^{1}

^{1}

^{1}

^{2}

^{1}

^{2}

The present paper treats a concept of

Among the asymptotic behaviors of discrete linear systems, an important role is played by the dichotomy property (see [

A natural generalization of both the uniform and nonuniform dichotomy is successfully modeled by the concept of

This is the direction in which the present paper intends to state the results, by defining the concept of

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., [

Finally, an illustrative example will show that the main assumption of

Let

We consider the linear difference system

The map

If the sequence

A map

If

A sequence

Let

As particular cases of the

If

If

If

Let

The system

In the particular case in which

As particular cases of (

If

If

If

If

If

If

Let

We say that the system

As we will see below, the

As particular cases of

if

if

if

if

if

if

if

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

The linear system

The_{2}) and (d_{3}). For the_{2}), and for _{3}).

The linear system

For the

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

Let

there exist

If

Let

By the hypothesis,

By choosing

Let

there exist

If

In contrast with the condition

Let

By setting

If the system

Let

If

Let

Let

By

It is easy to see that

From (

If the system

Indeed, if

If, for the system

In order to prove the

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

Consider the growth rates

Let

Moreover, if

we have that for all

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

Consider the sequence of operators

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

For

Thus we have

Consider now

The

Indeed, on one hand we have that

On the other hand, assume that there exist

Having in mind that

It follows that the system

In the preceding example, if we consider the family of projections

By similar arguments as in the previous example, we have that the system described by

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

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.