Let us denote by X ( s, r ) the transition matrix of the linear homogeneous system Under the conditions ( C 1) and ( C 2) there exist positive numbers N and λ such that ‖ X ( s, r ) ‖ ≤ Ne − λ ( $^{s}$−r ) for s ≥ r [6, 42]. The following conditions are also needed. (C5) NL$_{f}$ ( 1 λ + p ¯ δ 1 − e − λψ ( ω ) ) < 1 , where ¯ δ = max $^{0}$≤ $^{k}$≤$^{p}$− $_{1}$δ$_{k}$ ; (C6) − λ + NL$_{f}$ + p ψ ( ω ) ln ( 1 + NL$_{f}$ ¯ δ ) < 0; (C7) f ( t + ω, y ) = f ( t, y ) for all ( t, y ) ∈ T 0 × R $^{n}$. The next section is devoted to the bounded solutions of system (1.1). Under the conditions ( C 1) − ( C 5) , one can verify by using the results of [6, 42] that for a fixed sequence ζ = { ζ$_{k}$ } , k ∈ Z , there exists a unique bounded on R solution φ$_{ζ}$ ( s ) of (2.6), which satisfies the relation Moreover, sup $^{s}$∈ $_{R}$‖ φ$_{ζ}$ ( s ) ‖ ≤ K$_{0}$, where K$_{0}$ = N ( M$_{f}$ + M$_{F}$ ) ( 1 λ + p ¯ δ 1 − e − λψ ( ω ) ) . Therefore, for a fixed sequence ζ = { ζ$_{k}$ } , the function ϕ$_{ζ}$ ( t ) = φ$_{ζ}$ ( ψ ( t )) satisfying ϕ$_{ζ}$ ( θ$_{2}$$_{k}$$_{+1}$ ) = φ$_{ζ}$ ( s$_{k}$ +) , k ∈ Z , is the unique solution of (2.5), and hence of (1.1), which is bounded on T 0 such that sup $_{T$_{0}$}$‖ ϕ$_{ζ}$ ( t ) ‖ ≤ K$_{0}$. $^{t}$∈ We say that the bounded solution ϕ$_{ζ}$ ( t ) attracts a solution y ( t ) of (1.1) if ‖ y ( t ) − ϕ$_{ζ}$ ( t ) ‖ → 0 as t → ∞ , t ∈ T $_{0}$. The attractiveness feature of the bounded solutions of (1.1) is mentioned in the next assertion. Lemma 3.1 If the conditions ( C 1) − ( C 6) are valid, then for a fixed sequence ζ, the bounded solution ϕ$_{ζ}$ ( t ) attracts all other solutions of (1.1). Proof. Consider an arbitrary solution y ( t ) , y ( t $^{0}$) = y$_{0}$, of (1.1) for some t 0 ∈ T 0 and y$_{0}$ ∈ R $^{n}$. Assume without loss of generality that t 0 = $^{θ$_{2}$$_{k}$}$− 1 for any k ∈ Z . Let s 0 = ψ ( t $^{0}$) and x ( s ) = y ( ψ − $^{1}$( s )) . The relation x ( s ) − φ$_{ζ}$ ( s ) = X ( s, s $^{0}$)( y$_{0}$ − φ$_{ζ}$ ( s $^{0}$)) + ∫ s s 0 [ f ( $_{ψ}$− $^{1}$( r ) , x ( r ) − f ( $_{ψ}$− $^{1}$( r ) , φ$_{ζ}$ ( r ) )] dr 5 3 Bounded solutions x ′ ( s ) = Ax ( s ) , s = s$_{k}$, Δ x | s = s$_{k}$ = δ$_{k}$Ax ( s$_{k}$ ) . (2.7) φ$_{ζ}$ ( s ) = ∫ s −∞ X ( s, r ) [ f ( ψ − $^{1}$( r ) , φ$_{ζ}$ ( r ) + g ( ψ − $^{1}$( r ) , ζ )] dr + ∑ −∞ <s$_{k}$<s X ( s, s$_{k}$ +) f ψ − $^{1}$( s$_{k}$ ) , φ$_{ζ}$ ( s$_{k}$ ) + ζ$_{k}$ ] δ$_{k}$. (3.8)