^{1}

^{2}

^{2}

^{1}

^{2}

The necessary and sufficient conditions for solvability of the family of difference equations with periodic boundary condition were obtained using the notion of relative spectrum of linear bounded operator in the Banach space and the ergodic theorem. It is shown that when the condition of existence is satisfied, then such periodic solutions are built using the formula for the generalized inverse operator to the linear limited one.

The problem of existence of periodic solutions for classes of equations is well known. Though it is hard to mention all the contributors in a single paper, we would like to mark out well-developed Floke theory [

This paper is dedicated to obtaining analogous conditions for a family of difference equations in Banach space and to building representations of corresponding solutions. The proposed approach allows obtaining solutions for both critical and noncritical cases. Note that this problem can be approached using well-developed pseudoinverse techniques in theory of boundary value problems [

Let

We can represent [

Boundary value problem (

Following the paper [

Denote

In the sequel we assume that

The main result of this paper is contained in Theorem

Let

boundary value problem (

under condition (

Let us formulate and prove a number of auxiliary lemmas, which entail the theorem.

If

From the assumption above it follows that the conditions of statistical ergodic theorem hold [

Consider the following consequences of the assumptions above for further reasoning. Suppose that

Operator

Let us show that

Using the analyticity of the resolvent and well-known identity for points

Let us introduce some notation first before proving next statement.

Operator

It suffices to check conditions (1) and (2) of the Definition

According to general theory of linear equations solvability [

Under such a condition, all solutions of the problem (

Suppose

Supposing

Let us illustrate the statements proved above on example of two-dimensional systems.

(1) Consider equation

(2) We can search for periodic solutions of any period

To illustrate complexity of the set we did the following.

Recall that the length of vector

We can see how the trajectory of vector length densely fills rectangle or turns into a line (Figures

This allows us to conclude that behavior of the system is rather complex; it can undergo unpredictable changes with the slightest variations of a single parameter. We must admit that effects described need further theoretical investigation.

A. A. Boichuk was supported by the Grant 1/0090/09 of the Grant Agency of Slovak Republic (VEGA) and by the Project APVV-0700-07 of the Slovak Research and Development Agency.