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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 903518, 15 pages
http://dx.doi.org/10.1155/2012/903518
Research Article

Existence of Mild Solutions for a Class of Fractional Evolution Equations with Compact Analytic Semigroup

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received 2 May 2012; Revised 9 September 2012; Accepted 27 September 2012

Academic Editor: Dumitru Bǎleanu

Copyright © 2012 He Yang. 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 deals with the existence of mild solutions for a class of fractional evolution equations with compact analytic semigroup. We prove the existence of mild solutions, assuming that the nonlinear part satisfies some local growth conditions in fractional power spaces. An example is also given to illustrate the applicability of abstract results.

1. Introduction

The differential equations involving fractional derivatives in time have recently been proved to be valuable tools in the modeling of many phenomena in various fields of engineering, physics, economics, and science. Numerous applications can be found in electrochemistry, control, porous media, electromagnetic, see for example, [15] and references therein. Hence the study of such equations has become an object of extensive study during recent years, see [623] and references therein.

In this paper, we consider the existence of the following fractional evolution equation: where is the Caputo fractional derivative of order , is the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators, is the nonlinear term and will be specified later, and is a Volterra integral operator with integral kernel ,. Throughout this paper, we denote by

In some existing articles, the fractional differential equations were treated under the hypothesis that nonlinear term satisfies Lipschitz conditions or linear growth conditions. It is obvious that these conditions are not easy to be verified sometimes. To make the things more applicable, in this work, we will prove the existence of mild solutions for (1.1) under some new conditions. More precisely, the nonlinear term only satisfies some local growth conditions (see conditions and ). These conditions are much weaker than Lipschitz conditions and linear growth conditions. The main techniques used here are fractional calculus, theory of analytic semigroup, and Schauder fixed point theorem.

The rest of this paper is organized as follows. In Section 2, some preliminaries are given on the fractional power of the generator of a compact analytic semigroup and the definition of mild solutions of (1.1). In Section 3, we study the existence of mild solutions for (1.1). In Section 4, an example is given to illustrate the applicability of abstract results obtained in Section 3.

2. Preliminaries

In this section, we introduce some basic facts about the fractional power of the generator of a compact analytic semigroup and the fractional calculus that are used throughout this paper.

Let be a Banach space with norm . Throughout this paper, we assume that is the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operator in , that is, there exists such that for all . Without loss of generality, let , where is the resolvent set of . Then for any , we can define by

It follows that each is an injective continuous endomorphism of . Hence we can define by , which is a closed bijective linear operator in . It can be shown that each has dense domain and that for . Moreover, for every and with , where is the identity in . (For proofs of these facts, we refer to the literature [2426]).

We denote by the Banach space of equipped with norm for , which is equivalent to the graph norm of . Then we have for (with ), and the embedding is continuous. Moreover, has the following basic properties.

Lemma 2.1 (see [24]). has the following properties.(i) for each and . (ii) for each and . (iii)For every is bounded in and there exists such that (iv) is a bounded linear operator for in .

In the following, we denote by the Banach space of all continuous functions from into with supnorm given by for . From Lemma 2.1, since is a bounded linear operator for , there exists a constant such that for .

For any , denote by the restriction of to . From Lemma 2.1 and , for any , we have as . Therefore, is a strongly continuous semigroup in , and for all . To prove our main results, the following lemma is also needed.

Lemma 2.2 (see [27]). is an immediately compact semigroup in , and hence it is immediately norm-continuous.

Let us recall the following known definitions in fractional calculus. For more details, see [1620, 23].

Definition 2.3. 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 2.4. The Caputo derivative of a constant is equal to zero.
If is an abstract function with values in , then integrals which appear in Definition 2.3 are taken in Bochner's sense.

Lemma 2.5 (see [12]). A measurable function is Bochner integrable if is Lebesgue integrable.

For , we define two families and of operators by Where is a probability density function defined on , which has properties for all and The following lemma follows from the results in [7, 1113].

Lemma 2.6. The operators and have the following properties.(i)For fixed and any , we have (ii)The operators and are strongly continuous for all . (iii) and are norm-continuous in for .(iv) and are compact operators in for .(v)For every , the restriction of to and the restriction of to are norm-continuous.(vi)For every , the restriction of to and the restriction of to are compact operators in .

Based on an overall observation of the previous related literature, in this paper, we adopt the following definition of mild solution of (1.1).

Definition 2.7. By a mild solution of (1.1), we mean a function satisfying for all .

3. Existence of Mild Solutions

In this section, we give the existence theorems of mild solutions of (1.1). The discussions are based on fractional calculus and Schauder fixed point theorem. Our main results are as follows.

Theorem 3.1. Assume that the following condition on is satisfied.(H1) There exists a constant such that satisfies:(i) for each , the function is measurable;(ii) for each , the function is continuous;(iii) for any , there exists a function such that and there is a constant such that If and , then (1.1) has at least one mild solution.

Proof. Define an operator by It is not difficult to verify that . We will use Schauder fixed point theorem to prove that has fixed points in .
For any , let . We first show that there is a positive number such that . If this were not the case, then for each , there would exist and such that . Thus, from Lemma 2.6 and , we see that Dividing on both sides by and taking the lower limit as , we have which is a contradiction. Hence for some .
To complete the proof, we separate the rest of proof into the following three steps.
Step 1. is continuous.
Let with as . From the assumption , for each , we have as . Since , by the Lebesgue dominated convergence theorem, for each , we have as , which implies that is continuous.
Step 2. is relatively compact in for all .
It follows from (2.9) and (3.3) that is compact in . Hence it is only necessary to consider the case of . For each , and any , we define a set by where
Then the set is relatively compact in since by Lemma 2.2, the operator is compact in . For any and , from the following inequality: One can obtain that the set is relatively compact in for all . And since it is compact at , we have the relatively compactness of in for all .
Step 3. is equicontinuous.
For , by (3.3), we have Hence it is only necessary to consider the case of . For , by Lemma 2.1 and Lemma 2.6, we have From Lemma 2.6, we see that as independently of . From the expressions of and , it is clear that and as independently of . For any , we have
It follows from Lemma 2.6 that as and independently of . Therefore, we prove that is equicontinuous.
Thus, the Arzela-Ascoli theorem guarantees that is a compact operator. By the Schauder fixed point theorem, the operator has at least one fixed point in , which is a mild solution of (1.1). This completes the proof.

Remark 3.2. In assumption , if the function is independent of , then we can easily obtain a constant satisfying (3.2). For example, if there is a constant such that for all and , then for any , with , we have , where is independent of . Thus, is the constant in (3.2).
More generally, if satisfies the following condition:(H2) there is a constant such that satisfies:(i) for each , the function is measurable,(ii) for any , there exists a function such that for any with and , then we have the following existence and uniqueness theorem.

Theorem 3.3. Assume that the condition is satisfied. If and , then (1.1) has a unique mild solution.

Proof. For any , if with , then from , we have where . Therefore, the condition is satisfied with . By Theorem 3.1, (1.1) has at least one mild solution .
Let be the solutions of (1.1). We show that . Since and for all , we have By using the Gronwall-Bellman inequality (see [14, Theorem 1]), we can deduce that for all , which implies that . Hence (1.1) has a unique mild solution . This completes the proof.

Remark 3.4. In Theorem 3.3, we only assume that satisfies a local Lioschitz condition (see condition ), and an existence and uniqueness result is obtained. If , then the assumption deletes the linear growth condition of assumption in [12]. Therefore, the Theorem 3.3 extends and improves the main result in [12].

4. An Example

Assume that equipped with its natural norm and inner product defined, respectively, for all , by Consider the following fractional partial differential equation: where is a constant.

Let the operator be defined by It is well known that has a discrete spectrum with eigenvalues of the form , and corresponding normalized eigenfunctions given by . Moreover, generates a compact analytic semigroup in , and It is not difficult to verify that for all . Hence, we take .

The following results are also well known.(I)The operator can be written as for every .(II)The operator is given by for each and .

Lemma 4.1 (see [28]). If , then is absolutely continuous, and .

Let , where for all . Assume that satisfies the following conditions.(i)For each , the function is continuous.(ii)For each , the function is measurable.(iii)For each and , is differentiable, and .(iv).(v)There exist the functions such that for all .

Define . Then, for each , from assumptions and , we have

This implies from that . Moreover, for any , by Minkowski inequality, assumption and Lemma 4.1, we have

Therefore, satisfies the condition with . Thus, (4.2) has at least one mild solution provided that due to Theorem 3.1.

Assume furthermore that the function satisfies the following:(vi) for any , there exists a function such that for with and ,.

Then for each , by Lemma 4.1, we have This shows that satisfies the condition . Hence by Theorem 3.3, the mild solution of (4.2) is unique.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant no. 11261051), the Fundamental Research Funds for the Gansu Universities and the Project of NWNU-LKQN-11-3.

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