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Let us denote by X ( s, r ) the transition matrix of the linear homogeneous system 

<img src='content_image/323483.jpg'>

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$_{k}$ ; p 

(C6) − λ + NL$_{f}$ + 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). 

## 3 Bounded solutions 

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 

<img src='content_image/323484.jpg'>

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 $^{t}$∈ $_{T$_{0}$}$‖ ϕ$_{ζ}$ ( t ) ‖ ≤ K$_{0}$. 

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 

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