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)),𝑡∈𝐼,𝑢(0)=𝑢(𝜔),(1.1) where 𝐷𝛼 is the Caputo fractional derivative of order 0<𝛼<1, 𝐼=[0,𝜔], −𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is the infinitesimal generator of a 𝐶0-semigroup (i.e., strongly continuous semigroup) {𝑇(𝑡)}𝑡≥0 of uniformly bounded linear operators on 𝑋, and 𝑓∶𝐼×𝑋→𝑋 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 [1–3]. 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 [9–13], 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 [28–36]. 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),𝑡≥0,𝑦∈𝑅,(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 [37–41].

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+ğ‘žğ¼) generates a uniformly bounded 𝐶0-semigroup for ğ‘ž>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)}𝑡≥0. This means that there exists 𝑀≥1 such that ‖𝑇(𝑡)‖≤𝑀,𝑡≥0.(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  𝑓∈𝐴𝐶[0,∞)is defined as 𝐼𝛼1𝑓(𝑡)=Γ(𝛼)𝑡0𝑓(𝑠)(𝑡−𝑠)1−𝛼𝑑𝑠,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 𝑓∈𝐴𝐶[0,∞) can be written as 𝐿𝐷𝛼1𝑓(𝑡)=𝑑Γ(1−𝛼)𝑑𝑡𝑡0𝑓(𝑠)(𝑡−𝑠)𝛼𝑑𝑠,𝑡>0,0<𝛼<1.(2.3)

Definition 2.3 (see [9, 32]). The Caputo fractional derivative of order 𝛼 for a function 𝑓∈𝐴𝐶[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 𝑓∈𝐶1[0,∞), then 𝐷𝛼1𝑓(𝑡)=Γ(1−𝛼)𝑡0ğ‘“î…ž(𝑠)(𝑡−𝑠)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 2.1–2.3 are taken in Bochner’s sense.

For more fractional theories, one can refer to the books [9, 42–44].

Throughout this paper, let 𝑋 be an ordered Banach space with norm ‖⋅‖ and partial order ≤, whose positive cone 𝑃={𝑦∈𝑋∣𝑦≥𝜃} (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 ‖𝑢‖𝐶=max𝑡∈𝐼‖𝑢(𝑡)‖. For 𝑢,𝑣∈𝐶(𝐼,𝑋), 𝑢≤𝑣 if 𝑢(𝑡)≤𝑣(𝑡) for all 𝑡∈𝐼. For 𝑣,𝑤∈𝐶(𝐼,𝑋), denote the ordered interval [𝑣,𝑤]={𝑢∈𝐶(𝐼,𝑋)∣𝑣≤𝑢≤𝑤} and [𝑣(𝑡),𝑤(𝑡)]={𝑦∈𝑋∣𝑣(𝑡)≤𝑦≤𝑤(𝑡)}, 𝑡∈𝐼. Set 𝐶𝛼(𝐼,𝑋)={𝑢∈𝐶(𝐼,𝑋)∣𝐷𝛼𝑢 exists and 𝐷𝛼𝑢∈𝐶(𝐼,𝑋)}.

Definition 2.5. If  𝑣0∈𝐶𝛼(𝐼,𝑋)∩𝐶(𝐼,𝑋1)  and satisfies 𝐷𝛼𝑣0(𝑡)+𝐴𝑣0(𝑡)≤𝑓𝑡,𝑣0𝑣(𝑡),𝑡∈𝐼,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]). Ifâ€‰â€‰â„Žâˆˆğ¶(𝐼,𝑋),  by the mild solution of LIVP, 𝐷𝛼𝑢(𝑡)+𝐴𝑢(𝑡)=ℎ(𝑡),𝑡∈𝐼,𝑢(0)=𝑥0∈𝑋,(2.7) one means that the function 𝑢∈𝐶(𝐼,𝑋) and satisfies 𝑢(𝑡)=𝑈(𝑡)𝑥0+𝑡0(𝑡−𝑠)𝛼−1𝑉(𝑡−𝑠)ℎ(𝑠)𝑑𝑠,(2.8) where 𝑈(𝑡)=∞0𝜁𝛼(𝜃)𝑇(𝑡𝛼𝜃)𝑑𝜃,𝑉(𝑡)=ğ›¼âˆž0𝜃𝜁𝛼(𝜃)𝑇(𝑡𝛼𝜁𝜃)𝑑𝜃,(2.9)𝛼(1𝜃)=𝛼𝜃−1−1/𝛼𝜌𝛼𝜃−1/𝛼,𝜌𝛼1(𝜃)=ğœ‹âˆžî“ğ‘›=0(−1)𝑛−1𝜃−𝛼𝑛−1Γ(𝑛𝛼+1)𝑛!sin(𝑛𝜋𝛼),𝜃∈(0,∞),(2.10) and 𝜁𝛼(𝜃) is a probability density function defined on (0,∞).

Remark 2.7. (i) [29–31] 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 ∞0𝑒−𝑝𝜃𝜁𝛼(𝜃)𝑑𝜃=âˆžî“ğ‘›=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)}𝑡≥0 is an exponentially stable 𝐶0-semigroup, there are constants 𝑁≥1 and 𝛿>0, such that ‖𝑇(𝑡)‖≤𝑁𝑒−𝛿𝑡,𝑡≥0,(2.12) then the linear periodic boundary value problem (LPBVP) 𝐷𝛼𝑢(𝑡)+𝐴𝑢(𝑡)=ℎ(𝑡),𝑡∈𝐼,𝑢(0)=𝑢(𝜔)(2.13) has a unique mild solution (î€œğ‘ƒâ„Ž)(𝑡)=𝑈(𝑡)𝐵(ℎ)+𝑡0(𝑡−𝑠)𝛼−1𝑉(𝑡−𝑠)ℎ(𝑠)𝑑𝑠,(2.14) where 𝑈(𝑡) and 𝑉(𝑡) are given by (2.9) 𝐵(ℎ)=(𝐼−𝑈(𝜔))−1𝜔0(𝜔−𝑠)𝛼−1𝑉(𝜔−𝑠)ℎ(𝑠)𝑑𝑠.(2.15)

Proof. In 𝑋, give equivalent norm |⋅| by |𝑥|=sup𝑡≥0‖‖𝑒𝛿𝑡‖‖,𝑇(𝑡)𝑥(2.16) then ‖𝑥‖≤|𝑥|≤𝑁‖𝑥‖. By |𝑇(𝑡)|, we denote the norm of 𝑇(𝑡) in (𝑋,|⋅|), then for 𝑡≥0, ||||𝑇(𝑡)𝑥=sup𝑠≥0‖‖𝑒𝛿𝑠‖‖𝑇(𝑠)𝑇(𝑡)𝑥=𝑒−𝛿𝑡sup𝑠≥0‖‖𝑒𝛿(𝑠+𝑡)‖‖𝑇(𝑠+𝑡)𝑥=𝑒−𝛿𝑡sup𝜂≥𝑡‖‖𝑒𝛿𝜂‖‖𝑇(𝜂)𝑥≤𝑒−𝛿𝑡|𝑥|.(2.17) Thus, |𝑇(𝑡)|≤𝑒−𝛿𝑡. Then by Remark 2.7, ||𝑈||=||||(𝜔)∞0𝜁𝛼(𝜃)𝑇(𝜔𝛼||||≤𝜃)ğ‘‘ğœƒâˆž0𝜁𝛼(𝜃)𝑒−𝛿𝜔𝛼𝜃𝑑𝜃=𝐸𝛼(−𝛿𝜔𝛼)<1.(2.18) Therefore, 𝐼−𝑈(𝜔) has bounded inverse operator and (𝐼−𝑈(𝜔))−1=âˆžî“ğ‘›=0(𝑈(𝜔))𝑛.(2.19) Set 𝑥0=(𝐼−𝑈(𝜔))−1𝜔0(𝜔−𝑠)𝛼−1𝑉(𝜔−𝑠)ℎ(𝑠)𝑑𝑠,(2.20) then 𝑢(𝑡)=𝑈(𝑡)𝑥0+𝑡0(𝑡−𝑠)𝛼−1𝑉(𝑡−𝑠)ℎ(𝑠)𝑑𝑠(2.21) is the unique mild solution of LIVP (2.7) and satisfies 𝑢(0)=𝑢(𝜔). So set 𝐵(ℎ)=(𝐼−𝑈(𝜔))−1𝜔0(𝜔−𝑠)𝛼−1𝑉(𝜔−𝑠)ℎ(𝑠)𝑑𝑠,(ğ‘ƒâ„Ž)(𝑡)=𝑈(𝑡)𝐵(ℎ)+𝑡0(𝑡−𝑠)𝛼−1𝑉(𝑡−𝑠)ℎ(𝑠)𝑑𝑠,(2.22) then ğ‘ƒâ„Ž 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)}𝑡≥0 is called a compact semigroup if 𝑇(𝑡) is compact for 𝑡>0.

Definition 2.11. A 𝐶0-semigroup {𝑇(𝑡)}𝑡≥0 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)}𝑡≥0 is called a positive semigroup if 𝑇(𝑡)𝑥≥𝜃 for all 𝑥≥𝜃 and 𝑡≥0.

Remark 2.14. From Definition 2.13, if â„Žâ‰¥ğœƒ, 𝑥0≥𝜃, and 𝑇(𝑡)(𝑡≥0) is a positive 𝐶0-semigroup generated by −𝐴, the mild solution 𝑢∈𝐶(𝐼,𝑋) given by (2.8) satisfies 𝑢≥𝜃. For the applications of positive operators semigroup, we can see [37–41]. 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 𝐾𝑥≥𝜃 for all 𝑥≥𝜃.

Lemma 2.16. The operators  𝑈  and 𝑉 given by (2.9) have the following properties: (i)For any fixed 𝑡≥0, 𝑈(𝑡) and 𝑉(𝑡) are linear and bounded operators, that is, for any 𝑥∈𝑋, ‖𝑈(𝑡)𝑥‖≤𝑀‖𝑥‖,‖𝑉(𝑡)𝑥‖≤𝛼𝑀Γ(1+𝛼)‖𝑥‖,(2.23)(ii){𝑈(𝑡)}𝑡≥0 and {𝑉(𝑡)}𝑡≥0 are strongly continuous,(iii){𝑈(𝑡)}𝑡≥0 and {𝑉(𝑡)}𝑡≥0 are compact operators if {𝑇(𝑡)}𝑡≥0 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)}𝑡≥0 is an equicontinuous semigroup,(v)𝑈(𝑡) and 𝑉(𝑡) are positive for 𝑡≥0 if {𝑇(𝑡)}𝑡≥0 is a positive semigroup,(vi)(𝐼−𝑈(𝜔))−1 is a positive operator if {𝑇(𝑡)}𝑡≥0 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<𝑡1≤𝑡2, we have ‖‖𝑈𝑡2𝑡−𝑈1‖‖≤∞0𝜁𝛼‖‖𝑇𝑡(𝜃)𝛼2𝜃𝑡−𝑇𝛼1𝜃‖‖‖‖𝑉𝑡𝑑𝜃,2𝑡−𝑉1î€¸â€–â€–î€œâ‰¤ğ›¼âˆž0𝜃𝜁𝛼‖‖𝑇𝑡(𝜃)𝛼2𝜃𝑡−𝑇𝛼1𝜃‖‖𝑑𝜃.(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)}𝑡≥0 is compact.

Theorem 3.1. Assume that {𝑇(𝑡)}𝑡≥0 is a compact and positive semigroup in 𝑋, PBVP (1.1) has a lower solution 𝑣0 and an upper solution 𝑤0 with 𝑣0≤𝑤0 and satisfies the following. (H)There exists a constant 𝐶>0 such that 𝑓𝑡,𝑥2−𝑓𝑡,𝑥1𝑥≥−𝐶2−𝑥1,(3.1) for any 𝑡∈𝐼, and 𝑣0(𝑡)≤𝑥1≤𝑥2≤𝑤0(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)=𝑒−𝐶𝑡𝑇(𝑡). By (2.1), ‖𝑆(𝑡)‖≤𝑀. Let ∫Φ(𝑡)=∞0𝜁𝛼(𝜃)𝑆(𝑡𝛼∫𝜃)𝑑𝜃,Ψ(𝑡)=ğ›¼âˆž0𝜃𝜁𝛼(𝜃)𝑆(𝑡𝛼𝜃)𝑑𝜃. By Remark 2.7 (i), we have that 𝛼‖Φ(𝑡)‖≤𝑀,‖Ψ(𝑡)‖≤Γ(1+𝛼)𝑀,𝑡≥0.(3.2) From Lemma 2.8, (𝐼−Φ(𝜔)) has bounded inverse operator and (𝐼−Φ(𝜔))−1=âˆžî“ğ‘›=0(Φ(𝜔))𝑛.(3.3) By Lemma 2.16 (v) and (vi), Φ(𝑡) and Ψ(𝑡) are positive for 𝑡≥0, and (𝐼−Φ(𝜔))−1 is positive.
Let 𝐷=[𝑣0,𝑤0], then we define a mapping 𝑄∶𝐷→𝐶(𝐼,𝑋) by 𝑄𝑢(𝑡)=Φ(𝑡)𝐵1(𝑢)+𝑡0(𝑡−𝑠)𝛼−1[]Ψ(𝑡−𝑠)𝑓(𝑠,𝑢(𝑠))+𝐶𝑢(𝑠)𝑑𝑠,𝑡∈𝐼,(3.4) where 𝐵1(𝑢)=(𝐼−Φ(𝜔))−1𝜔0(𝜔−𝑠)𝛼−1[]Ψ(𝜔−𝑠)𝑓(𝑠,𝑢(𝑠))+𝐶𝑢(𝑠)𝑑𝑠.(3.5) By the continuity of 𝑓 and Lemma 2.16 (ii), 𝑄∶𝐷→𝐶(𝐼,𝑋) is continuous. By Lemma 2.8, 𝑢∈𝐷 is a mild solution of PBVP (1.1) if and only if 𝑢=𝑄𝑢.(3.6) For 𝑢1,𝑢2∈𝐷 and 𝑢1≤𝑢2, from (H), the positivity of operators (𝐼−Φ(𝜔))−1, Φ(𝑡), and Ψ(𝑡), we have that 𝑄𝑢1≤𝑄𝑢2.(3.7) Now, we show that 𝑣0≤𝑄𝑣0, 𝑄𝑤0≤𝑤0. Let 𝐷𝛼𝑣0(𝑡)+𝐴𝑣0(𝑡)+𝐶𝑣0(𝑡)â‰œğœŽ(𝑡), by Definition 2.5, the positivity of operator Ψ(𝑡), we have that 𝑣0(𝑡)=Φ(𝑡)𝑣0(0)+𝑡0(𝑡−𝑠)𝛼−1Ψ(𝑡−𝑠)ğœŽ(𝑠)𝑑𝑠≤Φ(𝑡)𝑣0(0)+𝑡0(𝑡−𝑠)𝛼−1𝑓Ψ(𝑡−𝑠)𝑠,𝑣0(𝑠)+𝐶𝑣0(𝑠)𝑑𝑠,𝑡∈𝐼.(3.8) In particular, 𝑣0(𝜔)≤Φ(𝜔)𝑣0(0)+𝜔0(𝜔−𝑠)𝛼−1𝑓Ψ(𝜔−𝑠)𝑠,𝑣0(𝑠)+𝐶𝑣0(𝑠)𝑑𝑠.(3.9) By Definition 2.5, 𝑣0(0)≤𝑣(𝜔), and by the positivity of operator (𝐼−Φ(𝜔))−1, we have that 𝑣0(0)≤(𝐼−Φ(𝜔))−1𝜔0(𝜔−𝑠)𝛼−1𝑓Ψ(𝜔−𝑠)𝑠,𝑣0(𝑠)+𝐶𝑣0(𝑠)𝑑𝑠=𝐵1𝑣0.(3.10) Then by (3.8) and the positivity of operator Φ(𝑡), 𝑣0(𝑡)≤Φ(𝑡)𝐵1𝑣0+𝑡0(𝑡−𝑠)𝛼−1𝑓Ψ(𝑡−𝑠)𝑠,𝑣0(𝑠)+𝐶𝑣0(=𝑠)𝑑𝑠𝑄𝑣0(𝑡),𝑡∈𝐼,(3.11) namely, 𝑣0≤𝑄𝑣0. Similarly, we can show that 𝑄𝑤0≤𝑤0. For 𝑢∈𝐷, in view of (3.7), then 𝑣0≤𝑄𝑣0≤𝑄𝑢≤𝑄𝑤0≤𝑤0. Thus, 𝑄∶𝐷→𝐷 is a continuous increasing monotonic operator. We can now define the sequences 𝑣𝑛=𝑄𝑣𝑛−1,𝑤𝑛=𝑄𝑤𝑛−1,𝑛=1,2,…,(3.12) and it follows from (3.7) that 𝑣0≤𝑣1≤⋯𝑣𝑛≤⋯≤𝑤𝑛≤⋯≤𝑤1≤𝑤0.(3.13)
In the following, we prove that {𝑣𝑛} and {𝑤𝑛} are convergent in 𝐶(𝐼,𝑋). First, we show that 𝑄𝐷={𝑄𝑢∣𝑢∈𝐷} is precompact in 𝐶(𝐼,𝑋). Let (𝑊𝑢)(𝑡)=𝑡0(𝑡−𝑠)𝛼−1[]Ψ(𝑡−𝑠)𝑓(𝑠,𝑢(𝑠))+𝐶𝑢(𝑠)𝑑𝑠,𝑡∈𝐼,(3.14)then we prove that for all 0<𝑡≤𝜔, (𝑊𝐷)(𝑡)={(𝑊𝑢)(𝑡)∣𝑢∈𝐷} is precompact in 𝑋. For 0<𝜀<𝑡, let 𝑊𝜀𝑢(𝑡)=0𝑡−𝜀(𝑡−𝑠)𝛼−1[]=Ψ(𝑡−𝑠)𝑓(𝑠,𝑢(𝑠))+𝐶𝑢(𝑠)𝑑𝑠0𝑡−𝜀(𝑡−𝑠)𝛼−1î‚µğ›¼î€œâˆž0𝜃𝜁𝛼(𝜃)𝑆((𝑡−𝑠)𝛼[]𝜃)𝑑𝜃𝑓(𝑠,𝑢(𝑠))+𝐶𝑢(𝑠)𝑑𝑠=𝑆(𝜀)0𝑡−𝜀(𝑡−𝑠)𝛼−1î‚µğ›¼î€œâˆž0𝜃𝜁𝛼(𝜃)𝑆((𝑡−𝑠)𝛼[]𝜃−𝜀)𝑑𝜃𝑓(𝑠,𝑢(𝑠))+𝐶𝑢(𝑠)𝑑𝑠.(3.15) For 𝑢∈𝐷, by (H), 𝑓(𝑡,𝑣0(𝑡))+𝐶𝑣0(𝑡)≤𝑓(𝑡,𝑢(𝑡))+𝐶𝑢(𝑡)≤𝑓(𝑡,𝑤0(𝑡))+𝐶𝑤0(𝑡) for 0≤𝑡≤𝜔. By the normality of the cone 𝑃, there is 𝑀1>0 such that ‖𝑓(𝑡,𝑢(𝑡))+𝐶𝑢(𝑡)‖≤𝑀1,0≤𝑡≤𝜔.(3.16) Thus, by (3.16) and Remark 2.7 (i), we have ‖‖‖0𝑡−𝜀(𝑡−𝑠)𝛼−1î‚µğ›¼î€œâˆž0𝜃𝜁𝛼(𝜃)𝑆((𝑡−𝑠)𝛼[]‖‖‖𝜃−𝜀)𝑑𝜃𝑓(𝑠,𝑢(𝑠))+𝐶𝑢(𝑠)𝑑𝑠≤𝑀10𝑡−𝜀(𝑡−𝑠)𝛼−1î‚µğ›¼î€œâˆž0𝜃𝜁𝛼(𝜃)‖𝑆((𝑡−𝑠)𝛼𝜃−𝜀)‖𝑑𝜃𝑑𝑠≤𝑀𝑀10𝑡−𝜀(𝑡−𝑠)𝛼−1î‚µğ›¼î€œâˆž0𝜃𝜁𝛼(𝜃)𝑑𝜃𝑑𝑠=𝑀𝑀1𝛼Γ(1+𝛼)0𝑡−𝜀(𝑡−𝑠)𝛼−1𝑑𝑠=𝑀𝑀1(𝑡𝛼−𝜀𝛼)Γ(1+𝛼),0<𝑡≤𝜔.(3.17) Then by (3.15), (3.17) and the compactness of 𝑆(𝜀), for 0<𝑡≤𝜔, (𝑊𝜀𝐷)(𝑡)={(𝑊𝜀𝑢)(𝑡)∣𝑢∈𝐷} is precompact in 𝑋. Furthermore, by (3.16) and Lemma 2.16 (i), we have ‖‖(𝑊𝑊𝑢)(𝑡)−𝜀𝑢(‖‖=‖‖‖𝑡)𝑡𝑡−𝜀(𝑡−𝑠)𝛼−1[]‖‖‖Ψ(𝑡−𝑠)𝑓(𝑠,𝑢(𝑠))+𝐶𝑢(𝑠)𝑑𝑠≤𝑀𝑀1𝛼Γ(1+𝛼)𝑡𝑡−𝜀(𝑡−𝑠)𝛼−1𝑑𝑠=𝑀𝑀1𝜀𝛼Γ.(1+𝛼)(3.18) Therefore, for 0<𝑡≤𝜔, (𝑊𝐷)(𝑡) is precompact in 𝑋. In particular, (𝑊𝐷)(𝜔) is precompact in 𝑋, and then 𝐵1(𝐷)=(𝐼−Φ(𝜔))−1(𝑊𝐷)(𝜔) is precompact. Then in view of Lemma 2.16 (i), (𝑄𝐷)(𝑡)={(𝑄𝑢(𝑡))∣𝑢∈𝐷}=Φ(𝑡)𝐵1(𝐷)+(𝑊𝐷)(𝑡) is precompact in 𝑋 for 0≤𝑡≤𝜔.
Furthermore, for 0≤𝑡1<𝑡2≤𝜔, by (3.16) and Lemma 2.16 (i) we have that ‖‖(𝑡𝑊𝑢)2𝑡−(𝑊𝑢)1‖‖=‖‖‖𝑡20𝑡2−𝑠𝛼−1Ψ𝑡2[]−−𝑠𝑓(𝑠,𝑢(𝑠))+𝐶𝑢(𝑠)𝑑𝑠𝑡10𝑡1−𝑠𝛼−1Ψ𝑡1[]‖‖‖−𝑠𝑓(𝑠,𝑢(𝑠))+𝐶𝑢(𝑠)𝑑𝑠≤𝑀1𝑡10‖‖𝑡2−𝑠𝛼−1Ψ𝑡2−𝑡−𝑠1−𝑠𝛼−1Ψ𝑡1‖‖−𝑠𝑑𝑠+𝑀𝑀1𝛼Γ(1+𝛼)𝑡2𝑡1𝑡2−𝑠𝛼−1𝑑𝑠≤𝑀1𝑡10𝑡2−𝑠𝛼−1‖‖Ψ𝑡2𝑡−𝑠−Ψ1‖‖−𝑠𝑑𝑠+𝑀1𝑡10‖‖𝑡2−𝑠𝛼−1−𝑡1−𝑠𝛼−1Ψ𝑡1‖‖−𝑠𝑑𝑠+𝑀𝑀1𝑡Γ(1+𝛼)2−𝑡1𝛼≤𝑀1𝑡2−𝑡1𝛼−1𝑡10‖‖Ψ𝑡2𝑡−𝑠−Ψ1‖‖+−𝑠𝑑𝑠𝑀𝑀1||𝑡Γ(1+𝛼)𝛼1+𝑡2−𝑡1𝛼−𝑡𝛼2||+𝑀𝑀1𝑡Γ(1+𝛼)2−𝑡1𝛼≤𝑀1𝑡2−𝑡1𝛼−1𝑡10‖‖Ψ𝑡2𝑡−𝑠−Ψ1‖‖−𝑠𝑑𝑠+2𝑀𝑀1𝑡Γ(1+𝛼)2−𝑡1𝛼+𝑀𝑀1𝑡Γ(1+𝛼)𝛼2−𝑡𝛼1.(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)}𝑡≥0 is strongly continuous. So, 𝑄𝐷 is equicontinuous in 𝐶(𝐼,𝑋).
Then by Ascoli-Arzela’s theorem, 𝑄𝐷={𝑄𝑢∣𝑢∈𝐷} 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=limğ‘›â†’âˆžğ‘£ğ‘›,𝑢=limğ‘›â†’âˆžğ‘¤ğ‘›.(3.20) Let ğ‘›â†’âˆž, by the continuity of 𝑄 and (3.12), we have 𝑢=𝑄𝑢,𝑢=𝑄𝑢.(3.21) By (3.7), if 𝑢∈𝐷 is a fixed-point of 𝑄, then 𝑣1=𝑄𝑣0≤𝑄𝑢=𝑢≤𝑄𝑤0=𝑤1. By induction, 𝑣𝑛≤𝑢≤𝑤𝑛. By (3.13) and taking the limit as ğ‘›â†’âˆž, we conclude that 𝑣0≤𝑢≤𝑢≤𝑢≤𝑤0. 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)}𝑡≥0 is a compact and positive semigroup in 𝑋, 𝑓(𝑡,𝜃)≥𝜃 for 𝑡∈𝐼. If there is 𝑦∈𝑋 such that 𝑦≥𝜃, 𝐴𝑦≥𝑓(𝑡,𝑦) for 𝑡∈𝐼, and 𝑓 satisfies the following: (H1)There exists a constant 𝐶1>0 such that 𝑓𝑡,𝑥2−𝑓𝑡,𝑥1≥−𝐶1𝑥2−𝑥1,(3.22) for any 𝑡∈𝐼, and 𝜃≤𝑥1≤𝑥2≤𝑦, that is, 𝑓(𝑡,𝑥)+𝐶1𝑥 is increasing in 𝑥 for 𝑥∈[𝜃,𝑦].
Then PBVP (1.1) has a positive mild solution 𝑢: 𝜃≤𝑢≤𝑦.

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

Case 2. {𝑇(𝑡)}𝑡≥0 is noncompact.

Theorem 3.3. Assume that the positive cone 𝑃 is regular, {𝑇(𝑡)}𝑡≥0 is an equicontinuous and positive semigroup in 𝑋, PBVP (1.1) has a lower solution 𝑣0 and an upper solution 𝑤0 with 𝑣0≤𝑤0, 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)}𝑡≥0 is strongly continuous. So, 𝑄𝐷 is equicontinuous in 𝐶(𝐼,𝑋). Thus, {𝑄𝑣𝑛} is equicontinuous in 𝐶(𝐼,𝑋).
For 0≤𝑡≤𝜔, 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=limğ‘›â†’âˆžğ‘£ğ‘›=limğ‘›â†’âˆžğ‘„ğ‘£ğ‘›âˆ’1,𝑢=limğ‘›â†’âˆžğ‘¤ğ‘›=limğ‘›â†’âˆžğ‘„ğ‘¤ğ‘›âˆ’1,(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)}𝑡≥0 is an equicontinuous and positive semigroup in 𝑋, PBVP (1.1) has a lower solution 𝑣0 and an upper solution 𝑤0 with 𝑣0≤𝑤0, 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)}𝑡≥0 is an equicontinuous and positive semigroup in 𝑋, PBVP (1.1) has a lower solution 𝑣0 and an upper solution 𝑤0 with 𝑣0≤𝑤0, 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)}𝑡≥0 is an equicontinuous and positive semigroup in 𝑋, 𝑓(𝑡,𝜃)≥𝜃 for 𝑡∈𝐼. If there is 𝑦∈𝑋 such that 𝑦≥𝜃, 𝐴𝑦≥𝑓(𝑡,𝑦) for 𝑡∈𝐼, 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: 𝜃≤𝑢≤𝑦.

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),𝑥∈Ω,(4.1) where 𝜕𝛼𝑡 is the Caputo fractional partial derivative with order 0<𝛼<1, 𝐼=[0,𝜔], Ω⊂ℝ𝑁 is a bounded domain with a sufficiently smooth boundary 𝜕Ω, 𝑔∶Ω×𝐼×ℝ→ℝ is continuous, 𝐵𝑢=𝑏0(𝑥)𝑢+𝛿(𝜕𝑢/𝜕𝑛) is a regular boundary operator on 𝜕Ω, and 𝐴(𝑥,𝐷)𝑢=−𝑁𝑁𝑖=1𝑗=1ğœ•ğœ•ğ‘¥ğ‘–î‚µğ‘Žğ‘–ğ‘—(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)(𝑝≥2), 𝑃={𝑣∣𝑣∈𝐿𝑝(Ω),𝑣(𝑥)≥0a.e.𝑥∈Ω}, then 𝑋 is a Banach space, and 𝑃 is a regular cone in 𝑋. Define the operator 𝐴 as follows: 𝐷(𝐴)=𝑢∈𝑊2,𝑝(Ω)∣𝐵𝑢=0,𝐴𝑢=𝐴(𝑥,𝐷)𝑢.(4.3) Then −𝐴 generates a uniformly bounded analytic semigroup 𝑇(𝑡)(𝑡≥0) in 𝑋 (see [39]). By the maximum principle, we can easily find that 𝑇(𝑡)(𝑡≥0) 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)),𝑡∈𝐼,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)𝑤0≥𝑔𝑥,𝑡,𝑤0,𝑤(𝑥,𝑡)∈Ω×𝐼,𝐵𝑤=0,(𝑥,𝑡)∈𝜕Ω×𝐼,0(𝑥,0)≥𝑤0(𝑥,𝜔),𝑥∈Ω,(4.5) and 𝑔 satisfies the following:(H4)there exists a constant 𝐶2≥0 such that 𝑔𝑥,𝑡,𝜉2−𝑔𝑥,𝑡,𝜉1≥−𝐶2𝜉2−𝜉1,(4.6) for any 𝑡∈𝐼, and 0≤𝜉1≤𝜉2≤𝑤0
Then PBVP (4.1) has a mild solution 𝑢∶0≤𝑢≤𝑤0.

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.