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. 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, 3 φ$_{ζ}$ ( t ) = ∫ t −∞$^{U}$$^{(}$$^{t, s}$$^{) [}$$^{f}$$^{(}$$^{s, φ$_{ζ}$}$$^{(}$$^{s}$$^{)) +}$$^{g}$$^{(}$$^{s, ζ}$$^{)]}$$^{ds}$ + ∑ −∞ <θ$_{k}$<t U ( t, θ$_{k}$ +) [ J$_{k}$ ( φ$_{ζ}$ ( θ$_{k}$ )) + ζ$_{k}$ ] . (2.4) ‖ J$_{k}$ ( x$_{1}$ ) − J$_{k}$ ( x$_{2}$ ) ‖ ≤ L$_{J}$ ‖ x$_{1}$ − x$_{2}$ ‖ ‖ f ( t, x$_{1}$ ) − f ( t, x$_{2}$ ) ‖ ≤ L$_{f}$ ‖ x$_{1}$ − x$_{2}$ ‖ sup ( t,x $^{)}$∈ R × R n ‖ f ( t, x ) ‖ ≤ M$_{f}$, sup $^{σ}$∈ $_{Λ}$‖ F ( σ ) ‖ ≤ M$_{F}$ , sup $^{k}$∈ Z $^{,x}$∈ R n ‖ J$_{k}$ ( x ) ‖ ≤ M$_{J}$ ; u ′ = A ( t ) u, t = θ$_{k}$, Δ u | t = θ$_{k}$ = B$_{k}$u ( θ$_{k}$ ) . (2.3) for all k ∈ Z , x$_{1}$, x$_{2}$ ∈ R $^{n}$; (C7) − ω + NL$_{f}$ + 1 θ ln(1 + NL$_{J}$ ) < 0 . (C6) N ( L$_{f}$ ω + L$_{J}$ 1 − e − ωθ ) < 1; (C5) There exist positive numbers L$_{f}$ and L$_{J}$ such that (C4) There exist positive numbers M$_{f}$, M$_{F}$ and M$_{J}$ such that (C3) There exist positive numbers N and ω such that ‖ U ( t, s ) ‖ ≤ Ne − ω ( $^{t}$−s ) for t ≥ s ; (C2) There exists a positive number θ such that θ$_{k}$$_{+1}$ − θ$_{k}$ ≥ θ for all k ∈ Z ; (C1) det ( I + B$_{k}$ ) = 0 for all k ∈ Z , where I is the n × n identity matrix; for all t ∈ R , x$_{1}$, x$_{2}$ ∈ R $^{n}$, and