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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 532589, 8 pages
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.
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).
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 ; 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.
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:(H0) is Lebesgue measurable with respect to on , and is continuous with respect to on . By condition and the technique used in , 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 , 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
Theorem 4. Assume that satisfies conditions and(H1)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: (H2) is a continuous function;(H3)there exist positive functions , such that for and every ;(H4)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 ,
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 ,
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.
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 , 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.
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.
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