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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 781404, 8 pages
http://dx.doi.org/10.1155/2013/781404
Research Article

Positive Solutions for the Initial Value Problems of Fractional Evolution Equation

Science College of Gansu Agricultural University, Lanzhou 730070, China

Received 27 July 2013; Accepted 22 September 2013

Academic Editor: S. A. Mohiuddine

Copyright © 2013 Yue Liang et al. 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

This paper discusses the existence of positive solutions for the initial value problem of fractional evolution equation with noncompact semigroup , ; in a Banach space , where denotes the Caputo fractional derivative of order , is a closed linear operator, generates an equicontinuous semigroup, and is continuous. In the case where satisfies a weaker measure of noncompactness condition and a weaker boundedness condition, the existence results of positive and saturated mild solutions are obtained. Particularly, an existence result without using measure of noncompactness condition is presented in ordered and weakly sequentially complete Banach spaces. These results are very convenient for application. As an example, we study the partial differential equation of parabolic type of fractional order.

1. Introduction

The theory of fractional differential equations is a new and important branch of differential equation theory, which has an extensive physical background and realistic mathematical model; see [16]. Correspondingly, the existence of solutions to fractional evolution equations in Banach space has also been studied by several authors; see [717]. In [7, 8], El-Borai first constructed the type of mild solutions to fractional evolution equations in terms of a probability density. And then they investigated the existence, uniqueness, and regularity of solutions to fractional integrodifferential equations in [9, 10]. Recently, this theory was developed by Zhou et al. [1114]. In [1517], the authors studied the existence of mild solutions to fractional impulsive evolutions equations. But as far as we know, there are seldom results on the existence of positive solutions to the fractional evolution equations; see [1820].

In this paper, we use the Sadovskii’s fixed point theorem and monotone iterative technique to discuss the existence of positive and saturated mild solutions for the initial value problem (IVP) of fractional evolution equations: in Banach space , where denotes the Caputo fractional derivative of order , is a closed linear operator, generates a -semigroup () in , and is continuous and will be specified later, .

In some existing articles, the fractional evolution equations were treated under the hypothesis that (I) generates a compact semigroup or (II) the nonlinearity is Lipschitz continuous in on a bounded set. For the case (I), the continuity of nonlinearity can guarantee the local existence of solutions. Hence it is convenient to apply to partial differential equations with compact resolvent. But for the case of noncompact semigroup, the condition (II) is not easy to verify sometimes. To make the things more applicable, in this work, we will prove the existence of mild solutions of the IVP(1) under the measure of noncompactness conditions. We will see that our conditions are weaker than the condition (II). In addition, we obtain the existence of positive mild solutions of the IVP(1) in this work, which is studied seldom before.

The rest of this paper is organized as follows. In Section 2, some preliminaries are given on the fractional calculus and the measure of noncompactness. In Section 3, we study the existence of positive and saturated mild solutions of the IVP(1). An example is given in Section 4 to illustrate the applicability of the abstract results obtained in Section 3.

2. Preliminaries

In this section, we introduce some basic facts about the fractional calculus and the measure of noncompactness that are used throughout this paper.

Let be a Banach space with norm , let be a closed linear operator, and generates a -semigroup () in . It is well known that there exist and such that Let be a constant. If , it follows from (2) that there exists a constant such that .

Let us recall the following known definitions in fractional calculus. For more details, see [7, 8, 1114, 16, 17] and the reference therein.

Definition 1. The fractional integral of order with the lower limits zero for a function is defined by where is the gamma function.
The Riemann-Liouville fractional derivative of order with the lower limits zero for a function can be written as Also the Caputo fractional derivative of order with the lower limits zero for a function can be written as

Remark 2. (1) The Caputo derivative of a constant is equal to zero.
(2) If is an abstract function with values in , then integrals which appear in Definition 1 are taken in Bochner’s sense.

Lemma 3 (see [12]). A measurable function is Bochner integrable if is Lebesgue integrable.

For , we define two families and of operators by where where is a probability density function defined on , which has properties for all and . Clearly, if the semigroup () is positive, then the operators and are also positive for all .

The following lemma is needed in the proof of the main results.

Lemma 4. The operators and have the following properties.(i)For any fixed and any , one has (ii)The operators and are strongly continuous for all . (iii)If () is a equicontinuous semigroup, then and are equicontinuous in for .

Proof. (i) and (ii) can be found in [12, 13], and we only need to prove (iii). For any , we have According to the equicontinuity of for , we see that and tend to zero as , which means that the operators and are equicontinuous in for .

Let be a constant and . We denote by the Banach space of all continuous value functions on interval with the norm . Let denote the Kuratowski measure of noncompactness of the bounded set in and . It is clear that . If , then the set is relatively compact. For more details of the definition and properties of the measure of noncompactness; see [21]. For any and , set . If is bounded in , then is bounded in , and . A mapping is said to be condensing if . The following Lemmas will be used in the proof of the main results.

Lemma 5 (see [22]). Let be bounded and equicontinuous. Then is continuous on and where .

Lemma 6 (see [23]). Let be countable. If there exists such that a.e. , , then is Lebesgue integral on and

Lemma 7 (see [24]). Let be bounded. Then there exists a countable subset of such that .

Lemma 8 (see [25] (Sadovskii’s fixed point theorem)). Let be a Banach space and let be a nonempty bounded convex closed set in . If is a condensing mapping, then has a fixed point in .

In the proof of the main results, we also need the following generalized Gronwall-Bellman inequality, which can be found in [26, Page 188].

Lemma 9. Suppose , , and is a nonnegative function locally integrable on (some ), and suppose is nonnegative and locally integrable on with on this interval, and then

Remark 10. In Lemma 9, if for all , we easily see that .

For any and , a function is called the mild solution of the initial value problem if satisfies the integral equation: Hence, for the IVP(1), we have the following definition.

Definition 11. By a mild solution of the IVP(1), we mean a function satisfying for all .

3. Existence of Positive Mild Solutions

In this section, we introduce the existence theorems of positive mild solutions of the IVP(1). The discussions are based on fractional calculus and fixed point theorems.

Let be the smallest positive real eigenvalue of the linear operator , and let be the positive eigenvector corresponding to . Our main results are as follows.

Theorem 12. Let be a Banach space, let be a closed linear operator, and generate a positive and equicontinuous -semigroup () in . Assume that and, for any , satisfies the following conditions.() There exist , such that () For any with , , we have where is a constant.() For any bounded set , there exists a constant such that
If and with , then the IVP(1) has at least one positive and saturated mild solution . And if , one has .

Proof. For any and with , we first prove that there exists a constant such that the initial value problem (IVP) has at least one positive mild solution on . For this purpose, we define an operator by Then is continuous, and the mild solutions of the IVP(20) are equivalent to the fixed point of the operator .
Let . Denote Then is a nonempty bounded convex closed set. Let , where , , . Then for any and , by Lemma 4(i), (), and (21), we have
Let . Then for any and By the positivity of semigroup (), the assumption () and (21), for any and , we have Thus, is continuous and it implies that for any and .
Now, we prove that the set is equicontinuous in . For any and , it follows from assumption () and (21) that By Lemma 4, it is easy to see that independently of as :
Hence independently of as : It follows that independently of as . For , , it is easy to see that . Let and be small enough, and we have Since Lemma 4 implies the continuity of for in the uniform operator topology, it is easy to see that independently of as and . Thus, independently of as , which means that the set is equicontinuous.
It remains to prove that is a condensing mapping. Let be a bounded set. By Lemma 7, there exists such that . Since is bounded and equicontinuous, by Lemma 5, it follows that . Thus, for any , by (21), one has Thus, , which means that is a condensing mapping. By Lemma 8, the operator has at least one fixed point in , and for all . Hence is a positive mild solution of the IVP(20).
Hence, for the IVP(1), there exists an interval such that the IVP(1) has a positive mild solution on . Now, by the extension theorem of initial value problem, can be extended to a saturated solution of the IVP(1), whose existence interval is , and if , one has .

For any and , define a set by If is increasing in , that is, satisfies the condition:() for any with , we have then we have for any and . Hence by Theorem 12, we have the following existence result.

Corollary 13. Let be a Banach space, let be a closed linear operator, and generates a positive and equicontinuous -semigroup () in . Assume that and, for any , satisfies the conditions (), (), and (). If and with , then the IVP(1) has at least one positive and saturated mild solution . And if , one has .

Noticing that the condition () is not easy to verify in applications, we can weaken or delete the condition () in ordered Banach space.

Theorem 14. Let be an ordered Banach space, whose positive cone is normal, let be a closed linear operator, and generates a positive and equicontinuous -semigroup () in . Assume that and for any , satisfies the conditions (), (), and() there exists a constant such that for any increasing sequence .
If and with , then the IVP(1) has at least one positive and saturated mild solution . And if , one has .

Proof. For any and with , we first prove that the IVP(20) has at least one positive mild solution on , where . Define an operator as in (21). Let . Write as in (22). A similar argument as in the proof of Theorem 12 shows that is continuous and the set is equicontinuous. From the assumption (), it is easy to see that is an increasing operator.
Let . Define a sequence by the iterative scheme Since , by the increasing property of the operator , we have By the equicontinuity property of the set , the set is equicontinuous. Next, we prove that the set is uniformly convergent on .
For convenience, let and . From , it follows that for any . Let . By Lemma 6, assumption (), and (21), we have Hence by Lemma 9, for any . By Lemma 5, , from which we obtain that the set is relatively compact. Thus, there is a subset such that . Combining this with the monotonicity (35), we easily prove that itself is convergent in , that is, as .
Letting in (34), by the continuity of the operator , we have and for all . Hence is a positive mild solution of the IVP(20).
Hence, for the IVP(1), there exists an interval such that the IVP(1) has a positive mild solution on . By the extension theorem of the initial value problem, can be extended to a saturated solution of the IVP(1), whose existence interval is , and if , then .

In Theorem 14, if is weakly sequentially complete, the condition () holds automatically. In fact, by [27, Theorem 2.2], any monotonic and order-bounded sequence is precompact. Let be an increasing sequence. Then by the conditions () and (), is a monotonic increasing and order-bounded sequence. By the property of the measure of noncompactness, we have Thus, the condition () holds. From Theorem 14, we have the following.

Corollary 15. Let be an ordered and weakly sequentially complete Banach space, whose positive cone is normal, let be a closed linear operator, and generates a positive and equicontinuous -semigroup () in . Assume that and, for any , satisfies the conditions () and (). If and with , then the IVP(1) has at least one positive and saturated mild solution . And if , one has .

4. Positive Mild Solutions of Parabolic Equations

Let be a bounded domain with a sufficiently smooth boundary , . We consider the following problem of parabolic type: where is a constant, and is the Laplace operator. Let . Then is an ordered Banach space with the norm for any and the partial order “”. is the positive cone in . Consider the operator defined by Then generates a positive and analytic semigroup () in (see [28, 29]). Let be the smallest positive real eigenvalue of operator under the Dirichlet boundary condition and let be the positive eigenvector corresponding to . Then and for . For any and , denote by where is a constant. Assume that is continuous with , , and satisfies the following conditions.() There exist , such that () For any with , we have

Let be defined by . Then with for and satisfies the assumptions () and (). Therefore, by Corollary 15, we have the following existence result for the problem (38).

Theorem 16. Assume that with for and satisfies the assumptions and . If with for any , then the problem (38) has at least one positive mild solution , satisfies for any and . And if , one has .

Remark 17. In Theorem 16, we do not use the property of compactness of the semigroup (), which is a key assumption in [79, 11, 12].

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