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

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 𝑋𝐷^𝛼𝑢(𝑡)+𝐴𝑢𝑢((𝑡0))==𝑓𝑢((𝑡𝜔,𝑢),(𝑡)),𝑡∈𝐼,(1.1) where 𝐷^𝛼 is the Caputo fractional derivative of order 0<𝛼<1,𝐼=[0,𝜔],−𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is the infinitesimal generator of an analytic semigroup {𝑇(𝑡)}_𝑡≥0 of uniformly bounded linear operators on 𝑋, and 𝑓∶𝐼×𝑋→𝑋 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),𝑡≥0,𝑦∈𝑅,(1.2) we can take 𝐴=𝜕^𝛽1_𝑦, for 𝛽₁∈(0,1], or 𝐴=𝜕_𝑦+𝜕^𝛽2_𝑦 for 𝛽₂∈(1,2], where 𝜕^𝛼𝑡, 𝜕^𝛽1_𝑦, 𝜕^𝛽2_𝑦 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 [13–18]).

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)}_𝑡≥0. 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 [20–23].

Definition 2.1. The fractional integral of order 𝛼 with the lower limit zero for a function 𝑓 is defined as: 𝐼^𝛼𝑓(𝑡)=1Γ(𝛼)^𝑡0𝑓(𝑠)(𝑡−𝑠)^1−𝛼𝑑𝑠,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: ^𝐿𝐷^𝛼𝑓(𝑡)=1Γ(𝑛−𝛼)𝑑𝑑𝑡^𝑛𝑛^𝑡0𝑓(𝑠)(𝑡−𝑠)^{𝛼+1−𝑛}𝑑𝑠,𝑡>0,𝑛−1<𝛼<𝑛.(2.2)

Definition 2.3. The Caputo fractional derivative of order 𝛼 for a function 𝑓 can be written as: 𝐷^𝛼𝑓(𝑡)=^𝐿𝐷^𝛼𝑓(𝑡)−^𝑛−1_𝑘=0𝑡^𝑘𝑘!𝑓^(𝑘)(0),𝑡>0,𝑛−1<𝛼<𝑛.(2.3)

Remark 2.4. (i) If 𝑓∈𝐶^𝑛[0,∞), then 𝐷^𝛼𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡0𝑓^𝑛(𝑠)(𝑡−𝑠)^{𝛼+1−𝑛}𝑑𝑠,𝑡>0,𝑛−1<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 2.1–2.3 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={𝑦∈𝑋∣𝑦≥𝜃} (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 ‖𝑢‖_𝐶=max_𝑡∈𝐼‖𝑢(𝑡)‖. Evidently, 𝐶(𝐼,𝑋) is also an ordered Banach space with the partial ≤ reduced by the positive function cone 𝑃_𝐶={𝑢∈𝐶(𝐼,𝑋)∣𝑢(𝑡)≥𝜃,𝑡∈𝐼}. 𝑃_𝐶 is also normal with the same constant 𝑁. 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} in 𝐶(𝐼,𝑋), and [𝑣(𝑡),𝑤(𝑡)]={𝑦∈𝑋∣𝑣(𝑡)≤𝑦≤𝑤(𝑡)}(𝑡∈𝐼) in 𝑋. Set 𝐶^𝛼(𝐼,𝑋)={𝑢∈𝐶(𝐼,𝑋)∣𝐷^𝛼𝑢 exists and 𝐷^𝛼𝑢∈𝐶(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)}_𝑡≥0. This means there exists 𝑀≥1 such that‖𝑇(𝑡)‖≤𝑀,𝑡≥0.(2.5)

Lemma 2.5 (see [4]). 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, 𝐷^𝛼𝑢(𝑡)+𝑢(𝐴0𝑢)(=𝑡)𝑥=₀ℎ(𝑡),𝑡∈𝐼,∈𝑋,(2.6) is given by 𝑢(𝑡)=𝑈(𝑡)𝑥₀+^𝑡0(𝑡−𝑠)^𝛼−1𝑉(𝑡−𝑠)ℎ(𝑠)𝑑𝑠,(2.7) where 𝑈(𝑡)=^∞0𝜁_𝛼(𝜃)𝑇(𝑡^𝛼𝜃)𝑑𝜃,𝑉(𝑡)=𝛼^∞0𝜃𝜁_𝛼(𝜃)𝑇(𝑡^𝛼𝜃)𝑑𝜃,(2.8)𝜁_𝛼(𝜃) is a probability density function defined on (0,∞).

Remark 2.6. (i) See [6, 7] 𝜁_𝛼(𝜃)=1𝛼𝜃^{−1−1/𝛼}𝜌_𝛼𝜌_𝛼𝜃^−1/𝛼,(𝜃)=1𝜋^∞_𝑛=0(−1)^𝑛−1𝜃^{−𝛼𝑛−1}Γ(𝑛𝛼+1)𝑛!sin(𝑛𝜋𝛼),𝜃∈(0,∞),(2.9)
(ii) see [6, 24], 𝜁_𝛼(𝜃)≥0,𝜃∈(0,∞),∫^∞0𝜁_𝛼(𝜃)𝑑𝜃=1, ∫^∞0𝜃𝜁_𝛼(𝜃)𝑑𝜃=1/Γ(1+𝛼), ∫^∞0𝜃^𝑣𝜁_𝛼(𝜃)𝑑𝜃=Γ(1+𝑣)/Γ(1+𝛼𝑣) for 𝑣∈(−1,∞),
(iii) see [4, 5], the Laplace transform of 𝜁_𝛼 is given by ^∞0𝑒^−𝑝𝜃𝜁_𝛼(𝜃)𝑑𝜃=^∞_𝑛=0(−𝑝)^𝑛Γ(1+𝑛𝛼)=𝐸^𝛼(−𝑝),(2.10) where 𝐸_𝛼(⋅) is the Mittag-Leffler function (see [20]),
(iv) see [24] by (i) and (ii), we can obtain that for 𝑝≥0^∞0𝑒^−𝑝𝜃𝜃𝜁_𝛼(𝜃)𝑑𝜃=1𝛼^∞_𝑛=0(−𝑝)^𝑛Γ(𝛼(𝑛+1))=𝛼1𝐸^𝛼,𝛼(−𝑝),(2.11) where 𝐸_𝛼(⋅), 𝐸_𝛼,𝛼(⋅) are the Mittag-Leffler functions.
(v) see [25] for 𝑝<0,0<𝐸_𝛼(𝑝)<𝐸_𝛼(0)=1,
(vi) see [10] if 𝛿>0 and 𝑡>0, then −(1/𝛿)(𝐸_𝛼(−𝛿𝑡^𝛼))^=𝑡^𝛼−1𝐸_𝛼,𝛼(−𝛿𝑡^𝛼).

Remark 2.7. See [6, 8], the operators 𝑈 and 𝑉, given by (2.8), 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‖,(2.12)(ii){𝑈(𝑡)}_𝑡≥0 and {𝑉(𝑡)}_𝑡≥0 are strongly continuous.

Definition 2.8. If ℎ∈𝐶(𝐼,𝑋), by the mild solution of IVP (2.6), we mean that the function 𝑢∈𝐶(𝐼,𝑋) satisfying the integral (2.7).

We also introduce some basic theories of the operator semigroups. For an analytic semigroup {𝑅(𝑡)}_𝑡≥0, there exist 𝑀₁>0 and 𝛿∈ℝ such that (see [26])‖𝑅(𝑡)‖≤𝑀₁𝑒^𝛿𝑡,𝑡≥0.(2.13) Then𝜈₀=inf𝛿∈ℝ∣thereexist𝑀₁>0suchthat‖𝑅(𝑡)‖≤𝑀₁𝑒^𝛿𝑡,∀𝑡≥0(2.14) is called the growth index of the semigroup {𝑅(𝑡)}_𝑡≥0. Furthermore, 𝜈₀ can also be obtained by the following formula:𝜈₀=limsup_𝑡→+∞ln‖𝑅(𝑡)‖𝑡.(2.15)

Definition 2.9 (see [26]). A 𝐶₀-semigroup {𝑇(𝑡)}_𝑡≥0 is called a compact semigroup if 𝑇(𝑡) is compact for 𝑡>0.

Definition 2.10. An analytic semigroup {𝑇(𝑡)}_𝑡≥0 is called positive if 𝑇(𝑡)𝑥≥𝜃 for all 𝑥≥𝜃 and 𝑡≥0.

Remark 2.11. For the applications of positive operators semigroup, we can see [27–31].

Definition 2.12. A bounded linear operator 𝐾 on 𝑋 is called to be positive if 𝐾𝑥≥𝜃 for all 𝑥≥𝜃.

Remark 2.13. By Remark 2.6(ii), we obtain that 𝑈(𝑡) and 𝑉(𝑡) are positive for 𝑡≥0 if {𝑇(𝑡)}_𝑡≥0 is a positive semigroup.

Lemma 2.14. Let 𝑋 be an ordered Banach space, whose positive cone 𝑃 is normal. If {𝑇(𝑡)}_𝑡≥0 is an exponentially stable analytic semigroup, that is, 𝜈₀=limsup_𝑡→+∞(ln‖𝑇(𝑡)‖/𝑡)<0. Then the linear periodic boundary value problem (LPBVP), 𝐷^𝛼𝑢(𝑡)+𝑢𝐴(𝑢0)(𝑡=)=𝑢(ℎ𝜔()𝑡,),𝑡∈𝐼,(2.16) has a unique mild solution 𝑢(𝑡)∶=(𝑄ℎ)(𝑡)=𝑈(𝑡)𝐵(ℎ)+^𝑡0(𝑡−𝑠)^𝛼−1𝑉(𝑡−𝑠)ℎ(𝑠)𝑑𝑠,(2.17) where 𝑈(𝑡) and 𝑉(𝑡) are given by (2.8), 𝐵(ℎ)=(𝐼−𝑈(𝜔))⁻¹^𝜔0(𝜔−𝑠)^𝛼−1𝑉(𝜔−𝑠)ℎ(𝑠)𝑑𝑠,(2.18)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,𝑡≥0.(2.19) In 𝑋, give the equivalent norm |⋅| by |𝑥|=sup_𝑡≥0‖‖𝑒^𝜈𝑡𝑇(𝑡)𝑥‖‖,(2.20) then ‖𝑥‖≤|𝑥|≤𝑀₁‖𝑥‖. By |𝑇(𝑡)| we denote the norm of 𝑇(𝑡) in (𝑋,|⋅|), then for 𝑡≥0, ||𝑇(𝑡)𝑥||==s𝑒up_{𝑠≥0−𝜈𝑡}‖𝑒^𝜈𝑠𝑇(𝑠)𝑇(𝑡)𝑥‖=𝑒_−𝜈𝑡sup_𝑠≥0‖‖𝑒^{𝜈(𝑠+𝑡)}𝑇(𝑠+𝑡)𝑥‖‖≤𝑒_−𝜈𝑡sup_𝜂≥𝑡‖𝑒^𝜈𝜂𝑇(𝜂)𝑥‖|𝑥|.(2.21) Thus, |𝑇(𝑡)|≤𝑒^−𝜈𝑡. Then by Remark 2.6, ||𝑈(𝑡)||≤=||||^∞0𝜁_𝛼(𝜃)𝑇(𝑡^𝛼𝜃)𝑑𝜃_∞0||||=𝐸_𝛼𝜁_𝛼(𝜃)𝑒^{−𝜈𝑡𝛼𝜃}𝑑𝜃(−𝜈𝑡^𝛼)<1.(2.22) Therefore, 𝐼−𝑈(𝜔) has bounded inverse operator and (𝐼−𝑈(𝜔))⁻¹=^∞_𝑛=0||(𝐼−𝑈(𝜔))₋₁(𝑈(𝜔))^𝑛,(2.23)||≤11−𝐸_𝛼(−𝜈𝜔^𝛼).(2.24) Set 𝑥₀=(𝐼−𝑈(𝜔))⁻¹^𝜔0(𝜔−𝑠)^𝛼−1𝑉(𝜔−𝑠)ℎ(𝑠)𝑑𝑠,(2.25) then 𝑢(𝑡)=𝑈(𝑡)𝑥₀+^𝑡0(𝑡−𝑠)^𝛼−1𝑉(𝑡−𝑠)ℎ(𝑠)𝑑𝑠(2.26) is the unique mild solution of LIVP (2.6) satisfing 𝑢(0)=𝑢(𝜔). So set 𝐵(ℎ)=(𝐼−𝑈(𝜔))⁻¹^𝜔0(𝑄ℎ)(𝑡)=𝑈(𝑡)𝐵(ℎ)+_𝑡0(𝜔−𝑠)^𝛼−1𝑉(𝜔−𝑠)ℎ(𝑠)𝑑𝑠,(𝑡−𝑠)^𝛼−1𝑉(𝑡−𝑠)ℎ(𝑠)𝑑𝑠,(2.27) 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)||=≤𝛼||||𝛼^∞0_∞0𝜃𝜁_𝛼(𝜃)𝑇(𝑡^𝛼𝜃)𝑑𝜃||||=𝐸_𝛼,𝛼𝜃𝜁_𝛼(𝜃)𝑒^{−𝜈𝑡𝛼𝜃}𝑑𝜃(−𝜈𝑡^𝛼).(2.28) By (2.24), (2.28) and Remark 2.6, for 𝑡≥0 we have that ||(𝑄ℎ)(𝑡)||≤||||𝑈(𝑡)(𝐼−𝑈(𝜔))⁻¹^𝜔0+||||(𝜔−𝑠)^𝛼−1𝑉(𝜔−𝑠)ℎ(𝑠)𝑑𝑠_𝑡0||||≤𝐸(𝑡−𝑠)^𝛼−1𝑉(𝑡−𝑠)ℎ(𝑠)𝑑𝑠_𝛼||||(−𝜈𝑡^𝛼)1−𝐸_𝛼(−𝜈𝜔^𝛼)||||^𝜔0(𝜔−𝑠)^𝛼−1𝑉(𝜔−𝑠)𝑑𝑠||||||ℎ||_𝐶+||||^𝑡0(𝑡−𝑠)^𝛼−1𝑉(𝑡−𝑠)𝑑𝑠||||||ℎ||_𝐶=𝐸^𝛼(−𝜈𝑡^𝛼)1−𝐸_𝛼(−𝜈𝜔^𝛼)𝜈1𝐸^𝛼(−𝜈(𝜔−𝑠)^𝛼)||||^𝜔0+1𝜈𝐸^𝛼(−𝜈(𝑡−𝑠)^𝛼)|||^𝑡0||ℎ||^𝐶=||ℎ||^𝐶𝜈,(2.29) where |⋅|_𝐶=max_𝑡∈𝐼|⋅(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 [32].

Remark 2.16. If {𝑇(𝑡)}_𝑡≥0 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)}_𝑡≥0. 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{Re𝜆∣𝜆∈𝜎(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)}_𝑡≥0 is a compact and positive analytic semigroup. Then by the characteristic of positive semigroups (see [31]), for sufficiently large 𝜆₀>−inf{Re𝜆∣𝜆∈𝜎(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{Re𝜆∣𝜆∈𝜎(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)}_𝑡≥0 is a compact and positive analytic semigroup, and its first eigenvalue of 𝐴 is 𝜆₁=inf{Re𝜆∣𝜆∈𝜎(𝐴)}>0,(2.32) 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)}_𝑡≥0 is 𝜈₀=−𝜆₁<0, that is, {𝑇(𝑡)}_𝑡≥0 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 Results

Theorem 3.1. Let 𝑋  be an ordered Banach space, whose positive cone 𝑃 is normal with normal constant 𝑁. If {𝑇(𝑡)}_𝑡≥0 is a positive analytic semigroup, 𝑓(𝑡,𝜃)≥𝜃forall𝑡∈𝐼, and the following conditions are satisfied.(H₁) For any 𝑅>0, there exists 𝐶=𝐶(𝑅)>0 such that 𝑓𝑡,𝑥₂−𝑓𝑡,𝑥₁≥−𝐶𝑥₂−𝑥₁,(3.1) for any 𝑡∈𝐼, 𝜃≤𝑥₁≤𝑥₂, ‖𝑥₁‖, ‖𝑥₂‖≤𝑅.(H₂) There exists 𝐿<−𝜈₀ (𝜈₀ is the growth index of {𝑇(𝑡)}_𝑡≥0), such that 𝑓𝑡,𝑥₂−𝑓𝑡,𝑥₁≤𝐿𝑥₂−𝑥₁,(3.2) for any 𝑡∈𝐼, 𝜃≤𝑥₁≤𝑥₂. Then PBVP (1.1) has a unique positive mild solution.

Proof. Let ℎ₀(𝑡)=𝑓(𝑡,𝜃), then ℎ₀∈𝐶(𝐼,𝑋), ℎ₀≥𝜃. Consider LPBVP 𝐷^𝛼𝑢(𝑡)+(𝐴−𝑢𝐿(0𝐼))=𝑢(𝑢𝑡)(𝜔=).ℎ₀(𝑡),𝑡∈𝐼,(3.3)−(𝐴−𝐿𝐼) generates a positive analytic semigroup 𝑒^𝐿𝑡𝑇(𝑡), whose growth index is 𝐿+𝜈₀<0. By Lemma 2.14 and Remark 2.16, LPBVP (3.3) has a unique mild solution 𝑤₀∈𝐶(𝐼,𝑋) and 𝑤₀≥𝜃.
Set 𝑅₀=𝑁‖𝑤₀‖+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 𝐷^𝛼𝑢(𝑡)+(𝐴+𝑢(𝐶0)𝐼)=𝑢𝑢(𝑡()𝜔=).ℎ(𝑡),𝑡∈𝐼,(3.4)−(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 ℎ∈𝐶(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)=ℎ₀≥𝜃. By (H₁), 𝐹 is an increasing operator on [𝜃,𝑤₀]. Set 𝑣₀=𝜃, we can define the sequences 𝑣_𝑛=𝑄₁⋅𝐹𝑣_𝑛−1,𝑤_𝑛=𝑄₁⋅𝐹𝑤_𝑛−1,𝑛=1,2,….(3.5) By (3.4), we have that 𝑤₀=𝑄₁ℎ₀+𝐿𝑤₀+𝐶𝑤₀.(3.6) In (H₂), we set 𝑥₁=𝜃,𝑥₂=𝑤₀(𝑡), then 𝜃≤𝐹𝑓(𝜃)𝑡,≤𝑤𝐹₀≤𝑤₀ℎ₀(𝑡)+𝐿𝑤₀(𝑡),(3.7)≤ℎ₀+𝐿𝑤₀+𝐶𝑤₀.(3.8) By (3.6) and (3.8), the definition and the positivity of 𝑄₁, we have that 𝑄₁𝜃=𝜃=𝑣₀≤𝑣₁≤𝑤₁≤𝑤₀.(3.9) Since 𝑄₁⋅𝐹 is an increasing operator on [𝜃,𝑤₀], in view of (3.5), we have that 𝜃≤𝑣₁≤⋯≤𝑣_𝑛≤⋯≤𝑤_𝑛≤⋯≤𝑤₁≤𝑤₀.(3.10) Therefore, we obtain that 𝜃≤𝑤_𝑛−𝑣_𝑛==𝑄𝑄₁₁𝐹𝑤_𝑛−1−𝐹𝑣_𝑛−1≤(𝐶+𝑓𝐿)⋅,𝑄𝑤_𝑛−11−𝑓⋅,𝑣_𝑛−1+𝐶𝑤_𝑛−1−𝑣_𝑛−1𝑤_𝑛−1−𝑣_𝑛−1.(3.11) By induction, 𝜃≤𝑤_𝑛−𝑣_𝑛≤(𝐶+𝐿)^𝑛𝑄^𝑛1𝑤₀−𝑣₀=(𝐶+𝐿)^𝑛𝑄^𝑛1𝑤₀.(3.12) In view of the normality of the cone 𝑃, we have that ‖‖𝑤_𝑛−𝑣_𝑛‖‖_𝐶≤𝑁(𝐶+𝐿)^𝑛‖‖𝑄^𝑛1(𝑤₀)‖‖_𝐶≤𝑁(𝐶+𝐿)^𝑛‖‖𝑄^𝑛1‖‖_𝐶‖‖𝑤₀‖‖_𝐶.(3.13) 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, lim^{𝑛→∞𝑛}‖𝑄^𝑛1‖_𝐶=𝑟(𝑄₁)≤1/(𝐶−𝜈₀). Then there exist 𝑁₀, for 𝑛≥𝑁₀, we have that ‖𝑄^𝑛1‖_𝐶≤1/(𝐶+𝐿+𝜀)^𝑛. By (3.13), we have that ‖‖𝑤_𝑛−𝑣_𝑛‖‖_𝐶≤𝑁‖‖𝑤₀‖‖_𝐶𝐶+𝐿𝐶+𝐿+𝜀^𝑛⟶0,(𝑛⟶∞).(3.14) By (3.10) and (3.14), similarly to the nested interval method, we can prove that there exists a unique 𝑢^∗∈⋂^∞𝑛=1[𝑣_𝑛,𝑤_𝑛], such that lim_𝑛→∞𝑣_𝑛=lim_𝑛→∞𝑤_𝑛=𝑢^∗.(3.15) By the continuity of the operator 𝑄₁⋅𝐹 and (3.5), we have that 𝑢^∗=𝑄₁⋅𝐹𝑢^∗.(3.16) 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 𝑤_𝑛=𝑄₁⋅𝐹(𝑢_𝑖)=𝑢_𝑖 (𝑖=1,2). By (3.14), we have that ‖‖𝑢_𝑖−𝑣_𝑛‖‖_𝐶⟶0,(𝑛⟶∞,𝑖=1,2).(3.17) Thus, 𝑢₁=𝑢₂=lim_𝑛→∞𝑣_𝑛, 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)}_𝑡≥0 is a compact and positive analytic semigroup, 𝑓(𝑡,𝜃)≥𝜃 for forall𝑡∈𝐼, 𝑓 satisfies (H₁) and the following condition:(H₂)^ There exist 𝐿<𝜆₁, where 𝜆₁ is the first eigenvalue of 𝐴, such that 𝑓𝑡,𝑥₂−𝑓𝑡,𝑥₁≤𝐿𝑥₂−𝑥₁,(3.18) for any 𝑡∈𝐼,𝜃≤𝑥₁≤𝑥₂. 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<𝜆₁” in (H₂)^ cannot be extended to “ 𝐿<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:𝜕^𝛼𝑡𝑢−Δ𝑢𝑢(0=,𝑥𝑓)(=𝑡𝑢,𝑢𝑢∣((𝜕𝑡𝜔Ω,,𝑥𝑥)=)),,0,(𝑥𝑡,∈𝑥)Ω∈,𝐼×Ω,(4.1) where 𝜕^𝛼𝑡 is the Caputo fractional partial derivative of order 0<𝛼<1,𝐼=[0,𝜔],Ω⊂ℝ^𝑁 is a bounded domain with a sufficiently smooth boundary 𝜕Ω, Δ is the Laplace operator, 𝑓∶𝐼×ℝ→ℝ is continuous.

Let 𝑋=𝐿²(Ω), 𝑃={𝑣∣𝑣∈𝐿²(Ω),𝑣(𝑥)≥0a.e.𝑥∈Ω}. 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)}_𝑡≥0 which is compact, analytic, and uniformly bounded. By the maximum principle, we can find that {𝑇(𝑡)}_𝑡≥0 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 𝑡∈𝐼, the partial derivative 𝑓^𝑢(𝑡,𝑢) is continuous on any bounded domain and sup𝑓^𝑢(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.