#### Abstract

The aim of this paper is to give characterizations in terms of Lyapunov functions for nonuniform exponential dichotomies of nonautonomous and noninvertible discrete-time systems.

#### 1. Introduction

The notion of (uniform) exponential dichotomy essentially 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 asymptotic behaviors theory of dynamical systems.

In some situations, particularly in the nonautonomous settings, the concept of uniform exponential dichotomy is too restrictive and it is important to consider more general behaviors.

One of the main reasons for weakening the assumption of uniform exponential dichotomy is that from the point of view of ergodic theory almost all variational equations in a finite-dimensional space admit a nonuniform exponential dichotomy. On the other hand it is important to treat the case of noninvertible systems because of their interest in applications (e.g., random dynamical systems, generated by random parabolic equations, are not invertible).

The importance of Lyapunov functions is well established in the study of linear and nonlinear systems in both continuous and discrete-time. Thus, after the seminal work of Lyapunov republished most recently in [3] relevant results using Lyapunov's direct method (or second method) are presented in the books due to LaSalle, Lefschetz [4], Hahn [5], Halanay, Wexler [6], and Maliso and Mazenc [7].

This paper considers the general notion of nonuniform exponential dichotomy for nonautonomous linear discrete-time systems in Banach spaces. The main objective is to give characterizations of nonuniform exponential dichotomy in terms of Lyapunov functions for the general case of noninvertible linear discrete-time systems, as a particular case is the concept of (nonuniform) exponential dichotomy for the discrete-time linear systems which are invertible in the unstable directions. This approach can be found in the works of Barreira and Valls (see [8, 9]), and for the uniform approach we can mention the paper of Papaschinopoulos (see [10]).

In the nonuniform exponential dichotomies of linear discrete-time systems presented in this paper we consider two types of projection sequences: invariant and strongly invariant, which are distinct even in the finite-dimensional case.

We remark that we consider linear discrete-time systems having the right hand side not necessarily invertible and the dichotomy concepts studied in this paper use the evolution operators in forward time. In the definition of nonuniform exponential dichotomy we assume the existence of invariant projections sequence. At a first view the existence of such sequence is a strong hypothesis. This impediment can be eliminated using the notion of admissibility (see [11]).

The main theme of our paper is the relation between the notion of nonuniform exponential dichotomy with invariant projection sequences and the notion of Lyapunov functions. The case of exponential dichotomy with strongly invariant projection sequences was considered by Barreira and Valls (see [8, 9]).

#### 2. Definitions, Notations, and Preliminary Results

We first fix the notions and introduce the basic concepts underlying this paper. By we denote the positive integers and denotes the set of positive real numbers. is a real or complex Banach space and is the Banach algebra of all bounded linear operators on . The norm on and will be denoted by . We denote by the identity operator on .

If then we will denote by and by , respectively, the kernel and range of ; that is, respectively, We also denote by the set of all pairs of natural numbers with .

Throughout this paper, we consider the linear discrete-time systems of the form
where is a sequence in . If for every the operator is invertible, then the linear discrete-time system is called * reversible.* Every solution of the system is given by
for all , where is defined by
The map is called the discrete * evolution operator* associated to the system .

*Remark 1. *The discrete evolution operator satisfies the propagator property; that is,
for all , .

*Definition 2. *A sequence in is called a projections sequence on , if

*Remark 3. *If is a projections sequence on the sequence defined by
is a projections sequence on , which is called the * complementary projections* of . We remark that , , and .

*Definition 4. *A projection sequence is called invariant for the system if
for all .

*Remark 5. *In the particular case when is autonomous, that is, and for all , then is invariant for if and only if .

*Remark 6. *The relation (8) from Definition 4 is also true for the complementary projection and, as a consequence of (8), we have that
respectively,
for all .

*Remark 7. *If is a projections sequence invariant for the reversible system then is invertible for all and
for all .

*Definition 8. *Let be a projections sequence which is invariant for the system . We say that is *strongly invariant* for if for every the linear operator is an isomorphism from to .

A characterization of strongly invariant projections sequence is given by the following.

Proposition 9. *Let be a projections sequence which is invariant for the system . Suppose that for all the evolution operator is injective on . Then is strongly invariant for if and only if
**
for all .*

*Proof. **Necessity.* If is strongly invariant for and then there is with and hence ).*Sufficiency.* We will prove that for every there exists with .

Let . Then and hence . Moreover, from the hypothesis there is such that . Then
where .

Corollary 10. *If the projections sequence is invariant for the reversible system then it is also strongly invariant for .*

An example of invariant projections sequence which is not strongly invariant is given by the following.

*Example 11. *Let with the norm
and let be the discrete-time system defined by the sequence
It is easy to verify that the sequence defined by
is a projections sequence which is invariant for the system . Moreover, the evolution operator associated to the system is given by
for all . We can see that the evolution operator is injective on . The sequence is not strongly invariant for because and .

*Remark 12. *If the projections sequence is strongly invariant for the system then there exist such that for all the evolution operator is an isomorphism from to .

Proposition 13. *The function has the following properties:**(b*_{1}*)*,
*(b*_{2}*)*,
*(b*_{3}*)*,
*(b*_{4}*)**
for all .*

*Proof. *The properties (b_{1}) and (b_{2}) are immediate.

(b_{3}) We observe that for every we have that
and hence
which implies (b_{3}).

(b_{4}) follows immediately from (b_{1}) and (b_{3}).

*Remark 14. *If the projections sequence is invariant for the reversible system then
for all .

#### 3. Nonuniform Exponential Dichotomies

In this section we investigate some dichotomy concepts of linear discrete-time systems with respect to a projections sequence invariant for .

*Definition 15. *We say that system admits a *nonuniform exponential dichotomy* (n.e.d.) with respect to the projections sequence , if there exist a constant and a function such that the following properties hold:
for all and all .

As particular cases of nonuniform exponential dichotomy, we have the following. (1)If with and then we say that system admits an *exponential dichotomy* (e.d.).(2)If for all then we say that system is * uniformly exponentially dichotomic* (u.e.d.).

*Remark 16. *If for all and is bounded (i.e., there exist such that ) then it can be easily checked that the concept of nonuniform exponential dichotomy considered in our paper is in fact a particular case of exponential forward splitting considered in [12], with .

*Remark 17. *If the system is uniformly exponentially dichotomic and is bounded then the sequence is uniformly bounded with respect to . (i.e., ). On the other hand, may be finite and system admits a nonuniform behavior. See [13] for various examples for nonuniform exponential contractions and [14, 15] for nonuniform dichotomies. For the differential equations case, see [16].

*Remark 18. *The connection between the dichotomy concepts considered in this paper can be synthesized as (u.e.d.) *⇒* (e.d.) *⇒* (n.e.d.). The following examples show that the converse implications are not valid.

*Example 19. *Let be the linear discrete-time system and the projections sequence considered in Example 11. By a simple computation we can see that
respectively,
for all and all . Hence, for and the system is nonuniform exponentially dichotomic. Obviously, the nonuniform part cannot be removed.

*Example 20. *Let be the Banach space of bounded real-valued sequences, endowed with the norm

Let be a sequence in defined by
It is a simple verification to see that is a projections sequence with the complementary

We consider the linear discrete-time system defined by the sequence given by
where
for all and all .

We have that the evolution operator associated to is
for all .

We observe that for all we obtain
hence
We can see that is strongly invariant for the system and
and, respectively,
for all and all . Finally, we observe that system is exponentially dichotomic.

On the other side, if we suppose that the system admits a uniform approach, taking into account (21) from Definition 15 with and (33) for and , we have that
for all , which shows that nonuniform part cannot be removed.

*Example 21. *Consider, on , the sequence in given by
for all , where
Then for in defined by
for all , we have that for and the system is nonuniform exponentially dichotomic. Obviously, system is neither exponentially dichotomic nor uniformly exponentially dichotomic.

*Remark 22. *The system is nonuniform exponentially dichotomic with respect to the projections sequence invariant for if and only if there exist a constant and a function such that
for all and all .

A characterization of nonuniform exponential dichotomy of reversible systems is given by the following.

Proposition 23. *The reversible system is nonuniform exponentially dichotomic if and only if there exist a projections sequence invariant for , a function , and a constant such that
**
for all and all .*

*Proof. *It is sufficient to prove the equivalence between (22) and (41).*Necessity.* If (22) holds then for all we have
*Sufficiency.* From (41) it results that for all we have

A characterization of nonuniform exponential dichotomy property with respect to strongly invariant projection sequences is given by the following.

Theorem 24. *Let be a projections sequence which is strongly invariant for the system . Then is nonuniform exponentially dichotomic with respect to if and only if there exist a function and a constant such that
**
for all and all .*

*Proof. *It is sufficient to prove the equivalence between (22) and (45).*Necessity.* We observe that from (22) and the properties (b_{1}) and (b_{3}) from Proposition 13, we obtain
for all .*Sufficiency.* By (45), using the property (b_{2}) from Proposition 13 we obtain
for all .

#### 4. Lyapunov Functions and Nonuniform Exponential Dichotomies

Let be a linear discrete-time system on a Banach space and a projections sequence which is invariant for .

*Definition 25. *We say that is a Lyapunov function for the system with respect to projections sequence if there exists a constant such that the following properties hold:
for all
for all ,
for all .

*Example 26. *Let be the linear discrete-time system and the projections sequence considered in Example 11. Let
for all . By a simple computation for we can see that
for all and
for all .

Moreover
for all .

The main result of this paper is as follows.

Theorem 27. *The linear discrete-time system is nonuniform exponentially dichotomic with respect to the projections sequence invariant for if and only if there exists a nondecreasing sequence such that
**
for all and all .*

*Proof. **Necessity.* Suppose that is nonuniform exponentially dichotomic with respect to the projections sequence . We define by
where and is given by Remark 22. First, we observe that
where , and thus (55) is verified.

On the other hand, for we have that
where . Moreover
From (58) and (59) we have that
Hence
for all .

In the same manner we can see that for we have that
Hence
*Sufficiency.* According to (48) for every we have that
which implies
From (65) we have that
Hence
for all .

In a completely analog way, from (49) and (50) for we have that
which implies
By (69) we have
Hence
for all . From (67) and (71) we obtain that system is nonuniform exponentially dichotomic. Thus, the proof is completed.

Corollary 28. *The linear discrete-time system is exponentially dichotomic with respect to the projections sequence if and only if there exist some constants , and a Lyapunov function with respect to such that
**
for all and all .*

Corollary 29. *The linear discrete-time system is uniformly exponentially dichotomic with respect to the projections sequence if and only if there exist a Lyapunov function with respect to and a constant such that
**
for all and all .*