Dynamical Aspects of Initial/Boundary Value Problems for Ordinary Differential EquationsView this Special Issue
Research Article | Open Access
Admissibility for Nonuniform Contraction and Dichotomy
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.
The study of the admissibility property has a fairly long history, and it goes back to the pioneering work of Perron  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  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 , Massera and Schäffer , and the work of Levitan and Zhikov  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  (see also ), 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  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 , 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 , 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 . Recently, Preda et al.  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 , also proposed in . 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 , 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 , 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 . Given and , we define a function by
We note that
Then, for each and , we have
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
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 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).
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 . 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
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