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Abstract and Applied Analysis

Volume 2012 (2012), Article ID 903518, 15 pages

http://dx.doi.org/10.1155/2012/903518

## 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, [1–5] and references therein. Hence the study of such equations has become an object of extensive study during recent years, see [6–23] 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 [24–26]).

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 [16–20, 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, 11–13].

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.*(H_{1}) *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.

#### References

- L. Gaul, P. Klein, and S. Kemple, “Damping description involving fractional operators,”
*Mechanical Systems and Signal Processing*, vol. 5, no. 2, pp. 81–88, 1991. View at Google Scholar · View at Scopus - W. G. Glockle and T. F. Nonnenmacher, “A fractional calculus approach to self-similar protein dynamics,”
*Biophysical Journal*, vol. 68, no. 1, pp. 46–53, 1995. View at Google Scholar · View at Scopus - R. Metzler, W. Schick, H. G. Kilian, and T. F. Nonnenmacher, “Relaxation in filled polymers: a fractional calculus approach,”
*Journal of Chemical Physics*, vol. 103, no. 16, pp. 7180–7186, 1995. View at Google Scholar · View at Scopus - F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in
*Fractals and Fractional Calculus in Continuum Mechanics*, vol. 378 of*CISM Courses and Lectures*, pp. 291–348, Springer, Vienna, Austria, 1997. View at Google Scholar · View at Zentralblatt MATH - R. Hilfer,
*Applications of Fractional Calculus in Physics*, World Scientific, River Edge, NJ, USA, 2000. View at Publisher · View at Google Scholar - I. Podlubny,
*Fractional Differential Equations*, vol. 198 of*Mathematics in Science and Engineering*, Academic Press, San Diego, Calif, USA, 1999. - M. M. El-Borai, “Some probability densities and fundamental solutions of fractional evolution equations,”
*Chaos, Solitons and Fractals*, vol. 14, no. 3, pp. 433–440, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. M. El-Borai, “Semigroups and some nonlinear fractional differential equations,”
*Applied Mathematics and Computation*, vol. 149, no. 3, pp. 823–831, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - V. Lakshmikantham, S. Leela J, and J. Vasundhara Devi,
*Theory of Fractional Dynamic Systems*, Cambridge Scientific, 2009. - R. P. Agarwal, V. Lakshmikantham, and J. J. Nieto, “On the concept of solution for fractional differential equations with uncertainty,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 6, pp. 2859–2862, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Zhou and F. Jiao, “Existence of mild solutions for fractional neutral evolution equations,”
*Computers & Mathematics with Applications*, vol. 59, no. 3, pp. 1063–1077, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Wang and Y. Zhou, “A class of fractional evolution equations and optimal controls,”
*Nonlinear Analysis: Real World Applications*, vol. 12, no. 1, pp. 262–272, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R.-N. Wang, T.-J. Xiao, and J. Liang, “A note on the fractional Cauchy problems with nonlocal initial conditions,”
*Applied Mathematics Letters*, vol. 24, no. 8, pp. 1435–1442, 2011. View at Publisher · View at Google Scholar - H. Ye, J. Gao, and Y. Ding, “A generalized Gronwall inequality and its application to a fractional differential equation,”
*Journal of Mathematical Analysis and Applications*, vol. 328, no. 2, pp. 1075–1081, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. M. El-Borai, “Semigroups and some nonlinear fractional differential equations,”
*Applied Mathematics and Computation*, vol. 149, no. 3, pp. 823–831, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, vol. 204 of*North-Holland Mathematics Studies*, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. - A. Debbouche and D. Baleanu, “Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems,”
*Computers & Mathematics with Applications*, vol. 62, no. 3, pp. 1442–1450, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Ashyralyev, “A note on fractional derivatives and fractional powers of operators,”
*Journal of Mathematical Analysis and Applications*, vol. 357, no. 1, pp. 232–236, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - I. Podlubny,
*Fractional Differential Equations*, vol. 198 of*Mathematics in Science and Engineering*, Academic Press, San Diego, Calif, USA, 1999. - A. Babakhani and V. Daftardar-Gejji, “On calculus of local fractional derivatives,”
*Journal of Mathematical Analysis and Applications*, vol. 270, no. 1, pp. 66–79, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. De la Sen, “About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory,”
*Fixed Point Theory and Applications*, vol. 2011, Article ID 867932, 2011. View at Google Scholar · View at Zentralblatt MATH - M. De la Sen, “Positivity and stability of the solutions of Caputo fractional linear time-invariant systems of any order with internal point delays,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 161246, 25 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus*, vol. 3 of*Series on Complexity, Nonlinearity and Chaos*, World Scientific, Hackensack, NJ, USA, 2012. View at Publisher · View at Google Scholar - A. Pazy,
*Semigroups of Linear Operators and Applications to Partial Differential Equations*, vol. 44 of*Applied Mathematical Sciences*, Springer, New York, NY, USA, 1983. View at Publisher · View at Google Scholar - H. Amann, “Periodic solutions of semilinear parabolic equations,” in
*Nonlinear Analysis*, pp. 1–29, Academic Press, New York, NY, USA, 1978. View at Google Scholar · View at Zentralblatt MATH - P. E. Sobolevskii, “Equations of parabolic type in a Banach space,”
*American Mathematical Society Translations Series 2*, vol. 49, pp. 1–62, 1966. View at Google Scholar - H. Liu and J.-C. Chang, “Existence for a class of partial differential equations with nonlocal conditions,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 9, pp. 3076–3083, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. C. Travis and G. F. Webb, “Existence, stability, and compactness in the $\alpha $-norm for partial functional differential equations,”
*Transactions of the American Mathematical Society*, vol. 240, pp. 129–143, 1978. View at Google Scholar