Abstract and Applied Analysis

Volume 2012 (2012), Article ID 741696, 23 pages

doi:10.1155/2012/741696

Research Article

Admissibility for Nonuniform Contraction and Dichotomy

Yongxin Jiang and Fang-fang Liao

Department of Mathematics, College of Science, Hohai University, Nanjing, Jiangsu 210098, China

Received 4 October 2012; Accepted 27 November 2012

Academic Editor: Juntao Sun

Copyright © 2012 Yongxin Jiang and Fang-fang Liao. 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.

The relation between the notions of nonuniform asymptotic stability and admissibility is considered. Using appropriate Lyapunov norms, it is showed that if any of their associated spaces, with , is admissible for a given evolution process, then this process is a nonuniform contraction and dichotomy. A collection of admissible Banach spaces for any given nonuniform contraction and dichotomy is provided.

1. Introduction

The study of the admissibility property has a fairly long history, and it goes back to the pioneering work of Perron [1] in 1930. Perron concerned originally the existence of bounded solutions of the equation

in for any bounded continuous perturbation . This property can be used to deduce the stability or the conditional stability under sufficiently small perturbations of a given linear equation:

His result served as a starting point for many works on the qualitative theory of the solutions of differential equations. Moreover, a simple consequence of one of the main results in that paper stated explicitly in [2, Theorem 1] is probably the first step in the literature concerning the study of the relation between admissibility and the notions of stability and conditional stability. We refer the reader to [2] for details. Relevant results concerning the extension of Perron's problem in the more general framework of the infinite-dimensional Banach spaces with bounded were obtained by Daleckij and Krein [3], Massera and Schäffer [4], and the work of Levitan and Zhikov [5] for certain cases of unbounded .

Over the last decades an increasing interest can be seen in the study of the asymptotic behavior of evolution equations in abstract spaces. In [6, 7], Latushkin et al. studied the dichotomy of linear skew-product semiflows defined on compact spaces. Using the so-called evolution semigroup, they expressed its dichotomy in terms of hyperbolicity of a family of weighted shift operators. In [8–10], Preda et al. considered related problems in the particular case of uniform exponential behavior. A large class of Schäffer spaces, which were introduced by Schäffer in [11] (see also [4]), acted as admissible spaces for the case of uniform exponential dichotomies. It is worth noting here the works by Huy [12–16] in the study of the existence of an exponential dichotomy for evolution equations.

In the case of nonuniform exponential dichotomies, Preda and Megan [17] obtained related results also for the class of Schäffer spaces, but using a notion of dichotomy which is different from the original one motivated by ergodic theory and the nonuniform hyperbolicity theory, as detailed, for example, in [18, 19]. In the more recent work [20], the authors consider the same weaker notion of exponential dichotomy and obtain sharper relations between admissibility and stability for perturbations and solutions in . Important contributions in this aspect have been made by Barreira et al. [2, 18, 19, 21–25]. Particularly, in [22], Barreira and Valls showed an equivalence between the admissibility of their associated spaces and the nonuniform exponential stability of certain evolution families by using appropriate adapted norms. They also establish a collection of admissible Banach spaces for any given nonuniform exponential dichotomy in [2]. Recently, Preda et al. [26] studied the connection between the (non)uniform exponential dichotomy of a non(uniform) exponentially bounded, strongly continuous evolution family and the admissibility of some function spaces, which extended those results established in [2, 22].

In the present paper, inspired by Barreira and Valls [2, 22], we give a characterization of nonuniform asymptotic stability in terms of admissibility property. We consider a more general type of dichotomy which is called dichotomy in [21], also proposed in [27]. In this dichotomy, not only the usual exponential behavior is replaced by an arbitrary, which may correspond, for example, to situations when the Lyapunov exponents are all infinity or are all zero, but also different growth rates for the uniform and nonuniform parts of the dichotomy are considered. It extended exponential dichotomy in various ways. In [21], it has also been showed that there is a large class of equations exhibiting this behavior. We emphasize that the characterization in our paper is a very general one; it includes as particular cases many interesting situations among them we can mention some results in previous references. To some extent, our results have a certain significance to study the theory of nonuniform hyperbolicity.

2. Admissibility for Nonuniform Contractions

We first concentrate on the simpler case of admissibility for nonuniform contractions, leaving the more elaborate case of admissibility for nonuniform dichotomies for the second part of the paper. This allows us to present the results and their proofs without some accessory technicalities. After the introduction of some basic notions, using appropriate adapted Lyapunov norms, we show that the admissibility with respect to some space with is sufficient for an evolution process to be a nonuniform contraction.

2.1. Basic Notions

We say that an increasing function is a growth rate if

We say that a family of linear operators , in a Banach space is an evolution process if:

(1) and ;(2) is continuous for and . In this section, we also assume that(3) there exist , and two growth rates , such that

We consider the new norms

These satisfy

Moreover, with respect to these norms the evolution process has the following bounded growth property.

Proposition 2.1. If is an evolution process, then

for every and .

Proof. We have

which yields the desired inequality.

Definition 2.2. We say that an evolution process is a nonuniform contraction in if there exist some constants , , and two growth rates , such that

When , we say that (1.2) has a uniform contraction or simply a contraction.

In the following, we introduce several Banach spaces that are used throughout the paper. We first set

for each , and

Respectively, with the norms

Then for each the set of the equivalence classes of functions such that Lebesgue-almost everywhere is a Banach space (again with the norms in (2.10)).

For each Banach space , with , we set

using the norms in (2.3), and we endow with the norm

Repeating arguments in the proof of Theorem 3 in [22], we obtain the following statement.

Lemma 2.3. For each and , the set is a Banach space with the norm in (2.12), and the convergence in implies the pointwise convergence Lebesgue-almost everywhere.

Definition 2.4. We say that a Banach space is admissible for the evolution process if for each the function defined by

is in (see (2.11)).

By Lemma 2.3 we know that is a Banach space with the norm

Lemma 2.5. There exists such that

Proof. We define a linear operator by . We use the closed graph theorem to show that is bounded. For this, let us take a sequence and such that in when and also such that in when . We need to show that Lebesgue-almost everywhere. For each and we have

According to Hölder's inequality, there exists such that

Therefore, for each , letting we find that . This shows that Lebesgue-almost everywhere, and by the closed graph theorem, we conclude that is a bounded operator. This completes the proof of the lemma.

2.2. Criterion for Nonuniform Contraction

Theorem 2.6. If for some the space is admissible for the evolution process , then is a nonuniform contraction.

Proof. We follow arguments in [22]. Given and , we define a function by

We note that

Then, for each and , we have

Therefore,

and in particular . On the other hand, according to (2.13) and (2.18), we have

for all , which implies that

By Lemma 2.5 and (2.21)–(2.23), we obtain

for all , , and . We claim that

for all . Indeed, for inequality (2.25) follows from (2.24), and for the inequality follows from (2.20).

Now given , , and , we define a function by

It follows from (2.25) that

and thus,

On the other hand, writing ,

Since

it follows from Lemma 2.5, (2.25), and (2.28) that

for all , , and ; we thus obtain

so

for all and . Since for , there exists sufficiently large such that

Setting for each , we have

By (2.25) and (2.33) we obtain

for . By (2.34) and

this implies that

where

We note that . Since

and by (2.4),

It follows from (2.38) that

for any . Therefore, the evolution process is a nonuniform contraction with and . This concludes the proof of Theorem 2.6.

2.3. Admissible Spaces for Nonuniform Contractions

We consider the spaces

for each , and

respectively, with the norms

In a similar manner to that in Lemma 2.3 these normed spaces induce Banach spaces and for each , the last one with norm

Theorem 2.7. If the evolution process is a nonuniform contraction, then for any the space is admissible for .

Proof. We first take . Then

Therefore,

and is admissible for .

Now we take for some . Using Hölder's inequality we obtain

where . We conclude that is also admissible for .

3. Admissibility for Nonuniform Dichotomies

We consider in this second part admissibility for nonuniform dichotomies. It generalizes the usual notion of exponential dichotomy in several ways: besides introducing a nonuniform term, causing that any conditional stability may be nonuniform, we consider rates that may not be exponential as well as different rates in the uniform and nonuniform parts. After introducing some basic notions, we show that the admissibility with respect to some space with is sufficient for an evolution process to be a nonuniform dichotomy. When compared to the case of contractions, this creates substantial complications. We also provide a collection of admissible Banach spaces for any given nonuniform dichotomy.

3.1. Basic Notions

We consider an evolution process , satisfied 1, 2 in Section 2.

We also consider a function , where is the set of bounded linear operators in , such that

(1), for every ;(2) is continuous in .

We will refer to as a projection function. Given an evolution process , we say that a projection function is compatible with if:

(1), for every ;(2) the map

 is invertible for every . We also assume that(3) there exist , and two growth rates , such that

We note that due to the invertibility assumption in condition (1.2), condition (3.3) is simply a version of (3.2) when time goes backwards.

We always consider in the paper an evolution process together with a projection function which is compatible with (and which satisfies (3.2) and (3.3)). We write

where for each . we consider the new norms

for each and , where denotes the inverse of the map in (3.1). We have

Moreover, by (3.2) and (3.3),

Definition 3.1. We say that an evolution process is a nonuniform dichotomy in if there exist a projection function compatible with , some constants , , and two growth rates , such that

When , we say that (1.2) has a uniform dichotomy or simply a dichotomy.

In the following, we still consider several spaces , , respectively, with the norms (2.10), which induce Banach spaces for each . We also set as in (2.11) but using the norms in (3.5), we endow with the norm in (2.12).

We also obtain easily the same statement in Lemma 2.3 for the set using the norms in (3.5).

Definition 3.2. We say that a Banach space is admissible for the evolution process if for each

(1) the function

 is in for each ;(2) the function defined by

 is in .

We note that since for every , and , any function in is also integrable in , and thus the first condition ensures that the function is well defined. By Lemma 2.3 we know that is a Banach space with the norm

Lemma 3.3. If for some the space is admissible for the evolution process , then there exists such that

Proof. We follow arguments in [2]. For each , we define a map by

Clearly, is linear. We use the closed graph theorem to show that is bounded. For this, let us take a sequence and such that in when , and also such that in when . We need to show that Lebesgue-almost everywhere. By Lemma 2.3, the sequence converges pointwise Lebesgue-almost everywhere. Therefore,

when , for Lebesgue-almost every . On the other hand, since in when , we also have when , for Lebesgue-almost every . This shows that Lebesgue-almost everywhere, and is bounded for each .

We define a linear operator by . We use again the closed graph theorem to show that is bounded. For this, let us take a sequence and such that in when , and also such that in when . We write

Using the similar proof of Lemma 2.5, for each and we have

According to Hölder's inequality, there exists such that

Furthermore, we have

It follows from (3.17) and (3.18) that

Therefore, for each , letting we find that . This shows that Lebesgue-almost everywhere, and by the closed graph theorem, we conclude that is a bounded operator. This completes the proof of the lemma.

3.2. Criterion for Nonuniform Dichotomy

Theorem 3.4. If for some the space is admissible for the evolution process , then is a nonuniform dichotomy.

Proof. We first consider the space . Given and , repeating argument of the proof in Theorem 2.6, except limiting on , we obtain

where

We note that . For each , we have

and by (3.2),

It follows from (3.5) and (3.20) that

for any .

Now we consider the space . Given and , we define a function by

Clearly, for every . Moreover, for each (this interval may be empty), we have

and it follows from Lemma 3.3 that

for .

On the other hand, for each , we have

So

and in particular . We thus have

for every , and . This implies that

for all . Indeed, for inequality (3.31) follows from (3.30), and for the inequality follows from (3.28).

Now given , , and , we define a function by

It follows from (3.31) that

and thus,

On the other hand, in a similar manner to that in (2.29),

Since

It follows from Lemma 3.3, (3.31), and (3.34) that

for all , , and ; we thus obtain

so

for all and . Taking the same as before, and setting for each , we have

By (3.31) and (3.39) we obtain

for , where . Since

this implies that

where

We note that . By (3.6)

and by (3.3),

for . It follows from (3.43) that

for any . To show that is a nonuniform exponential dichotomy, we note that setting in (3.2) and (3.3) yields

for every . Together with (3.24) and (3.47) this implies that

This shows that is a nonuniform dichotomy with , and .

3.3. Admissible Spaces for a Nonuniform Dichotomy

We consider the spaces

for each , and

respectively, with the norms

In a similar manner to Lemma 2.3 these normed spaces induce Banach spaces and for each , the last one with norm

Theorem 3.5. If the evolution process is a nonuniform dichotomy, then for any the space is admissible for .

Proof. We first take . Then

Therefore,

and is admissible for .

Now we take for some . Using Hölder's inequality we obtain

where . We conclude that is also admissible for .

Acknowledgments

This work is partly supported by the National Natural Science Foundation of China under Grant no. 11171090 and the Fundamental Research Funds for the Central Universities. The authors would like to show their great thanks to Professor Jifeng Chu for his useful suggestions and comments.

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