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## 2 Preliminaries 

In the sequel, we will make use of the usual Euclidean norm for vectors and the norm induced by the Euclidean norm for matrices [16]. 

Let us denote by U ( t, s ) the transition matrix of the linear homogeneous system 

The following conditions are required. 

(C1) det ( I + B$_{k}$ ) ̸ = 0 for all k ∈ Z , where I is the n × n identity matrix; 

(C2) There exists a positive number θ such that θ$_{k}$$_{+1}$ − θ$_{k}$ ≥ θ for all k ∈ Z ; 

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(C3) There exist positive numbers N and ω such that ‖ U ( t, s ) ‖ ≤ Ne − ω ( $^{t}$−s ) for t ≥ s ; 

(C4) There exist positive numbers M$_{f}$, M$_{F}$ and M$_{J}$ such that 

(C5) There exist positive numbers L$_{f}$ and L$_{J}$ such that 

for all t ∈ R , x$_{1}$, x$_{2}$ ∈ R $^{n}$, and 

for all k ∈ Z , x$_{1}$, x$_{2}$ ∈ R $^{n}$; 

(C6) N ( L$_{f}$ ω + L$_{J}$ 1 − e − ωθ ) < 1; 

(C7) − ω + NL$_{f}$ + 1 θ ln(1 + NL$_{J}$ ) < 0 . 

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Let Θ be the set of all sequences ζ = { ζ$_{k}$ } , k ∈ Z , obtained by equation (1.2). By using the results of [1, 21] one can show under the conditions ( C 1) − ( C 6) that for a fixed sequence ζ ∈ Θ the system (1.1) possesses a unique bounded on R solution φ$_{ζ}$ ( t ) , which satisfies the following relation, 

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