜔𝑠)(𝑠)𝑑𝑠,(2.20) then 𝑢(𝑡)=𝑈(𝑡)𝑥0+𝑡0(𝑡𝑠)𝛼1𝑉(𝑡𝑠)(𝑠)𝑑𝑠(2.21) is the unique mild solution of LIVP (2.7) and satisfies 𝑢(0)=𝑢(𝜔). So set 𝐵()=(𝐼𝑈(𝜔))1𝜔0(𝜔𝑠)𝛼1𝑉(𝜔𝑠)(𝑠)𝑑𝑠,(𝑃)(𝑡)=𝑈(𝑡)𝐵()+𝑡0(𝑡𝑠)𝛼1𝑉(𝑡𝑠)(𝑠)𝑑𝑠,(2.22) then 𝑃 is the unique mild solution of LPBVP (2.13).

Remark 2.9. For sufficient conditions of exponentially stable 𝐶0-semigroup, one can see [49].

Definition 2.10. A 𝐶0-semigroup {𝑇(𝑡)}𝑡0 is called a compact semigroup if 𝑇(𝑡) is compact for 𝑡>0.

Definition 2.11. A 𝐶0-semigroup {𝑇(𝑡)}𝑡0 is called an equicontinuous semigroup if 𝑇(𝑡) is continuous in the uniform operator topology (i.e., uniformly continuous) for 𝑡>0.

Remark 2.12. Compact semigroups, differential semigroups, and analytic semigroups are equicontinuous semigroups, see [50]. In the applications of partial differential equations, such as parabolic and strongly damped wave equations, the corresponding solution semigroups are analytic semigroups.

Definition 2.13. A 𝐶0-semigroup {𝑇(𝑡)}𝑡0 is called a positive semigroup if 𝑇(𝑡)𝑥𝜃 for all 𝑥𝜃 and 𝑡0.

Remark 2.14. From Definition 2.13, if 𝜃, 𝑥0𝜃, and 𝑇(𝑡)(𝑡0) is a positive 𝐶0-semigroup generated by 𝐴, the mild solution 𝑢𝐶(𝐼,𝑋) given by (2.8) satisfies 𝑢𝜃. For the applications of positive operators semigroup, we can see [3741]. It is easy to see that positive operators semigroup can play the role as the comparison principles.

Definition 2.15. A bounded linear operator 𝐾 on 𝑋 is called to be positive if 𝐾𝑥𝜃 for all 𝑥𝜃.

Lemma 2.16. The operators  𝑈  and 𝑉 given by (2.9) 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,(2.23)(ii){𝑈(𝑡)}𝑡0 and {𝑉(𝑡)}𝑡0 are strongly continuous,(iii){𝑈(𝑡)}𝑡0 and {𝑉(𝑡)}𝑡0 are compact operators if {𝑇(𝑡)}𝑡0 is a compact semigroup,(iv)𝑈(𝑡) and 𝑉(𝑡) are continuous in the uniform operator topology (i.e., uniformly continuous) for 𝑡>0 if {𝑇(𝑡)}𝑡0 is an equicontinuous semigroup,(v)𝑈(𝑡) and 𝑉(𝑡) are positive for 𝑡0 if {𝑇(𝑡)}𝑡0 is a positive semigroup,(vi)(𝐼𝑈(𝜔))1 is a positive operator if {𝑇(𝑡)}𝑡0 is an exponentially and positive semigroup.

Proof. For the proof of (i)–(iii), one can refer to [29, 31]. We only check (iv), (v), and (vi) as follows.(iv) For 0<𝑡1𝑡2, we have 𝑈𝑡2𝑡𝑈10𝜁𝛼𝑇𝑡(𝜃)𝛼2𝜃𝑡𝑇𝛼1𝜃𝑉𝑡𝑑𝜃,2𝑡𝑉1𝛼0𝜃𝜁𝛼𝑇𝑡(𝜃)𝛼2𝜃𝑡𝑇𝛼1𝜃𝑑𝜃.(2.24) Since 𝑇(𝑡) is continuous in the uniform operator topology for 𝑡>0, by Lebesque-dominated convergence theorem and Remark 2.7 (i), 𝑈(𝑡) and 𝑉(𝑡) are continuous in the uniform operator topology for 𝑡>0.(v) By Remark 2.7 (i), the proof is then complete.(vi) By (𝑣), (2.18), and (2.19), the proof is then complete.

3. Main Results

Case 1. {𝑇(𝑡)}𝑡0 is compact.

Theorem 3.1. Assume that {𝑇(𝑡)}𝑡0 is a compact and positive semigroup in 𝑋, PBVP (1.1) has a lower solution 𝑣0 and an upper solution 𝑤0 with 𝑣0𝑤0 and satisfies the following. (H)There exists a constant 𝐶>0 such that 𝑓𝑡,𝑥2𝑓𝑡,𝑥1𝑥𝐶2𝑥1,(3.1) for any 𝑡𝐼, and 𝑣0(𝑡)𝑥1𝑥2𝑤0(𝑡), that is, 𝑓(𝑡,𝑥)+𝐶𝑥 is increasing in 𝑥 for 𝑥[𝑣0(𝑡),𝑤0(𝑡)].
Then PBVP (1.1) has the minimal and maximal mild solutions between 𝑣0 and 𝑤0, which can be obtained by a monotone iterative procedure starting from 𝑣0 and 𝑤0, respectively.

Proof. It is easy to see that (𝐴+𝐶𝐼) generates an exponentially stable and positive compact semigroup 𝑆(𝑡)=𝑒𝐶𝑡𝑇(𝑡). By (2.1), 𝑆(𝑡)𝑀. Let Φ(𝑡)=0𝜁𝛼(𝜃)𝑆(𝑡𝛼𝜃)𝑑𝜃,Ψ(𝑡)=𝛼0𝜃𝜁𝛼(𝜃)𝑆(𝑡𝛼𝜃)𝑑𝜃. By Remark 2.7 (i), we have that 𝛼Φ(𝑡)𝑀,Ψ(𝑡)Γ(1+𝛼)𝑀,𝑡0.(3.2) From Lemma 2.8, (𝐼Φ(𝜔)) has bounded inverse operator and (𝐼Φ(𝜔))1=𝑛=0(Φ(𝜔))𝑛.(3.3) By Lemma 2.16 (v) and (vi), Φ(𝑡) and Ψ(𝑡) are positive for 𝑡0, and (𝐼Φ(𝜔))1 is positive.
Let 𝐷=[𝑣0,𝑤0], then we define a mapping 𝑄𝐷𝐶(𝐼,𝑋) by 𝑄𝑢(𝑡)=Φ(𝑡)𝐵1(𝑢)+𝑡0(𝑡𝑠)𝛼1[]Ψ(𝑡𝑠)𝑓(𝑠,𝑢(𝑠))+𝐶𝑢(𝑠)𝑑𝑠,𝑡𝐼,(3.4) where 𝐵1(𝑢)=(𝐼Φ(𝜔))1𝜔0(𝜔𝑠)𝛼1[]Ψ(𝜔𝑠)𝑓(𝑠,𝑢(𝑠))+𝐶𝑢(𝑠)𝑑𝑠.(3.5) By the continuity of 𝑓 and Lemma 2.16 (ii), 𝑄𝐷𝐶(𝐼,𝑋) is continuous. By Lemma 2.8, 𝑢𝐷 is a mild solution of PBVP (1.1) if and only if 𝑢=𝑄𝑢.(3.6) For 𝑢1,𝑢2𝐷 and 𝑢1𝑢2, from (H), the positivity of operators (𝐼Φ(𝜔))1, Φ(𝑡), and Ψ(𝑡), we have that 𝑄𝑢1𝑄𝑢2.(3.7) Now, we show that 𝑣0𝑄𝑣0, 𝑄𝑤0𝑤0. Let 𝐷𝛼𝑣0(𝑡)+𝐴𝑣0(𝑡)+𝐶𝑣0(𝑡)𝜎(𝑡), by Definition 2.5, the positivity of operator Ψ(𝑡), we have that 𝑣0(𝑡)=Φ(𝑡)𝑣0(0)+𝑡0(𝑡𝑠)𝛼1Ψ(𝑡𝑠)𝜎(𝑠)𝑑𝑠Φ(𝑡)𝑣0(0)+𝑡0(𝑡𝑠)𝛼1𝑓Ψ(𝑡𝑠)𝑠,𝑣0(𝑠)+𝐶𝑣0(𝑠)𝑑𝑠,𝑡𝐼.(3.8) In particular, 𝑣0(𝜔)Φ(𝜔)𝑣0(0)+𝜔0(𝜔𝑠)𝛼1𝑓Ψ(𝜔𝑠)𝑠,𝑣0(𝑠)+𝐶𝑣0(𝑠)𝑑𝑠.(3.9) By Definition 2.5, 𝑣0(0)𝑣(𝜔), and by the positivity of operator (𝐼Φ(𝜔))1, we have that 𝑣0(0)(𝐼Φ(𝜔))1𝜔0(𝜔𝑠)𝛼1𝑓Ψ(𝜔𝑠)𝑠,𝑣0(𝑠)+𝐶𝑣0(𝑠)𝑑𝑠=𝐵1𝑣0.(3.10) Then by (3.8) and the positivity of operator Φ(𝑡), 𝑣0(𝑡)Φ(𝑡)𝐵1𝑣0+𝑡0(𝑡𝑠)𝛼1𝑓Ψ(𝑡𝑠)𝑠,𝑣0(𝑠)+𝐶𝑣0(=𝑠)𝑑𝑠𝑄𝑣0(𝑡),𝑡𝐼,(3.11) namely, 𝑣0𝑄𝑣0. Similarly, we can show that 𝑄𝑤0𝑤0. For 𝑢𝐷, in view of (3.7), then 𝑣0𝑄𝑣0𝑄𝑢𝑄𝑤0𝑤0. Thus, 𝑄𝐷𝐷 is a continuous increasing monotonic operator. We can now define the sequences 𝑣𝑛=𝑄𝑣𝑛1,𝑤𝑛=𝑄𝑤𝑛1,𝑛=1,2,,(3.12) and it follows from (3.7) that 𝑣0𝑣1𝑣𝑛𝑤𝑛𝑤1𝑤0.(3.13)
In the following, we prove that {𝑣𝑛} and {𝑤𝑛} are convergent in 𝐶(𝐼,𝑋). First, we show that 𝑄𝐷={𝑄𝑢𝑢𝐷} is precompact in 𝐶(𝐼,𝑋). Let (𝑊𝑢)(𝑡)=𝑡0(𝑡𝑠)𝛼1[]Ψ(𝑡𝑠)𝑓(𝑠,𝑢(𝑠))+𝐶𝑢(𝑠)𝑑𝑠,𝑡𝐼,(3.14)then we prove that for all 0<𝑡𝜔, (𝑊𝐷)(𝑡)={(𝑊𝑢)(𝑡)𝑢𝐷} is precompact in 𝑋. For 0<𝜀<𝑡, let 𝑊𝜀𝑢(𝑡)=0𝑡𝜀(𝑡𝑠)𝛼1[]=Ψ(𝑡𝑠)𝑓(𝑠,𝑢(𝑠))+𝐶𝑢(𝑠)𝑑𝑠0𝑡𝜀(𝑡𝑠)𝛼1𝛼0𝜃𝜁𝛼(𝜃)𝑆((𝑡𝑠)𝛼[]𝜃)𝑑𝜃𝑓(𝑠,𝑢(𝑠))+𝐶𝑢(𝑠)𝑑𝑠=𝑆(𝜀)0𝑡𝜀(𝑡𝑠)𝛼1𝛼0𝜃𝜁𝛼(𝜃)𝑆((𝑡𝑠)𝛼[]𝜃𝜀)𝑑𝜃𝑓(𝑠,𝑢(𝑠))+𝐶𝑢(𝑠)𝑑𝑠.(3.15) For 𝑢𝐷, by (H), 𝑓(𝑡,𝑣0(𝑡))+𝐶𝑣0(𝑡)𝑓(𝑡,𝑢(𝑡))+𝐶𝑢(𝑡)𝑓(𝑡,𝑤0(𝑡))+𝐶𝑤0(𝑡) for 0𝑡𝜔. By the normality of the cone 𝑃, there is 𝑀1>0 such that 𝑓(𝑡,𝑢(𝑡))+𝐶𝑢(𝑡)𝑀1,0𝑡𝜔.(3.16) Thus, by (3.16) and Remark 2.7 (i), we have 0𝑡𝜀(𝑡𝑠)𝛼1𝛼0𝜃𝜁𝛼(𝜃)𝑆((𝑡𝑠)𝛼[]𝜃𝜀)𝑑𝜃𝑓(𝑠,𝑢(𝑠))+𝐶𝑢(𝑠)𝑑𝑠𝑀10𝑡𝜀(𝑡𝑠)𝛼1𝛼0𝜃𝜁𝛼(𝜃)𝑆((𝑡𝑠)𝛼𝜃𝜀)𝑑𝜃𝑑𝑠𝑀𝑀10𝑡𝜀(𝑡𝑠)𝛼1𝛼0𝜃𝜁𝛼(𝜃)𝑑𝜃𝑑𝑠=𝑀𝑀1𝛼Γ(1+𝛼)0𝑡𝜀(𝑡𝑠)𝛼1𝑑𝑠=𝑀𝑀1(𝑡𝛼𝜀𝛼)Γ(1+𝛼),0<𝑡𝜔.(3.17) Then by (3.15), (3.17) and the compactness of 𝑆(𝜀), for 0<𝑡𝜔, (𝑊𝜀𝐷)(𝑡)={(𝑊𝜀𝑢)(𝑡)𝑢𝐷} is precompact in 𝑋. Furthermore, by (3.16) and Lemma 2.16 (i), we have (𝑊𝑊𝑢)(𝑡)𝜀𝑢(=𝑡)𝑡𝑡𝜀(𝑡𝑠)𝛼1[]Ψ(𝑡𝑠)𝑓(𝑠,𝑢(𝑠))+𝐶𝑢(𝑠)𝑑𝑠𝑀𝑀1𝛼Γ(1+𝛼)𝑡𝑡𝜀(𝑡𝑠)𝛼1𝑑𝑠=𝑀𝑀1𝜀𝛼Γ.(1+𝛼)(3.18) Therefore, for 0<𝑡𝜔, (𝑊𝐷)(𝑡) is precompact in 𝑋. In particular, (𝑊𝐷)(𝜔) is precompact in 𝑋, and then 𝐵1(𝐷)=(𝐼Φ(𝜔))1(𝑊𝐷)(𝜔) is precompact. Then in view of Lemma 2.16 (i), (𝑄𝐷)(𝑡)={(𝑄𝑢(𝑡))𝑢𝐷}=Φ(𝑡)𝐵1(𝐷)+(𝑊𝐷)(𝑡) is precompact in 𝑋 for 0𝑡𝜔.
Furthermore, for 0𝑡1<𝑡2𝜔, by (3.16) and Lemma 2.16 (i) we have that (𝑡𝑊𝑢)2𝑡(𝑊𝑢)1=𝑡20𝑡2𝑠𝛼1Ψ𝑡2[]𝑠𝑓(𝑠,𝑢(𝑠))+𝐶𝑢(𝑠)𝑑𝑠𝑡10𝑡1𝑠𝛼1Ψ𝑡1[]𝑠𝑓(𝑠,𝑢(𝑠))+𝐶𝑢(𝑠)𝑑𝑠𝑀1𝑡10𝑡2𝑠𝛼1Ψ𝑡2𝑡𝑠1𝑠𝛼1Ψ𝑡1𝑠𝑑𝑠+𝑀𝑀1𝛼Γ(1+𝛼)𝑡2𝑡1𝑡2𝑠𝛼1𝑑𝑠𝑀1𝑡10𝑡2𝑠𝛼1Ψ𝑡2𝑡𝑠Ψ1𝑠𝑑𝑠+𝑀1𝑡10𝑡2𝑠𝛼1𝑡1𝑠𝛼1Ψ𝑡1𝑠𝑑𝑠+𝑀𝑀1𝑡Γ(1+𝛼)2𝑡1𝛼𝑀1𝑡2𝑡1𝛼1𝑡10Ψ𝑡2𝑡𝑠Ψ1+𝑠𝑑𝑠𝑀𝑀1||𝑡Γ(1+𝛼)𝛼1+𝑡2𝑡1𝛼𝑡𝛼2||+𝑀𝑀1𝑡Γ(1+𝛼)2𝑡1𝛼𝑀1𝑡2𝑡1𝛼1𝑡10Ψ𝑡2𝑡𝑠Ψ1𝑠𝑑𝑠+2𝑀𝑀1𝑡Γ(1+𝛼)2𝑡1𝛼+𝑀𝑀1𝑡Γ(1+𝛼)𝛼2𝑡𝛼1.(3.19) By Remark 2.12 and Lemma 2.16 (iv), Ψ(𝑡) is continuous in the uniform operator topology for 𝑡>0. Then by Lebesque-dominated convergence theorem, 𝑊𝐷 is equicontinuous in 𝐶(𝐼,𝑋). By Lemma 2.16 (ii), {Ψ(𝑡)}𝑡0 is strongly continuous. So, 𝑄𝐷 is equicontinuous in 𝐶(𝐼,𝑋).
Then by Ascoli-Arzela’s theorem, 𝑄𝐷={𝑄𝑢𝑢𝐷} is precompact in 𝐶(𝐼,𝑋). By (3.12) and (3.13), {𝑣𝑛} has a convergent subsequence in 𝐶(𝐼,𝑋). Combining this with the monotonicity of {𝑣𝑛}, it is itself convergent in 𝐶(𝐼,𝑋). Using a similar argument to that for {𝑣𝑛}, we can prove that {𝑤𝑛} is also convergent in 𝐶(𝐼,𝑋). Set 𝑢=lim𝑛𝑣𝑛,𝑢=lim𝑛𝑤𝑛.(3.20) Let 𝑛, by the continuity of 𝑄 and (3.12), we have 𝑢=𝑄𝑢,𝑢=𝑄𝑢.(3.21) By (3.7), if 𝑢𝐷 is a fixed-point of 𝑄, then 𝑣1=𝑄𝑣0𝑄𝑢=𝑢𝑄𝑤0=𝑤1. By induction, 𝑣𝑛𝑢𝑤𝑛. By (3.13) and taking the limit as 𝑛, we conclude that 𝑣0𝑢𝑢𝑢𝑤0. This means that 𝑢,𝑢 are the minimal and maximal fixed-points of 𝑄 on [𝑣0,𝑤0], respectively. By (3.6), they are the minimal and maximal mild solutions of PBVP (1.1) on [𝑣0,𝑤0], respectively.

Theorem 3.2. Assume that {𝑇(𝑡)}𝑡0 is a compact and positive semigroup in 𝑋, 𝑓(𝑡,𝜃)𝜃 for 𝑡𝐼. If there is 𝑦𝑋 such that 𝑦𝜃, 𝐴𝑦𝑓(𝑡,𝑦) for 𝑡𝐼, and 𝑓 satisfies the following: (H1)There exists a constant 𝐶1>0 such that 𝑓𝑡,𝑥2𝑓𝑡,𝑥1𝐶1𝑥2𝑥1,(3.22) for any 𝑡𝐼, and 𝜃𝑥1𝑥2𝑦, that is, 𝑓(𝑡,𝑥)+𝐶1𝑥 is increasing in 𝑥 for 𝑥[𝜃,𝑦].
Then PBVP (1.1) has a positive mild solution 𝑢: 𝜃𝑢𝑦.

Proof. Let 𝑣0=𝜃 and 𝑤0=𝑦, by Theorem 3.1, PBVP (1.1) has mild solution on [𝑣0,𝑤0].

Case 2. {𝑇(𝑡)}𝑡0 is noncompact.

Theorem 3.3. Assume that the positive cone 𝑃 is regular, {𝑇(𝑡)}𝑡0 is an equicontinuous and positive semigroup in 𝑋, PBVP (1.1) has a lower solution 𝑣0 and an upper solution 𝑤0 with 𝑣0𝑤0, and (H) holds, then PBVP (1.1) has the minimal and maximal mild solutions between 𝑣0 and 𝑤0, which can be obtained by a monotone iterative procedure starting from 𝑣0 and 𝑤0, respectively.

Proof. By the proof of Theorem 3.1, (3.2)–(3.13) and (3.19) are valid. By Lemma 2.16 (iv), Ψ(𝑡) is continuous in the uniform operator topology for 𝑡>0. Then by Lebesque-dominated convergence theorem, 𝑊𝐷 is equicontinuous in 𝐶(𝐼,𝑋). From Lemma 2.16 (ii), {Ψ(𝑡)}𝑡0 is strongly continuous. So, 𝑄𝐷 is equicontinuous in 𝐶(𝐼,𝑋). Thus, {𝑄𝑣𝑛} is equicontinuous in 𝐶(𝐼,𝑋).
For 0𝑡𝜔, by (3.7) and (3.13), {(𝑄𝑣𝑛)(𝑡)} is monotone in 𝑋. Since the cone 𝑃 is regular, then {(𝑄𝑣𝑛)(𝑡)} is convergent in 𝑋.
By Ascoli-Arzela’s theorem, {𝑄𝑣𝑛} is precompact in 𝐶(𝐼,𝑋) and {𝑄𝑣𝑛} has a convergent subsequence in 𝐶(𝐼,𝑋). Combining this with the monotonicity of {𝑄𝑣𝑛}, it is itself convergent in 𝐶(𝐼,𝑋). Using a similar argument to that for {𝑄𝑤𝑛}, we can prove that {𝑄𝑤𝑛} is also convergent in 𝐶(𝐼,𝑋). Let 𝑢=lim𝑛𝑣𝑛=lim𝑛𝑄𝑣𝑛1,𝑢=lim𝑛𝑤𝑛=lim𝑛𝑄𝑤𝑛1,(3.23) then it is similar to the proof of Theorem 3.1 that 𝑢 and 𝑢 are the minimal and maximal mild solutions of PBVP (1.1) on [𝑣0,𝑤0], respectively.

Corollary 3.4. Let 𝑋 be an ordered and weakly sequentially complete Banach space. Assume that {𝑇(𝑡)}𝑡0 is an equicontinuous and positive semigroup in 𝑋, PBVP (1.1) has a lower solution 𝑣0 and an upper solution 𝑤0 with 𝑣0𝑤0, and (H) holds, then PBVP (1.1) has the minimal and maximal mild solutions between 𝑣0 and 𝑤0, which can be obtained by a monotone iterative procedure starting from 𝑣0 and 𝑤0, respectively.

Proof. In an ordered and weakly sequentially complete Banach space, the normal cone 𝑃 is regular. Then the proof is complete.

Corollary 3.5. Let 𝑋 be an ordered and reflective Banach space. Assume that {𝑇(𝑡)}𝑡0 is an equicontinuous and positive semigroup in 𝑋, PBVP (1.1) has a lower solution 𝑣0 and an upper solution 𝑤0 with 𝑣0𝑤0, and (H) holds, then PBVP (1.1) has the minimal and maximal mild solutions between 𝑣0 and 𝑤0, which can be obtained by a monotone iterative procedure starting from 𝑣0 and 𝑤0, respectively.

Proof. In an ordered and reflective Banach space, the normal cone 𝑃 is regular. Then the proof is complete.

By Theorem 3.3, Corollaries 3.4 and 3.5, we have the following.

Corollary 3.6. Assume that {𝑇(𝑡)}𝑡0 is an equicontinuous and positive semigroup in 𝑋, 𝑓(𝑡,𝜃)𝜃 for 𝑡𝐼. If there is 𝑦𝑋 such that 𝑦𝜃, 𝐴𝑦𝑓(𝑡,𝑦) for 𝑡𝐼, 𝑓 satisfies (H1) and one of the following conditions: (i)𝑋 is an ordered Banach space, whose positive cone 𝑃 is regular,(ii)𝑋 is an ordered and weakly sequentially complete Banach space,(iii)𝑋 is an ordered and reflective Banach space.
then PBVP (1.1) has positive mild solution 𝑢: 𝜃𝑢𝑦.

4. Examples

Example 4.1. Consider the following periodic boundary value problem for fractional parabolic partial differential equations in 𝑋:  𝜕𝛼𝑡𝑢𝑢+𝐴(𝑥,𝐷)𝑢=𝑔(𝑥,𝑡,𝑢),(𝑥,𝑡)Ω×𝐼,𝐵𝑢=0,(𝑥,𝑡)𝜕Ω×𝐼,(𝑥,0)=𝑢(𝑥,𝜔),𝑥Ω,(4.1) where 𝜕𝛼𝑡 is the Caputo fractional partial derivative with order 0<𝛼<1, 𝐼=[0,𝜔], Ω𝑁 is a bounded domain with a sufficiently smooth boundary 𝜕Ω, 𝑔Ω×𝐼× is continuous, 𝐵𝑢=𝑏0(𝑥)𝑢+𝛿(𝜕𝑢/𝜕𝑛) is a regular boundary operator on 𝜕Ω, and 𝐴(𝑥,𝐷)𝑢=𝑁𝑁𝑖=1𝑗=1𝜕𝜕𝑥𝑖𝑎𝑖𝑗(𝑥)𝜕𝑢𝜕𝑦𝑖(4.2) is a symmetrical strong elliptic operator of second order, whose coefficient functions are Hölder continuous in Ω.
Let 𝑋=𝐿𝑝(Ω)(𝑝2), 𝑃={𝑣𝑣𝐿𝑝(Ω),𝑣(𝑥)0a.e.𝑥Ω}, then 𝑋 is a Banach space, and 𝑃 is a regular cone in 𝑋. Define the operator 𝐴 as follows: 𝐷(𝐴)=𝑢𝑊2,𝑝(Ω)𝐵𝑢=0,𝐴𝑢=𝐴(𝑥,𝐷)𝑢.(4.3) Then 𝐴 generates a uniformly bounded analytic semigroup 𝑇(𝑡)(𝑡0) in 𝑋 (see [39]). By the maximum principle, we can easily find that 𝑇(𝑡)(𝑡0) is positive (see [39]). Let 𝑢(𝑡)=𝑢(,𝑡), 𝑓(𝑡,𝑢)=𝑔(,𝑡,𝑢(,𝑡)), then the problem (4.1) can be transformed into the following problem: 𝐷𝛼𝑢(𝑡)+𝐴𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑡𝐼,𝑢(0)=𝑢(𝜔).(4.4)

Theorem 4.2. Let 𝑓(𝑥,𝑡,0)0. If there exists 𝑤0(𝑥,𝑡)𝐶2,𝛼(Ω×𝐼) such that 𝜕𝛼𝑡𝑤0+𝐴(𝑥,𝐷)𝑤0𝑔𝑥,𝑡,𝑤0,𝑤(𝑥,𝑡)Ω×𝐼,𝐵𝑤=0,(𝑥,𝑡)𝜕Ω×𝐼,0(𝑥,0)𝑤0(𝑥,𝜔),𝑥Ω,(4.5) and 𝑔 satisfies the following:(H4)there exists a constant 𝐶20 such that 𝑔𝑥,𝑡,𝜉2𝑔𝑥,𝑡,𝜉1𝐶2𝜉2𝜉1,(4.6) for any 𝑡𝐼, and 0𝜉1𝜉2𝑤0
Then PBVP (4.1) has a mild solution 𝑢0𝑢𝑤0.

Proof. Set 𝑣0=0, by Theorem 3.3, PBVP (4.1) has the minimal and maximal solutions between 0 and 𝑤0.

Acknowledgments

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

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