Parameter Dependence of Stable Invariant Manifolds for Delay Differential Equations under ([mu],[nu])-Dichotomies

Abstract

We obtain the existence of stable invariant manifolds for the nonlinear equation [superscript]x[variant prime][/superscript] =L(t)[subscript]xt[/subscript] +f(t,[subscript]xt[/subscript] ,λ) provided that the linear delay equation [superscript]x[variant prime][/superscript] =L(t)[subscript]xt[/subscript] admits a nonuniform (μ,ν)-dichotomy and f is a sufficiently small Lipschitz perturbation. We show that the stable invariant manifolds are dependent on parameter λ. Namely, the stable invariant manifolds are Lipschitz in the parameter λ. In addition, we also show that nonuniform (μ,ν)-contraction persists under sufficiently small nonlinear perturbations.

Academic Editor:Alfred Peris

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

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

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.

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 Schaffer [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 [figure omitted; refer to PDF] -dichotomy, which includes the traditional exponential dichotomy and the polynomial dichotomy [17-19] and the [figure omitted; refer to PDF] -nonuniform exponential dichotomy [20-22]. Since the nonuniform [figure omitted; refer to PDF] -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 [figure omitted; refer to PDF] -dichotomy provided that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] were two differentiable functions. Authors of [24] showed the existence of invariant manifolds for sufficiently small Lipschitz perturbations of a linear equation with nonuniform [figure omitted; refer to PDF] -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 [figure omitted; refer to PDF] is a sufficiently Lipschitz perturbation.

Our main aim is to show that the general behavior exhibited by a linear nonuniform [figure omitted; refer to PDF] -dichotomy persists under nonlinear perturbed equations with parameter [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are two increasing functions. More precisely, we establish the existence of Lipschitz stable invariant for nonlinear perturbed equations with parameter [figure omitted; refer to PDF] provided that the linear part has a nonuniform [figure omitted; refer to PDF] -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 [figure omitted; refer to PDF] . In Section 3, we consider the case of nonuniform [figure omitted; refer to PDF] -contraction. In Section 4, we show that the asymptotic stability of a nonuniform [figure omitted; refer to PDF] -dichotomy persists under sufficiently small nonlinear perturbations with parameter [figure omitted; refer to PDF] . Finally, an example is provided to illustrate our theorems in Section 5.

2. Preliminaries

Given [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be the Banach space of continuous functions [figure omitted; refer to PDF] endowed with the norm [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the norm in [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be the set of functions [figure omitted; refer to PDF] such that for each [figure omitted; refer to PDF] the limits [figure omitted; refer to PDF] and [figure omitted; refer to PDF] exist, and [figure omitted; refer to PDF] . By [26], we see that [figure omitted; refer to PDF] is a Banach space with the norm.

Given [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we consider the initial value problem [figure omitted; refer to PDF] where [figure omitted; refer to PDF] for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are linear operators, and there is a [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . For each [figure omitted; refer to PDF] , there is a unique solution [figure omitted; refer to PDF] of the initial value problem (2) with [figure omitted; refer to PDF] . Define the evolution operator [figure omitted; refer to PDF] associated with (2) by [figure omitted; refer to PDF]

If [figure omitted; refer to PDF] is written in the form [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is [figure omitted; refer to PDF] matrices and is measurable in [figure omitted; refer to PDF] and continuous from the left in [figure omitted; refer to PDF] , by [26], each linear operator [figure omitted; refer to PDF] can be extended to [figure omitted; refer to PDF] with the help of the integral in (5) in case that the Riemann-Stieltjes sums take value [figure omitted; refer to PDF] for each subinterval [figure omitted; refer to PDF] . Thus for each [figure omitted; refer to PDF] , there is a unique solution [figure omitted; refer to PDF] of the integral equation obtained from (2) with [figure omitted; refer to PDF] and the corresponding evolution operator [figure omitted; refer to PDF] is defined by [figure omitted; refer to PDF] It is easy to see that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . We also consider the perturbed equation [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is an open subset of a Banach space (the parameter space).

3. Stability for Nonuniform [figure omitted; refer to PDF] -Contraction

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

Definition 1.

Equation (2) is said to admit a nonuniform [figure omitted; refer to PDF] -contraction if there exist constants [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

We assume that the following conditions hold:

[figure omitted; refer to PDF] : [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ;

[figure omitted; refer to PDF] : there exist constants [figure omitted; refer to PDF] such that for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] [figure omitted; refer to PDF]

[figure omitted; refer to PDF] : there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] ;

[figure omitted; refer to PDF] : [figure omitted; refer to PDF] .

Theorem 2.

Assume that (2) admits a nonuniform [figure omitted; refer to PDF] -contraction. If [figure omitted; refer to PDF] hold, then for each [figure omitted; refer to PDF] , the solutions [figure omitted; refer to PDF] of (7) with initial condition [figure omitted; refer to PDF] satisfy [figure omitted; refer to PDF] for every [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Proof.

We see that the solution of (7) satisfies the following variation-of-parameters formula: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] We consider the space [figure omitted; refer to PDF] of function [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] It is easy to see that [figure omitted; refer to PDF] is a complete metric space. Set [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Clearly, [figure omitted; refer to PDF] is continuous and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] . By [figure omitted; refer to PDF] , for each [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] Thus we have [figure omitted; refer to PDF] Then the operator [figure omitted; refer to PDF] becomes a contraction. Furthermore, by [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and (8), we get [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] . Therefore, there exists a unique function [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . By (18), we have [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] . By (19) and [figure omitted; refer to PDF] , we get [figure omitted; refer to PDF] Therefore, it follows that [figure omitted; refer to PDF] which yields that [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] -dichotomy.

Definition 3.

Equation (2) is said to admit a nonuniform [figure omitted; refer to PDF] -dichotomy if there exist projections [figure omitted; refer to PDF] and constants [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] such that for each [figure omitted; refer to PDF]

(1) [figure omitted; refer to PDF] ;

(2) letting [figure omitted; refer to PDF] , the map [figure omitted; refer to PDF]

is invertible;

(3) [figure omitted; refer to PDF]

Setting the linear subspace [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] be the space of continuous functions [figure omitted; refer to PDF] such that, for each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] with the norm [figure omitted; refer to PDF] Moreover, we denote by [figure omitted; refer to PDF] the space of continuous functions [figure omitted; refer to PDF] , which can be written by [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Given [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we consider the set [figure omitted; refer to PDF] Assume that (2) admits a nonuniform [figure omitted; refer to PDF] -dichotomy, for each [figure omitted; refer to PDF] , the solution [figure omitted; refer to PDF] of (7) with initial condition [figure omitted; refer to PDF] satisfies [figure omitted; refer to PDF] We consider the semiflow [figure omitted; refer to PDF] which is generated by the following equation: [figure omitted; refer to PDF]

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

[figure omitted; refer to PDF] : [figure omitted; refer to PDF] ;

[figure omitted; refer to PDF] : there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] ;

[figure omitted; refer to PDF] : [figure omitted; refer to PDF] ;

[figure omitted; refer to PDF] : [figure omitted; refer to PDF] ;

[figure omitted; refer to PDF] : [figure omitted; refer to PDF] .

Theorem 4.

Assume that (2) admits a nonuniform [figure omitted; refer to PDF] -dichotomy. If [figure omitted; refer to PDF] and [figure omitted; refer to PDF] hold, then for each [figure omitted; refer to PDF] sufficiently small, there exists a unique function [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] Furthermore, for each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] [figure omitted; refer to PDF]

Proof.

We replace (30) by [figure omitted; refer to PDF]

Lemma 5.

For each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , given [figure omitted; refer to PDF] , there is a unique function [figure omitted; refer to PDF] with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , satisfying (36). Moreover, for [figure omitted; refer to PDF] [figure omitted; refer to PDF]

Proof.

We consider the operator [figure omitted; refer to PDF] defined by [figure omitted; refer to PDF] in the space [figure omitted; refer to PDF] of functions [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is given by (13). We see that [figure omitted; refer to PDF] is a complete metric space with the norm [figure omitted; refer to PDF] . For each [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , by [figure omitted; refer to PDF] and (26), we obtain [figure omitted; refer to PDF] By (39), (40), and (24), we have [figure omitted; refer to PDF] Therefore [figure omitted; refer to PDF] By [figure omitted; refer to PDF] and (26), we obtain [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] The remainder of proof is proceeded the same as Theorem 2; here we omit it.

Lemma 6.

Given [figure omitted; refer to PDF] , for each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , one has [figure omitted; refer to PDF]

Proof.

By [figure omitted; refer to PDF] and (26), for each [figure omitted; refer to PDF] , we get [figure omitted; refer to PDF] By (39), (46), and (24), we have [figure omitted; refer to PDF] which implies that inequality (45) holds.

Lemma 7.

Given [figure omitted; refer to PDF] , for each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , one has [figure omitted; refer to PDF]

Proof.

By [figure omitted; refer to PDF] and (26), for each [figure omitted; refer to PDF] , we get [figure omitted; refer to PDF] Then, by (24), (49), and Lemma 5, we obtain [figure omitted; refer to PDF] Therefore, inequality (48) holds for [figure omitted; refer to PDF] .

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

Lemma 8.

Given [figure omitted; refer to PDF] , the following properties hold.

(1) If for each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] [figure omitted; refer to PDF]

then [figure omitted; refer to PDF]

(2) If identity (52) holds for each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , then (51) holds for each [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] .

Proof.

By [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , (26), and Lemma 5, we have [figure omitted; refer to PDF] Hence [figure omitted; refer to PDF] Therefore, the right-hand of (52) is well defined. We assume that (51) holds for each [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] ; then it follows that [figure omitted; refer to PDF] By (24), (26), and Lemma 5, we have [figure omitted; refer to PDF] Letting [figure omitted; refer to PDF] in (55), it follows from [figure omitted; refer to PDF] that for each [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , (52) holds.

If (52) holds for every [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF]

Replacing [figure omitted; refer to PDF] by [figure omitted; refer to PDF] in (52), we have [figure omitted; refer to PDF] From (57) and (58), we see that (51) holds for [figure omitted; refer to PDF] .

Lemma 9.

For each [figure omitted; refer to PDF] , there exists a unique function [figure omitted; refer to PDF] such that (52) holds for every [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

Proof.

We consider the operator [figure omitted; refer to PDF] defined for each [figure omitted; refer to PDF] by [figure omitted; refer to PDF] for every [figure omitted; refer to PDF] . When [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF] . By (24), (46), and Lemma 6, we have [figure omitted; refer to PDF] By [figure omitted; refer to PDF] , for each [figure omitted; refer to PDF] , we can conclude that [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] .

The next step is done that [figure omitted; refer to PDF] is a contraction. Given [figure omitted; refer to PDF] , for each [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , by (49), Lemmas 5, and 7, we have [figure omitted; refer to PDF] By [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we see that the operator [figure omitted; refer to PDF] becomes a contraction. Therefore, given [figure omitted; refer to PDF] , for each [figure omitted; refer to PDF] , there exists a unique function [figure omitted; refer to PDF] satisfying (51). By (26) and Lemma 6, we obtain [figure omitted; refer to PDF]

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

Lemma 10.

Given [figure omitted; refer to PDF] , for each [figure omitted; refer to PDF] , one has [figure omitted; refer to PDF] for every [figure omitted; refer to PDF] .

Proof.

By (26), (28), [figure omitted; refer to PDF] , and Lemma 5, we have [figure omitted; refer to PDF] Thus, it follows from (24) and (65) that [figure omitted; refer to PDF] This proves that inequality (64) holds.

Therefore, by (28) and Lemma 10, we get [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Then, we have [figure omitted; refer to PDF] Setting [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] This shows that the linear part of (68) admits a nonuniform [figure omitted; refer to PDF] -dichotomy with [figure omitted; refer to PDF] Furthermore, we see that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] Taking suitable [figure omitted; refer to PDF] and sufficiently small [figure omitted; refer to PDF] , we conclude that (68) has a stable invariant manifold provided that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] hold.

Conflict of Interests

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

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