Periodic Boundary Value Problems for Semilinear Fractional Differential Equations

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 ,  where is the Caputo fractional derivative of order , , is the infinitesimal generator of a -semigroup (i.e., strongly continuous semigroup) of uniformly bounded linear operators on , and 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 can turn to semilinear fractional differential equations in Banach spaces; they always generate an unbounded closed operator term , such as the time fractional diffusion equation of order , namely, where 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 . 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: 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 is the infinitesimal generator of a -semigroup in a Banach space, then generates a uniformly bounded -semigroup for large enough. This allows us to reduce the general case in which is the infinitesimal generator of a -semigroup to the case in which the semigroup is uniformly bounded. Hence, for convenience, throughout this paper, we suppose that is the infinitesimal generator of a uniformly bounded -semigroup . This means that there exists such that

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 with the lower limit zero for a function  is defined as provided the right side is pointwise defined on , where is the gamma function.

Definition 2.2 (see [9, 32]). The Riemann-Liouville derivative of order with the lower limit zero for a function can be written as

Definition 2.3 (see [9, 32]). The Caputo fractional derivative of order for a function can be written as

Remark 2.4 (see [32]). (i) If , then
(ii) The Caputo derivative of a constant is equal to zero.
(iii) If is an abstract function with values in , 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 be an ordered Banach space with norm and partial order ≤, whose positive cone ( is the zero element of ) is normal with normal constant . denotes the Banach space with the graph norm . Let be the Banach space of all continuous -value functions on interval with norm . For , if for all . For , denote the ordered interval and , . Set exists and .

Definition 2.5. If    and satisfies then 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  ,  by the mild solution of LIVP, one means that the function and satisfies where and is a probability density function defined on .

Remark 2.7. (i) [29–31] , , and .
(ii) [33, 34, 46, 47] The Laplace transform of is given by where is Mittag-Leffler function (see [42]).
(iii) [48] For , .

Lemma 2.8. If is an exponentially stable -semigroup, there are constants and , such that then the linear periodic boundary value problem (LPBVP) has a unique mild solution where and are given by (2.9)

Proof. In , give equivalent norm by then . By , we denote the norm of in , then for , Thus, . Then by Remark 2.7, Therefore, has bounded inverse operator and Set then is the unique mild solution of LIVP (2.7) and satisfies . So set then is the unique mild solution of LPBVP (2.13).

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

Definition 2.10. A -semigroup is called a compact semigroup if is compact for .

Definition 2.11. A -semigroup is called an equicontinuous semigroup if is continuous in the uniform operator topology (i.e., uniformly continuous) for .

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 -semigroup is called a positive semigroup if for all and .

Remark 2.14. From Definition 2.13, if , , and is a positive -semigroup generated by , the mild solution given by (2.8) satisfies . 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 on is called to be positive if for all .

Lemma 2.16. The operators    and given by (2.9) have the following properties: (i)For any fixed , and are linear and bounded operators, that is, for any , (ii) and are strongly continuous,(iii) and are compact operators if is a compact semigroup,(iv) and are continuous in the uniform operator topology (i.e., uniformly continuous) for if is an equicontinuous semigroup,(v) and are positive for if is a positive semigroup,(vi) is a positive operator if 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 , we have Since is continuous in the uniform operator topology for , by Lebesque-dominated convergence theorem and Remark 2.7 (i), and are continuous in the uniform operator topology for .(v) By Remark 2.7 (i), the proof is then complete.(vi) By , (2.18), and (2.19), the proof is then complete.

3. Main Results

Case 1. is compact.

Theorem 3.1. Assume that is a compact and positive semigroup in , PBVP (1.1) has a lower solution and an upper solution with and satisfies the following. (H)There exists a constant such that for any , and , that is, is increasing in for .
Then PBVP (1.1) has the minimal and maximal mild solutions between and , which can be obtained by a monotone iterative procedure starting from and , respectively.

Proof. It is easy to see that generates an exponentially stable and positive compact semigroup . By (2.1), . Let . By Remark 2.7 (i), we have that From Lemma 2.8, has bounded inverse operator and By Lemma 2.16 (v) and (vi), and are positive for , and is positive.
Let , then we define a mapping by where By the continuity of and Lemma 2.16 (ii), is continuous. By Lemma 2.8, is a mild solution of PBVP (1.1) if and only if For and , from (H), the positivity of operators , , and , we have that Now, we show that , . Let , by Definition 2.5, the positivity of operator , we have that In particular, By Definition 2.5, , and by the positivity of operator , we have that Then by (3.8) and the positivity of operator , namely, . Similarly, we can show that . For , in view of (3.7), then . Thus, is a continuous increasing monotonic operator. We can now define the sequences and it follows from (3.7) that
In the following, we prove that and are convergent in . First, we show that is precompact in . Let then we prove that for all , is precompact in . For , let For , by (H), for . By the normality of the cone , there is such that Thus, by (3.16) and Remark 2.7 (i), we have Then by (3.15), (3.17) and the compactness of , for , is precompact in . Furthermore, by (3.16) and Lemma 2.16 (i), we have Therefore, for , is precompact in . In particular, is precompact in , and then is precompact. Then in view of Lemma 2.16 (i), is precompact in for .
Furthermore, for , by (3.16) and Lemma 2.16 (i) we have that By Remark 2.12 and Lemma 2.16 (iv), is continuous in the uniform operator topology for . Then by Lebesque-dominated convergence theorem, is equicontinuous in . By Lemma 2.16 (ii), is strongly continuous. So, is equicontinuous in .
Then by Ascoli-Arzela’s theorem, is precompact in . By (3.12) and (3.13), has a convergent subsequence in . Combining this with the monotonicity of , it is itself convergent in . Using a similar argument to that for , we can prove that is also convergent in . Set Let , by the continuity of and (3.12), we have By (3.7), if is a fixed-point of , then . By induction, . By (3.13) and taking the limit as , we conclude that . This means that are the minimal and maximal fixed-points of on , respectively. By (3.6), they are the minimal and maximal mild solutions of PBVP (1.1) on , respectively.

Theorem 3.2. Assume that is a compact and positive semigroup in , for . If there is such that , for , and satisfies the following: (H₁)There exists a constant such that for any , and , that is, is increasing in for .
Then PBVP (1.1) has a positive mild solution : .

Proof. Let and , by Theorem 3.1, PBVP (1.1) has mild solution on .

Case 2. is noncompact.

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

Proof. By the proof of Theorem 3.1, (3.2)–(3.13) and (3.19) are valid. By Lemma 2.16 (iv), is continuous in the uniform operator topology for . Then by Lebesque-dominated convergence theorem, is equicontinuous in . From Lemma 2.16 (ii), is strongly continuous. So, is equicontinuous in . Thus, is equicontinuous in .
For , by (3.7) and (3.13), is monotone in . Since the cone is regular, then is convergent in .
By Ascoli-Arzela’s theorem, is precompact in and has a convergent subsequence in . Combining this with the monotonicity of , it is itself convergent in . Using a similar argument to that for , we can prove that is also convergent in . Let then it is similar to the proof of Theorem 3.1 that and are the minimal and maximal mild solutions of PBVP (1.1) on , respectively.

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

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

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

Proof. In an ordered and reflective Banach space, the normal cone 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 is an equicontinuous and positive semigroup in , for . If there is such that , for , satisfies (H₁) and one of the following conditions: (i) is an ordered Banach space, whose positive cone is regular,(ii) is an ordered and weakly sequentially complete Banach space,(iii) is an ordered and reflective Banach space.
then PBVP (1.1) has positive mild solution : .

4. Examples

Example 4.1. Consider the following periodic boundary value problem for fractional parabolic partial differential equations in :  where is the Caputo fractional partial derivative with order , , is a bounded domain with a sufficiently smooth boundary , is continuous, is a regular boundary operator on , and is a symmetrical strong elliptic operator of second order, whose coefficient functions are Hölder continuous in .
Let , , then is a Banach space, and is a regular cone in . Define the operator as follows: Then generates a uniformly bounded analytic semigroup in (see [39]). By the maximum principle, we can easily find that is positive (see [39]). Let , , then the problem (4.1) can be transformed into the following problem: