International Journal of Differential Equations

Volume 2013 (2013), Article ID 761680, 7 pages

http://dx.doi.org/10.1155/2013/761680

## On ()-Dichotomies for Nonautonomous Linear Difference Equations in Banach Spaces

^{1}Department of Mathematics, Faculty of Mathematics and Computer Science, West University of Timişoara, V. Pârvan Boulevard. no. 4, 300223 Timişoara, Romania^{2}Academy of Romanian Scientists, Independenţei 54, 050094 Bucharest, Romania

Received 30 April 2013; Revised 2 August 2013; Accepted 5 August 2013

Academic Editor: Timothy R. Marchant

Copyright © 2013 Mihai Gabriel Babuţia et al. 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

This paper considers two general concepts of dichotomy for noninvertible and nonautonomous linear discrete-time systems in Banach spaces. These concepts use two types of dichotomy projections sequences (invariant and strongly invariant) and generalize some well-known dichotomy concepts (uniform, nonuniform, exponential, and polynomial). In the particular case of strongly invariant dichotomy projections, we present characterizations of these sequences and connections with other dichotomy concepts existent in the literature. Some illustrative examples clarify the implications between these concepts.

#### 1. Introduction

In the last few years, the theory of difference equations has witnessed an impressive development. The notion of (uniform) exponential dichotomy introduced by Perron in [1] for differential equations and by Li in [2] for difference equations plays a central role in a substantial part of the theory of differential equations and dynamical systems.

The notion of dichotomy for differential equations has gained prominence since the appearance of two fundamental monographs of Massera and Schäffer [3] and Daleckii and Krein [4]. These were followed by the important book of Coppel [5] who synthesized and improved the results that existed in the literature up to 1978. Early works in the counterpart results of difference equations appeared in the paper of Coffman and Schäffer [6] and later, in 1981, when Henry included discrete dichotomies in his book [7]. This was followed by the classical monographs due to Agarwal [8] where ordinary and exponential dichotomy properties of difference equations are studied and various applications are provided. Significant work was reported by Pötzsche in [9].

One important and useful concept of dichotomy in the study of difference equations is the so-called (uniform) -dichotomy concept introduced by Pinto in [10]. Since then, this concept has been extensively studied and applied; see, for example, Naulin and Pinto [11], Megan [12], Fenner [13], and Lin [14] where several examples are presented.

A new notion called nonuniform -dichotomy is proposed by Bento and Silva in [15] for the continuous case and in [16, 17] for discrete time settings, for invertible systems, with growth rates given by increasing functions (sequences) which go to infinity. In the last years, this subject (resp., nonuniform exponential dichotomy and nonuniform polynomial dichotomy) became one of the subjects of large interest, significant results being obtained (see, e.g., [18–24]).

On the other hand, we can consider the case of noninvertible systems. From this point of view the paper of Aulbach and Kalkbrenner [25] is of interest, where the notion of exponential forward splitting is introduced, motivated by the fact that there are equations whose backward solutions are not guaranteed to exist. Indeed, this study is of interest in applications; see, for example, random dynamical systems, generated by random parabolic equations, which are not invertible (for more details, see Zhou et al. [26]).

Motivated by the above studies, in this paper, we consider two concepts of nonuniform -dichotomies for difference equations in Banach spaces. These concepts use two concepts of sequences of projections: invariant and strongly invariant for the respective difference systems. Thus, we offer characterizations for these concepts and present connections between them. It is worth to mention the fact that if a sequence of projections is invariant for a reversible system then it is also strongly invariant for that system, and in this way the concepts merge. In this way, our study is related to the case of noninvertible systems.

#### 2. Preliminaries

Let be a Banach space and the Banach space of all bounded linear operators on . The norms on and on will be denoted by . The identity operator on is denoted by . If , then we will denote by the kernel of ; that is, respectively. We also denote by the set of all pairs of all natural numbers with ; that is, .

We consider the linear difference system
where is a given sequence. For , we define
It is obvious that
and every solution of satisfies
If for every the operator is invertible, then the system is called * reversible. *

*Definition 1. * A sequence is called a *sequence of projections* if

*Remark 2. *If is a sequence of projections, then is also a sequence of projections (which is called *the complementary sequence of projections of *) with and for every .

#### 3. -Dichotomy with Invariant Projections

*Definition 3. * A projection sequence is called *invariant* for the system if

*Remark 4. *If is invariant for , then its complementary is also invariant for . Furthermore, as a consequence of (7), we have
for all .

*Remark 5. *If is invariant for , then(i) for all .(ii) and for all .

*Definition 6. * An nondecreasing sequence is said to be a *growth rate sequence* if and .

Let be two growth rate sequences.

*Definition 7. * We say that the linear difference system admits a *strong *-*dichotomy* if there exist a sequence of projections invariant for and constants , and such that
for all .

As particular cases of -dichotomy, we have(1)If , then we say that admits a * strong exponential dichotomy*.(2)If and , then we say that system admits a * strong uniform exponential dichotomy*.(3)If , then we say that system admits a * strong polynomial dichotomy*.(4)If and , then we say that system admits a * strong uniform polynomial dichotomy*.

*Remark 8. * If the linear difference system admits a strong -dichotomy, then for all the following assertion holds:

*Definition 9. * We say that the linear difference system admits an dichotomy if there exist a sequence of projections invariant for and constants , and such that
for all .

If the system admits a -dichotomy and we take into account the particular cases from above, we obtain the notions of * exponential dichotomy, uniform exponential dichotomy, polynomial dichotomy*, and * uniform polynomial dichotomy*, respectively.

From the previous definition, we have that

*Definition 10. *If is a given growth rate sequence and is a sequence of projections that verifies (14), then we say that is -*bounded*.

As particular cases of *k*-boundedness, we have the following.(i)If there exist and such that for all , then we say that is * exponentially bounded*. (ii)If there exist and such that for all , then we say that is * polynomially bounded*. (iii)If there exist such that for all , then we say that is * bounded*.

*Remark 11. *It is worth to mention that the concept that we name “strong -dichotomy,” in the exponential case with bounded sequences of projections, is in fact the notion of exponential forward splitting considered in [25, Definition 2.1]. The notion of strong (uniform) exponential dichotomy has been also considered in [27]. The name of “strong dichotomy” derives from the fact that it implies the other dichotomy concept in this paper (under the -boundedness assumption of the sequence of the projections). It is important to state that the “strong dichotomy” from this paper differs from the “strong dichotomy” defined in [18].

*Remark 12. *Through the following two examples we will show that for a difference system the concepts of strong -dichotomy and -dichotomy are distinct, neither of them implying the other one.

*Example 13. * Consider the Banach space of all real valued sequences endowed with the norm . Fix two growth rate sequences . Define, for every , by given by
One can see that for all , is a sequence of projections on and if we pick satisfying and , for all , a simple computation gives us that
from where we deduce that
In addition, we have the following properties:
for all .

Consider the linear difference system
where is the complementary sequence of projections of .

Some computations show us that is given by
for all .

One can check that the sequence of projections is invariant for the system . From the fact that
for all , we have that the system admits a strong -dichotomy.

Indeed, assume that the system admits a -dichotomy. Then, by (14) and (17), we get the inequalities
Because the sequence is a growth rate, choose such that
Hence, the preceding inequality implies that
which, by making , contradicts the fact that is a growth rate. Thus, we can see that the system fails to admit a -dichotomy.

*Example 14. * On consider the sequence defined, for and , by
By considering the linear difference system associated to the above-defined sequence of bounded linear operators, we obtain that for all we have
We define the sequence of projections by
for all and . One can see that is invariant for the system and
for all . Moreover, for , if we pick , then we get that
from where we obtain that
for all .

By (28) and (30), we conclude that the system admits a -dichotomy.

Because and , by Remark 8, it follows that the system does not admit a strong -dichotomy.

#### 4. -Dichotomy with Strongly Invariant Projections

*Definition 15. * A sequence of projections is called *strongly invariant* for the system if is invariant for and for all the restriction of to is an isomorphism from to .

*Remark 16. * is strongly invariant for the system if and only if for all the restriction of to is an isomorphism from to .

*Remark 17. *In the exponential setting, under the hypotheses that is a strongly invariant for a system and it is bounded, it can be easily checked that the concept of strong exponential dichotomy defined in our paper is in fact the concept of *exponential splitting* defined in [25, Definition 4.1] or the concept of *exponential dichotomy* from [28, Definition 1.1].

*Remark 18. *If the sequence of projections is invariant for the reversible system , then it is also strongly invariant for .

Indeed, if is invariant for the reversible system , then for every we have with .

There are linear difference systems which admit -dichotomy with respect to projection sequences which are invariant but are not strongly invariant for the system. This phenomenon is illustrated by the following example.

*Example 19. *On , with , we consider the system generated by the sequence defined by
for all and . It is immediate to see that
Define the sequence of projections by
We have that the sequence is invariant for the system . Moreover, one can see that
and hence
Moreover, for , , and , we have that
On the other hand, it is easy to check that
hence
This shows us that we have a system which is (uniformly) exponentially dichotomic with the invariant projection sequence . But is not strongly invariant because, for example, .

*Remark 20. *Our second concept of -dichotomy introduced in this paper (see Definition 9) can be viewed in the light of the general dichotomies presented in [17], by making the choice and in the case of strongly invariant sequences of projections and by observing that the inequality from [17], namely, , implies the corresponding instability inequality from our dichotomy concept by observing that
Moreover, under the assumption of -boundedness of the sequence of the projections and strong invariance, the notion of strong -dichotomy from our paper is a particular case of the general dichotomies from [17], by making the same choice of the sequences and .

The above-mentioned concept is useful in input-output techniques, and we mention the work of B. Sasu and A. L. Sasu [29], where, in the uniform exponential setting, from a sufficient condition of admissibility of pairs of function spaces to an associated control system of , they deduce the property of uniform exponential dichotomy and the boundedness of the sequence of projections.

Even in the particular case of exponential dichotomy, the implication that gives (39) from the preceding remark fails to hold, showing that there exist reversible systems that verify Definition 9 and do not satisfy the dichotomic behavior pointed out by (39), even in the case of bounded sequences of projections.

*Remark 21. * If the sequence of projections is strongly invariant for the system , then (i)for every there is an isomorphism from the to such that ;(ii)for all there is an isomorphism from to with
for all .

In particular case when is invertible and is invariant for , then for every and for all .

Proposition 22. * Let and be growth rate sequences and a sequence of projections that is strongly invariant for a system . Then the system admits a strong -dichotomy if and only if there exist , , and such that
**
for all , where are the images of the function defined in Remark 21. *

*Proof. * We have to prove the equivalence of (10) and (43). Let . For the necessity, by (40), we have
For the sufficiency, by (41), we have
which concludes our proof.

Proposition 23. * Let and be growth rate sequences and a sequence of projections that is strongly invariant for a system . If there exist , , and such that the system has the properties
**
for all , where are the images of the function defined in Remark 21. Then, admits a -dichotomy.*

*Proof. *We have to show that (47) implies (13). Let . The conclusion follows from

The converse of the preceding proposition is not generally valid, as shown in the following example.

*Example 24. *On , consider () an arbitrary growth rate, for , and . Define the sequence by , where
Then, we have that for , with
Define, for every , the projection by where
It is easy to check that the sequence of projections is strongly invariant for the system , and for , is given by
A simple computation shows us that
for all and by choosing with and , it follows that
Hence, the system admits a -dichotomy.

Assume now that there exist , , and with
for all . In our case, the preceding inequality becomes
for all .

Consider that , , and given by
It follows that
from where we get that
Combining the above inequality with (56), we get that
for all , which is a contradiction.

Moreover, the converse of the implication from Proposition 23 does not hold not even in the finite dimensional case, as we can see from the following example.

*Example 25. *Consider that , . Consider, on , the sequence is defined by
A simple computation shows us that
for all and . Define the (constant) sequence of projections by
We have that is strongly invariant to the system with
for all and for all . A straightforward computation shows us that
for all ; hence, satisfies the conditions from Definition 9 (in the exponential case).

Assume by a contradiction that there exist , , and such that
for all . Choose . From the fact that
From (67) and (68), we get the contradiction

Proposition 26. * Let and be growth rate sequences and a sequence of projections that is strongly invariant for a system . If there exist , , and such that the system verifies (46) and (47), then admits a strong -dichotomy. *

*Proof. *Let . It follows that
which, by Proposition 22, concludes the proof.

Theorem 27. *Let and be growth rates and a sequence of projections that is strongly invariant for a system . Then, the system verifies (46) and (47) if and only if admits a strong -dichotomy and is -bounded. *

*Proof. * The necessity follows from Proposition 26 and (14).

For the sufficiency, let and such that
Let . From
the conclusion follows.

Corollary 28. *Let be a sequence of projections that is strongly invariant for a system . *(i)*If , then the system verifies (46) and (47) if and only if it is exponentially dichotomic and is exponentially bounded. *(ii)*If , then the system verifies (46) and (47) if and only if it is polynomially dichotomic and is polynomially bounded. *(iii)*If (resp., and , then the system verifies (46) and (47) if and only if it is uniformly exponentially (polynomially) dichotomic and is bounded.*

#### Acknowledgments

The authors would like to thank the referees for carefully reading the paper and for their valuable suggestions and comments, which led to the current improved form of the paper. Mihai Gabriel Babuia is supported by a grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, Project no. PN-II-RU-TE-2011-3-0103.

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