Journal of Mathematics
Volume 2014 (2014), Article ID 989526, 9 pages
http://dx.doi.org/10.1155/2014/989526

Research Article

Parameter Dependence of Stable Invariant Manifolds for Delay Differential Equations under -Dichotomies

Lijun Pan

School of Mathematics, Jia Ying University, Meizhou, Guangdong 514015, China

Received 20 May 2014; Accepted 14 September 2014; Published 27 October 2014

Academic Editor: Alfred Peris

Copyright © 2014 Lijun Pan. 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.

We obtain the existence of stable invariant manifolds for the nonlinear equation provided that the linear delay equation admits a nonuniform -dichotomy and is a sufficiently small Lipschitz perturbation. We show that the stable invariant manifolds are dependent on parameter . Namely, the stable invariant manifolds are Lipschitz in the parameter . In addition, we also show that nonuniform -contraction persists under sufficiently small nonlinear perturbations.

1. Introduction

The theory of exponential dichotomy is attracting much attention in recent years. This is mostly because the concept of exponential dichotomy plays an important role in the obtention of invariant manifolds of differential equations. The notion of exponential dichotomy can be traced back to Perron in [1]. Since then, Sacker and Sell [2–5] investigated sufficient conditions for the existence of an exponential dichotomy, also in the infinite-dimensional setting. In [6], Barreira and Valls discussed the much weaker notion of nonuniform exponential dichotomy. This theory of nonuniform hyperbolicity was introduced in the seventies by Barreira and Pesin [7]. For background material on exponential dichotomy for differential equations, the books [8–10] may be consulted.

We note that the study of robustness in the case of exponential behavior has a long history. For the early work we can refer to Perron [1] and Massera and Schäffer [11]. For more recent works we refer to [12–15]. In particular, in [12], Barreira and Valls studied the existence of stable invariant manifold for any sufficiently small nonlinear perturbation of the linear equation. In [13], the authors discussed the parameter dependence of stable manifolds under nonuniform exponential dichotomy.

Recently, general stable and unstable behaviors with growth rates given by two increasing functions are exhibited by Bento and Silva [16]. The results in this work rely on the notion of nonuniform -dichotomy, which includes the traditional exponential dichotomy and the polynomial dichotomy [17–19] and the -nonuniform exponential dichotomy [20–22]. Since the nonuniform -dichotomy is more general than nonuniform dichotomy before, so it is an emerging field drawing attention from both theoretical and applied disciplines. For example, in [23], the authors showed the robustness of nonuniform -dichotomy provided that and were two differentiable functions. Authors of [24] showed the existence of invariant manifolds for sufficiently small Lipschitz perturbations of a linear equation with nonuniform -dichotomy.

In the implementation of differential equations, time delay is a common phenomenon due to instantaneous perturbations. We can refer to the book [25] for functional differential equations. Recently, Barreira et al. [26] investigated the parameter dependence of stable manifolds for delay equations with polynomial dichotomies. It is assumed that is a sufficiently Lipschitz perturbation.

Our main aim is to show that the general behavior exhibited by a linear nonuniform -dichotomy persists under nonlinear perturbed equations with parameter , where , are two increasing functions. More precisely, we establish the existence of Lipschitz stable invariant for nonlinear perturbed equations with parameter provided that the linear part has a nonuniform -dichotomy and the nonlinear perturbations are sufficiently small. Our method is inspired in the former work in [26]. Meanwhile, an example is given to illustrate the applicability of the results.

The organization of this paper is as follows. In Section 2, we introduce delay linear differential equations and the perturbed equation with parameter . In Section 3, we consider the case of nonuniform -contraction. In Section 4, we show that the asymptotic stability of a nonuniform -dichotomy persists under sufficiently small nonlinear perturbations with parameter . Finally, an example is provided to illustrate our theorems in Section 5.

2. Preliminaries

Given , let be the Banach space of continuous functions endowed with the norm

where is the norm in . Let be the set of functions such that for each the limits and exist, and . By [26], we see that is a Banach space with the norm.

Given and , we consider the initial value problem

where for , are linear operators, and there is a such that

for all . For each , there is a unique solution of the initial value problem (2) with . Define the evolution operator associated with (2) by

If is written in the form

where is matrices and is measurable in and continuous from the left in , by [26], each linear operator can be extended to with the help of the integral in (5) in case that the Riemann-Stieltjes sums take value for each subinterval . Thus for each , there is a unique solution of the integral equation obtained from (2) with and the corresponding evolution operator is defined by

It is easy to see that and for . We also consider the perturbed equation

where and is an open subset of a Banach space (the parameter space).

3. Stability for Nonuniform -Contraction

In this section, we show that the asymptotic stability of a nonuniform contraction persists under sufficiently small nonlinear perturbations. We say that an increasing function is said to be a growth rate if and . Let be growth rates. Then we recall the definition of nonuniform -contraction for (2).

Definition 1. Equation (2) is said to admit a nonuniform -contraction if there exist constants , , and such that

for .

We assume that the following conditions hold:

for all and ; there exist constants such that for , ,

 there exists such that ; .

Theorem 2. Assume that (2) admits a nonuniform -contraction. If hold, then for each , the solutions of (7) with initial condition satisfy

for every and .

Proof. We see that the solution of (7) satisfies the following variation-of-parameters formula:

where

We consider the space of function such that , for each and , where

It is easy to see that is a complete metric space. Set

for each , , and . Clearly, is continuous and and for each . By , for each , we have

Thus we have

Then the operator becomes a contraction. Furthermore, by , , and (8), we get

Thus

which implies that . Therefore, there exists a unique function such that . By (18), we have

for each . By (19) and , we get

Therefore, it follows that

which yields that

This completes the proof of the theorem.

4. Stable Invariant Manifolds

In this section, we establish stable manifolds theorem under sufficiently small perturbation of nonuniform -dichotomy.

Definition 3. Equation (2) is said to admit a nonuniform -dichotomy if there exist projections and constants , , and such that for each

 (1); (2)letting , the map

is invertible;

 (3)

Setting the linear subspace

Let be the space of continuous functions such that, for each , , and

with the norm

Moreover, we denote by the space of continuous functions , which can be written by , such that

for each , , and . Given and , we consider the set

Assume that (2) admits a nonuniform -dichotomy, for each , the solution of (7) with initial condition satisfies

We consider the semiflow

which is generated by the following equation:

To obtain our results, we also need the following assumptions:

 ; there exists such that ; ; ; .

Theorem 4. Assume that (2) admits a nonuniform -dichotomy. If and hold, then for each sufficiently small, there exists a unique function such that

Furthermore, for each , , and

Proof. We replace (30) by

Lemma 5. For each , , given , there is a unique function with and , satisfying (36). Moreover, for

Proof. We consider the operator defined by

in the space of functions such that , for each and , where is given by (13). We see that is a complete metric space with the norm . For each and , by and (26), we obtain

By (39), (40), and (24), we have

Therefore

By and (26), we obtain

Thus

The remainder of proof is proceeded the same as Theorem 2; here we omit it.

Lemma 6. Given , for each , , one has

Proof. By and (26), for each , we get

By (39), (46), and (24), we have

which implies that inequality (45) holds.

Lemma 7. Given , for each , , , one has

Proof. By and (26), for each , we get

Then, by (24), (49), and Lemma 5, we obtain

Therefore, inequality (48) holds for .

Next, we transform (37) into an equivalent problem.

Lemma 8. Given , the following properties hold.

 (1)If for each , , 

then

 (2)If identity (52) holds for each , , then (51) holds for each , , and .

Proof. By , , (26), and Lemma 5, we have

Hence

Therefore, the right-hand of (52) is well defined. We assume that (51) holds for each , and ; then it follows that

By (24), (26), and Lemma 5, we have

Letting in (55), it follows from that for each , and , (52) holds.
If (52) holds for every , and , we have

Replacing by in (52), we have

From (57) and (58), we see that (51) holds for .

Lemma 9. For each , there exists a unique function such that (52) holds for every , .

Proof. We consider the operator defined for each by

for every . When , we have ; then . By (24), (46), and Lemma 6, we have

By , for each , we can conclude that

which implies that .
The next step is done thatis a contraction. Given , for each , and , by (49), Lemmas 5, and 7, we have

By and , we see that the operator becomes a contraction. Therefore, given , for each , there exists a unique function satisfying (51). By (26) and Lemma 6, we obtain

Finally, we turn to the proof of inequality (35) of the theorem. The following lemma is necessary for our establishing (35).

Lemma 10. Given , for each , one has

for every .

Proof. By (26), (28), , and Lemma 5, we have

Thus, it follows from (24) and (65) that

This proves that inequality (64) holds.
Therefore, by (28) and Lemma 10, we get

This completes the proof of Theorem 4.

5. Example

In this section, we provide an example to demonstrate the derived results. Consider the delay system

where and . Then, we have

Setting and , we have

which implies that

This shows that the linear part of (68) admits a nonuniform -dichotomy with

Furthermore, we see that and

Taking suitable and sufficiently small , we conclude that (68) has a stable invariant manifold provided that and hold.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publishing of this paper.

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