Periodic Problems of Difference Equations and Ergodic Theory

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 and The Main Statement
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 welldeveloped Floke theory 1 , which is used in analysis of linear differential equation systems by the means of monodromy matrix. Operator analogy of such theory in noncritical case when there is single solution for differential equations in Banach space was developed by Daletskyi and Krein 2 . 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 3 . In this paper we firstly build a new representation of the pseudoinverse operator based on results of ergodic theory, and then we provide the necessary and sufficient conditions that guarantee the existence of the corresponding solutions.
Let B-complex Banach space with norm · and zero-element 0; L B -Banach space of bounded linear operators from B to B. In this paper we consider existence of periodic solutions of the equation x n 1 λA n 1 x n h n 1 , n 0, 1.1 with periodicity condition where A n ∈ L B , A n m A n , for all n 0, λ is a complex parameter, and {h n } ∞ n 0 is a sequence in B. The solution of the corresponding homogeneous equation to 1.1 has the following form 4 : Operator U m, λ is traditionally called monodromy operator.
We can represent 4 the solution 1.1 with arbitrary initial condition x 0, λ x 0 , x 0 ∈ B in the form If we substitute this representation in boundary condition 1.2 , we obtain operator equation According to notations, we get operator equation In the sequel we assume that B is reflexive for simplicity 6 .
The main result of this paper is contained in Theorem 1.1.
b under condition 1.9 , solutions of boundary value problem 1.1 , 1.2 have the following form: where c is an arbitrary element of Banach space B, G n, λ -generalized Green operator of boundary value 1.1 , 1.2 , which is defined by equality 1.11

Auxiliary Result
Let us formulate and prove a number of auxiliary lemmas, which entail the theorem. Consider the following consequences of the assumptions above for further reasoning. Suppose that λ ∈ ρ NS E − U m, λ and λ is right stable point of the monodromy operator, such that λ define eigenspace N E − U m, λ , which coincides with the values set of operator U 0 λ x lim n → ∞ U n m, λ x. This operator satisfies the following conditions 6 :

Proof. Let us show that Ker
Since I − U m, λ x ∈ Im I − U m, λ and U 0 λ x ∈ Ker I − U m, λ 6 , subspaces Im I − I − U m, λ y U 0 λ z, where y, z ∈ B, which proves that Im I − U m, λ U 0 λ B. Hence according to the Banach theorem 6 original operator has inverse since it bijectively maps B to itself. Therefore point μ 1 is regular 6 for the operator μI − U m, λ U 0 λ . Since powers of the operator U m, λ are uniformly bounded and spectral radius r U m,λ 1 n U m, λ n n √ c, then r U m,λ lim n → ∞ n U m, λ n lim n → ∞ n √ c 1 . It is well known 6 that resolvent set of a bounded operator is open. Number μ 1 ∈ ρ U m, λ − U 0 λ ; thus there exist a neighborhood of μ such that each point from the neighborhood belongs to resolvent set. For any point μ > r U m,λ −U 0 λ that belongs to the neighborhood there exists a resolvent 6 , which has the form of converging in the norm series Using the analyticity of the resolvent and well-known identity for points μ > 1 such that Finally, by substituting the series in the equation above, we get 2.5 , which proves the lemma. Let us introduce some notation first before proving next statement.
or in the form of converging operator series for all μ > 1 : Proof. It suffices to check conditions 1 and 2 of the Definition 2.3. We use both representations 2.10 , 2.11 and the expression 2.4 for operator U 0 λ . Consider the following product: Note that U 0 λ U m, λ − U 0 λ l 0 for any l ∈ N this directly follows from 2.4 using formula of binominal coefficient . Now, prove that