- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2013 (2013), Article ID 532589, 8 pages

http://dx.doi.org/10.1155/2013/532589

## Uniform Asymptotic Stability of Solutions of Fractional Functional Differential Equations

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

Received 30 September 2013; Accepted 15 October 2013

Academic Editor: Massimiliano Ferrara

Copyright © 2013 Yajing Li and Yejuan Wang. 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

Some global existence and uniform asymptotic stability results for fractional functional differential equations are proved. It is worth mentioning that when the initial value problem reduces to a classical dissipative differential equation with delays as in Caraballo et al.'s work (2005).

#### 1. Introduction

Consider the initial value problem (IVP for short) of the following fractional functional differential equation: where is the Caputo fractional derivative, , , where , is a given function satisfying some assumptions that will be specified later, , and . If , then for any , define by

The study of retarded differential equations is an important area of applied mathematics due to physical reasons, noninstant transmission phenomena, memory processes, and specially biological motivations (see, e.g., [1–4]). Fractional differential equations have attracted much attention recently (see, e.g., [5–11] and the references cited therein for the applications in various sciences such as physics, mechanics, chemistry, and engineering).

Some attractive results for fractional functional differential equations and nonlinear functional integral equations are obtained by using the fixed point theory; see [12–16] and references therein. Global asymptotic stability of solutions of a functional integral equation is discussed in [17]; however, there is no work on uniform asymptotic stability of solutions of fractional functional differential equation. It is our intention here to show the global existence and uniform asymptotic stability of the fractional functional differential equation (1).

We organize the paper as follows. In Section 2, we recall some necessary concepts and results. In Section 3 we give the global existence and uniform asymptotic stability of fractional functional differential equations. Finally, two examples are given to illustrate our main results.

#### 2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.

We consider the Banach space of all bounded and continuous functions from into with the norm Let for .

Throughout this paper, we always assume that satisfies the following condition:(*H*_{0}) is Lebesgue measurable with respect to on , and is continuous with respect to on . By condition and the technique used in [7], we get the equivalent form of IVP (1) as
where is the gamma function.

*Definition 1. *We say that the solutions and of IVP (1) are uniformly asymptotically stable if for any bounded subset of and , there exists a such that

We recall the following generalization of Gronwall’s lemma for singular kernels [18], which will be used in the sequel.

Lemma 2. *Let be a real function and is a nonnegative, locally integrable function on and there are constants and such that
**
Then there exists a constant such that
**
for every .*

Theorem 3 (Leray-Schauder fixed-point theorem). *Let be a continuous and compact mapping of a Banach space into itself, such that the set
**
is bounded. Then, has a fixed point.*

#### 3. FDEs of Fractional Order

In this section, we will investigate the IVP (1). Our first global existence and uniform asymptotic stability result for the IVP (1) is based on the Banach contradiction principle and Lemma 2.

Theorem 4. *Assume that satisfies conditions and**(H*_{1}*)**there exists such that
**for and every . Moreover, the function is bounded with .**If
**
then the IVP (1) has a unique solution in the space . Moreover, solutions of IVP (1) are uniformly asymptotically stable.*

*Proof. *We divide the proof into two steps.*Step **1.* We define the operator by
The operator maps into itself. Indeed for each , and for each , it follows from that
For each , we have
and consequently .

Since is a Banach space with norm , we will show that is a contraction map. Let . Then, we have for each ,

Therefore, for any ,
and for ,
and thus
Hence, (10) and (17) imply that the operator is a contraction. Therefore, has a unique fixed point by Banach’s contraction principle.*Step **2.* For any two solutions and of IVP (1) corresponding to initial values and , by (4) we can deduce that for all and all ,
Then, it follows that
Let . Then, we have
Applying Lemma 2, one can see that there exists a constant such that
Hence, we obtain
and thus for all ,
which implies that the solutions of IVP (1) are uniformly asymptotically stable.

Now we give global existence and uniform asymptotic stability results based on the nonlinear alternative of Leray-Schauder type.

Theorem 5. *Assume that the following hypotheses hold: *(H_{2})* is a continuous function;*(H_{3})*there exist positive functions , such that
**for and every ;*(H_{4})*moreover, assume that
**Then the IVP (1) admits a solution in the space . Moreover, solutions of IVP (1) are uniformly asymptotically stable.*

*Proof. *Let be defined as in (11). First, we show that maps into itself. Let , . Indeed, the map is continuous on for each , and for each , implies that
for each , we have
and for any ,
Thus,
and consequently .

Next, we show that the operator is continuous and completely continuous, and there exists an open set with for and .*Step **1 ( ** is continuous).* Let be a sequence such that in . Then, there exist and such that
Let be given. Since holds, there is a real number such that
for all . Now we consider the following two cases.*Case **1*. If , then it follows from and (30)-(31) that for sufficiently large
*Case *2. If , since is a continuous function, one has
Note that in . Hence, (32) and (33) imply that
*Step **2 (P maps bounded sets into bounded sets in **).* Indeed, it is enough to show that for any , there exists a positive constant such that for each one has . Let . Then, we have for each ,
and for each with ,

Hence, .*Step **3* (* maps bounded sets into equicontinuous sets on every compact subset ** of *). Let , , and let be a bounded set of as in Step 2. Let . Then, we have

Observing that
from Taylor’s theorem, we obtain
where . By (37)–(39), we can conclude that
As , the right-hand side of the above inequality tends to zero. The equicontinuity for the cases and is obvious.*Step **4* (* maps bounded sets into equiconvergent sets*). Let . Then
Therefore, implies that uniformly (with respect to ) converges to as . As a consequence of Steps 1–4, we can conclude that is continuous and completely continuous.*Step **5 (a priori bounds)*. We now show that there exists an open set with for and .

Let and for some . Then, for each , we obtain
By , we have that for all and ,
and thus
From the arguments in (26)-(27), we can conclude that for each ,
Hence,
Let . Then, from Lemma 2, there exists such that we have for all ,
Since , there exists such that

Set
is continuous and completely continuous. From the choice of , there is no such that , for . As a consequence of Leray-Schauder fixed-point theorem, we deduce that has a fixed point in .*Step **6 (uniform asymptotic stability of solutions).* Let be bounded; that is, there exists such that
From the similar arguments in Step 4, we can deduce that there exists such that for all solutions of IVP (1) with initial data , we have

Now we consider two solutions and of IVP (1) corresponding to the initial values and . Note that for all ,
Then, the proof of uniform asymptotic stability of solutions can be done by making use of and (52).

The proof of Theorem 5 is completed.

#### 4. Examples

*Example 1. *Consider the fractional functional differential equation
where . It is clear that condition holds. Let , . Then for all , we have
On the other hand, note that for each and . Hence, conditions and (10) hold. By Theorem 4, we conclude that IVP (53) has a unique solution in the space , and the solution of IVP (53) is uniformly asymptotically stable.

*Example 2. *Consider the fractional functional differential equation
where . It is easy to see that condition holds. Let . Then, for all , we find that
where with and
Thus, conditions and hold, and the global existence and the uniform asymptotic stability of solutions of IVP (55) can be obtained by applying Theorem 5.

By using the algorithm given in [19], we numerically simulate Example 1 with the initial conditions , and Example 2 with ; see Figures 1 and 2. From the numerical results, it can be noted that both of the solutions of Examples 1 and 2 converge uniformly, and the solutions of Example 1 converge faster than the ones of Example 2. The numerical results confirm the theoretical analysis.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This research is supported by the National Natural Science Foundation of China under Grants nos. 10801066 and 11271173 and the Fundamental Research Funds for the Central Universities under Grant nos. lzujbky-2011-47 and lzujbky-2012-k26. The project is sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

#### References

- T. Caraballo, P. Marín-Rubio, and J. Valero, “Autonomous and non-autonomous attractors for differential equations with delays,”
*Journal of Differential Equations*, vol. 208, no. 1, pp. 9–41, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. K. Hale and S. M. Verduyn Lunel,
*Introduction to Functional Differential Equations*, Springer, New York, NY, USA, 1993. - Y. Kuang,
*Delay Differential Equations with Applications in Population Dynamics*, vol. 191 of*Mathematics in Science and Engineering*, Academic Press, New York, NY, USA, 1993. View at MathSciNet - M. C. Mackey and L. Glass, “Oscillation and chaos in physiological control systems,”
*Science*, vol. 197, pp. 287–289, 1977. View at Google Scholar - M. Benchohra, J. Henderson, S. K. Ntouyas, and A. Ouahab, “Existence results for fractional order functional differential equations with infinite delay,”
*Journal of Mathematical Analysis and Applications*, vol. 338, no. 2, pp. 1340–1350, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Benchohra and B. A. Slimani, “Partial neutral functional hyperbolic differential equations with Caputo fractional derivative,”
*Nonlinear Analysis Forum*, vol. 15, pp. 143–151, 2010. View at Google Scholar · View at MathSciNet - K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 265, no. 2, pp. 229–248, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Diethelm,
*The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type*, vol. 2004 of*Lecture Notes in Mathematics*, Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems. II,”
*Applicable Analysis*, vol. 81, no. 2, pp. 435–493, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. A. Messaoudi, B. Said-Houari, and N. Tatar, “Global existence and asymptotic behavior for a fractional differential equation,”
*Applied Mathematics and Computation*, vol. 188, no. 2, pp. 1955–1962, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. Podlubny,
*Fractional Differential Equations*, vol. 198 of*Mathematics in Science and Engineering*, Academic Press, San Diego, Calif, USA, 1999. View at MathSciNet - F. Chen and Y. Zhou, “Attractivity of fractional functional differential equations,”
*Computers & Mathematics with Applications*, vol. 62, no. 3, pp. 1359–1369, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Chen, J. J. Nieto, and Y. Zhou, “Global attractivity for nonlinear fractional differential equations,”
*Nonlinear Analysis. Real World Applications*, vol. 13, no. 1, pp. 287–298, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. C. Dhage, “Local asymptotic attractivity for nonlinear quadratic functional integral equations,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 70, no. 5, pp. 1912–1922, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. C. Dhage, “Global attractivity results for nonlinear functional integral equations via a Krasnoselskii type fixed point theorem,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 70, no. 7, pp. 2485–2493, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. C. Dhage and V. Lakshmikantham, “On global existence and attractivity results for nonlinear functional integral equations,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 72, no. 5, pp. 2219–2227, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Banaś and B. C. Dhage, “Global asymptotic stability of solutions of a functional integral equation,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 69, no. 7, pp. 1945–1952, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Henry,
*Geometric Theory of Semilinear Parabolic Partial Differential Equations*, Springer, Berlin, Germany, 1989. - W. H. Deng, “Numerical algorithm for the time fractional Fokker-Planck equation,”
*Journal of Computational Physics*, vol. 227, no. 2, pp. 1510–1522, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet