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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.

The notion of (uniform) exponential dichotomy essentially introduced by Perron in [

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 [

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 [

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 [

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 [

We first fix the notions and introduce the basic concepts underlying this paper. By

If

Throughout this paper, we consider the linear discrete-time systems of the form

The discrete evolution operator

A sequence

If

A projection sequence

In the particular case when

The relation (

If

Let

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

Let

Let

If the projections sequence

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

Let

If the projections sequence

The function

for all

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

(b_{3}) We observe that for every _{3}).

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

If the projections sequence

In this section we investigate some dichotomy concepts of linear discrete-time systems

We say that system

As particular cases of nonuniform exponential dichotomy, we have the following.

If

If

If

If the system

The connection between the dichotomy concepts considered in this paper can be synthesized as (u.e.d.)

Let

Let

Let

We consider the linear discrete-time system

We have that the evolution operator associated to

We observe that for all

On the other side, if we suppose that the system

Consider, on

The system

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

The reversible system

It is sufficient to prove the equivalence between (

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

Let

It is sufficient to prove the equivalence between (

_{1}) and (b_{3}) from Proposition

_{2}) from Proposition

Let

We say that

Let

Moreover

The main result of this paper is as follows.

The linear discrete-time system

On the other hand, for

In the same manner we can see that for

In a completely analog way, from (

The linear discrete-time system

The linear discrete-time system