#### Abstract

The present paper treats a concept of -dichotomy for linear discrete systems. Sufficient conditions for the -boundedness of the projection sequences that give the dichotomy are presented and an illustrative example shows the connection between the growth of the system and the bound of the sequence of projections. Thus the growth of the system that is assumed in the theorems is essential.

#### 1. Introduction

Among the asymptotic behaviors of discrete linear systems, an important role is played by the dichotomy property (see [1–12] and the references therein).

A natural generalization of both the uniform and nonuniform dichotomy is successfully modeled by the concept of -dichotomy, where a significant number of papers containing many interesting results were published, from which we mention [5, 13–15]. In recent years, several papers, also dedicated to asymptotic behavior of discrete dynamical systems, appeared and hence the generalization to -behavior is present even for the trichotomy property (see [3, 8, 9, 15–17]). Among the results that were obtained in this field, we concentrate our attention on the property of boundedness of the sequence of the projections that give the dichotomic behavior. For example, in [10], in the case of discrete variational systems, the uniform exponential dichotomy property of a discrete cocycle which has a uniform exponential growth implies the boundedness of the dichotomic sequence of projections (here, the proof being more direct, compared with those found in the literature, without using the angular distance between the dichotomy subspaces). Further results concerning the boundedness of the dichotomy sequence of the projections are also presented in [4]. In [12], such results are proved both in the case of uniformly bounded coefficients of the discrete linear system (see Remark 3.3) and through techniques of admissibility of pairs of function spaces to the associated control system of the discrete linear system (Theorem 3.3).

This is the direction in which the present paper intends to state the results, by defining the concept of -dichotomy for linear discrete systems and showing that, under similar hypotheses as stated in [10], we obtain the -boundedness of the sequence of projections that give the dichotomic behavior.

Bounded and exponentially bounded sequences of projections, although not explicitly stated, are widely used in the study of the exponential dichotomy, and this intrinsic property is assumed from the beginning, in the definition of the asymptotic behavior (see, e.g., [2]). Having in mind the above reasons, we present sufficient conditions of the -boundedness of a sequence of projections that give the -dichotomy for a given discrete linear system.

Finally, an illustrative example will show that the main assumption of -growth is essential, and we present an example that, besides the property mentioned above, also shows how closely connected is the growth of the system to the boundedness of the sequence of projections.

#### 2. Basic Definitions and Notations

Let be a Banach space and the Banach algebra of all bounded linear operators on . The norms on and on will be denoted by . The identity operator on is denoted by . We also denote by the set of positive integers and by the set of nonnegative real numbers. We also denote by

We consider the linear difference system where is a given sequence. For we define

*Definition 1. *The map defined above is called the* evolution operator* associated to the system .

*Remark 2. *If the sequence is a solution of , then

*Definition 3. *A map is called a* sequence of projections* if

If is a sequence of projections, then the sequence defined by
is also a sequence of projections, called the* complementary sequence of *.

*Definition 4. *A sequence is called a* growth rate* if the following conditions are satisfied:(gr_{1}) is nondecreasing,(gr_{2}),(gr_{3}).

*Definition 5. *Let be a growth rate and let be a sequence of projections. We say that is -*bounded* if there exist and such that

*Remark 6. *As particular cases of the -boundedness, we give the following. (i)If , that is, there exist and such that
then we say that is* exponentially bounded*.(ii)If , that is, there exist and such that
then we say that is* polynomially bounded*.(iii)If , that is, there exists such that
then we say that is* bounded*.

Let be two growth rates.

*Definition 7. *The system is said to have a -*growth* if there exist and such that
for all .

In the particular case in which , we say that the system has a* uniform *-*growth* or an -*growth*.

*Remark 8. *As particular cases of (10), we have the following. (i)If , then we say that the system has an* exponential growth*.(ii)If , then we say that the system has a* nonuniform exponential growth*.(iii)If and , then we say that the system has a* uniform exponential growth*.(iv)If , then we say that the system has a* polynomial growth*.(v)If , then we say that the system has a* nonuniform polynomial growth*.(vi)If and , then we say that the system has a* uniform polynomial growth*.

#### 3. -Dichotomy and Main Results

Let be two growth rates.

*Definition 9. *We say that the system admits a -*dichotomy* if there exists a sequence of projections and there exist and such that (d_{1}),(d_{2}),(d_{3}), for all .

As we will see below, the -dichotomy covers a wide range of dichotomy concepts, which are widely used throughout the papers from the bibliography and the references therein (e.g., [13, 14]).

*Remark 10. *As particular cases of -dichotomy, having in mind the above definition, we have the following:(i)if , we have the concept of* uniform *-*dichotomy*;(ii)if , we have the concept of* nonuniform exponential dichotomy*;(iii)if , we have the concept of* exponential dichotomy*;(iv)if and , we have the concept of* uniform exponential dichotomy*;(v)if , we have the concept of* nonuniform polynomial dichotomy*;(vi)if , we have the concept of* polynomial dichotomy*;(vii)if and , we have the concept of* uniform polynomial dichotomy*.

Two simple characterizations of the introduced dichotomy concept are given in what follows.

Proposition 11. *The linear system admits a -dichotomy if and only if there exists a sequence of projections and there exist and such that for all and
**
for all .*

*Proof. *The* necessity* follows immediately by adding up (d_{2}) and (d_{3}). For the* sufficiency*, let . For , we obtain (d_{2}), and for we obtain (d_{3}).

Proposition 12. *The linear system admits a -dichotomy if and only if there exists a sequence of projections and there exist and such that
**
for all , and for all .*

*Proof. *For the* necessity*, let and and . We have
The* sufficiency* follows from choosing .

In what follows we will present the main results of our paper.

Theorem 13. *Let be a linear system such that there exist two growth rates with the following properties:*(a)* has a -growth;*(b)* admits a uniform -dichotomy;*(c)*there exist with such that
for all .**
If is a family of projections that gives the -dichotomy of the system , then is -bounded.*

*Proof. *Let and be given from the uniform -dichotomy property and , , the constants from the -growth property of . Let and be fixed and consider . From the fact that
by using (c), we obtain that

By the hypothesis, , for all . Because , it follows that and therefore
hence there exists such that

By choosing , taking into account (16) we finally have that

Corollary 14. *Let be a linear system such that there exist two growth rates with the following properties: *(a)* has a -growth;*(b)* admits a uniform -dichotomy;*(c)*there exist such that
for all .**
If is a family of projections that gives the -dichotomy of the system , then is -bounded.*

*Remark 15. *In contrast with the condition from Theorem 13, the condition from Corollary 14 is rather restrictive; namely, such a pair is quite particular. Indeed, from , having in mind that is a growth rate, it follows that, for all ,
hence

Let and . It follows that from where

By setting we obtain that
showing us that the behavior of the system is exponential.

Corollary 16. *If the system has exponential growth and it is uniformly exponentially dichotomic, then there exists a sequence of projections that gives the specified dichotomic behavior of the system and it is also exponentially bounded.*

Theorem 17. *Let be a linear system such that there exist two growth rates with the following properties:*(a)* has a -growth;*(b)* admits a uniform -dichotomy;*(c)* .**
If is a family of projections that gives the -dichotomy of the system , then is -bounded.*

*Proof. *Let and be given from the uniform -dichotomy property and the constants form the -growth property of . Let and be fixed. As in the proof from Theorem 13 we obtain the inequality
valid for all .

Let , be such that

By we may choose such that

It is easy to see that because if we assume the contrary, from , using , we get and which, from the fact that , gives us , which contradicts the choice of .

From (25), (26), and (27), it follows that
for all and for all ; hence is -bounded.

Corollary 18. *If the system has polynomial growth and it is uniformly polynomially dichotomic, then there exists a sequence of projections that gives the assumed polynomial behavior of the system which is polynomially bounded.**
Indeed, if , it is easy to see that
**
and by applying the preceding result, we obtain the desired conclusion.*

*Remark 19. *If, for the system , the -growth is uniform and the -dichotomy is as well uniform, then the given sequence of projections that gives the dichotomy property is bounded.

*Remark 20. *In order to prove the -boundedness of the sequence of projections in the case of uniform asymptotic behavior, the -growth is essential, as we will see in the following example. Moreover, by the following choice of the system to depend on the sequence of projections, we also intend to point out how the growth and the dichotomy are strongly connected to each other, through the same growth rate .

We will give a particular example, in the exponential framework.

*Example 21. *Consider the growth rates defined by
On endowed with the norm
define the sequence of projections by

Let and . We observe that
hence

Moreover, if is the complementary sequence of , defined by

we have that for all
hence

As a consequence of the above, we also have that for all

In addition, another property that we will use in what follows is given by the fact that if and , we have that

Consider the sequence of operators given by
and the corresponding linear system .

Next we will give the expression of the evolution operator which governs .

For with , one can see that

Thus we have

Consider now and . We will prove that is uniformly -dichotomic with , , , and . Consider the following:
from where the conclusion follows.

The -boundedness of the sequence of projections fails to hold, mainly because the -growth of the system does not hold.

Indeed, on one hand we have that
from where the -boundedness fails.

On the other hand, assume that there exist , , and such that the system has a -growth. Let and with
Then, by (10), we have in particular that

Having in mind that
by (47) it follows that
which leads us to the contradicting inequality

It follows that the system does not admit a -growth.

*Remark 22. *In the preceding example, if we consider the family of projections
and denote , then the associated discrete system has the evolution operator defined by

By similar arguments as in the previous example, we have that the system described by is also uniformly -dichotomic and
hence is -bounded. Thus, along with the counterexample presented from above with not being -bounded, there are systems which verify the same property and the sequence of projections is -bounded. More examples of discrete systems and -(un)bounded sequences of projections can be found in [13].

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

Mihai-Gabriel Babuţia was supported by a Grant of the Romanian National Authority for Scientific Research, CNCS UEFISCDI, Project no. PN-II-RU-TE-2011-3-0103. The authors would like to thank the referees for their suggestions that led to the final form of the paper.