Uniform Exponential Stability of Discrete Evolution Families on Space of p-Periodic Sequences

and Applied Analysis 3 Proof. Let z(n) ∈ W. By virtue of (9),


Introduction
The investigation of difference equations  +1 = A    or  +1 = A    +   leads to the idea of discrete evolution family.The main interest is the asymptotic behavior of the solutions and different types of stabilities in the study of such systems.There are a number of spectral criteria for the characterizations of stability of such systems.
New difficulties appear in the study of nonautonomous systems, especially because the part of the solution generated by the forced term (  ), that is, ∑  =] U(, )  , is not a convolution in the classical sense.These difficulties may be passed by using the so-called evolution semigroups.The evolution semigroups were exhaustively studied in [1].Clark et al. [2] developed this efficient method to the study of continuous case.So far, there are few related results regarding the investigation of discrete systems.Recently, the discrete versions of [3,4] were obtained in [5,6], respectively.
Bus ¸e et al. [7] considered the uniform exponential stability of discrete nonautonomous systems on the space of sequences denoted by  00 (Z + , ).The objective of this paper is to extend results obtained in [7] to the space of -periodic sequences denoted by P  0 (Z + , ).The results reported in this paper develop the theory of discrete evolution families on a space of bounded sequences.Similar results of this type for the continuous case may be found in the paper by Bus ¸e and Jitianu [8] and the references cited therein.

Notation and Preliminaries
Let  be a real or complex Banach space and let B() be the Banach algebra of all linear and bounded operators acting on .
We denote by ‖ ⋅ ‖ the norms of operators and vectors.Denote by R the set of real numbers and by Z + the set of all nonnegative integers.
Throughout this paper, A ∈ B(), (A) denotes the spectrum of A, and (A) := sup{|| :  ∈ (A)} denotes the spectral radius of A. It is well known that (A) = lim  → ∞ ‖A  ‖ 1/ .The resolvent set of A is defined as (A) := C \ (A), that is, the set of all  ∈ C for which A −  is an invertible operator in B().
We give some results in the framework of general Banach space and spaces of sequences as defined above.
The family U := {U(, ) : ,  ∈ Z + ,  ≥ } of bounded linear operators is called -periodic discrete evolution family, for a fixed integer  ∈ {2, 3, . ..}, if it satisfies the following properties: It is well known that any -periodic discrete evolution family U is exponentially bounded; that is, there exist  ∈ R and an   ≥ 0 such that ‖U (, )‖ ≤    (−) , ∀ ≥  ∈ Z + . ( When a family U is exponentially bounded, its growth bound  0 (U) is the infimum of all  ∈ R for which there exists an   ≥ 1 such that inequality (2) is fulfilled.It is known that As a matter of fact, A family U is termed uniformly exponentially stable if  0 (U) is negative or, equivalently, there exist an  > 0 and  > 0 such that ‖U(, )‖ ≤  −(−) , for all  ≥  ∈ Z + .Therefore, we have the following lemma.
Lemma 2. The discrete evolution family U is uniformly exponentially stable if and only if (U(, 0)) < 1.
The map U(, 0) is also called the Poincare map of the evolution family U.
It is clear that With the help of partition (8), we construct the space W which consists of all the sequences of the form That is, W := { () :  () has the property (9)} .
Obviously, W is the subspace of P  0 (Z + , ).Our result is stated as follows.
Theorem 3. Let U := {U(, ) : ,  ∈ Z + ,  ≥ } be a discrete evolution family on .If the sequence is bounded for each real number  and each -periodic sequence () ∈ W, then U is uniformly exponentially stable.