The connection between the stable sets of the solutions of (1.1) and (1.2) is provided in the next assertion. Lemma 3.1 Suppose that the conditions ( C 1) − ( C 7) are fulfilled, and let ζ and η be elements of Θ . If η ∈ W $^{s}$( ζ ) , then φ$_{η}$ ( t ) ∈ W $^{s}$( φ$_{ζ}$ ( t )) . Proof. Fix an arbitrary positive number ϵ, and denote α = ω − NL$_{f}$ − 1 θ ln(1 + NL$_{J}$ ) . Assume without loss of generality that ϵ ≤ 2 M$_{F}$ . Let γ be a real number such that Because the sequence η = { η$_{k}$ } , k ∈ Z , belongs to the stable set W $^{s}$( ζ ) of ζ = { ζ$_{k}$ } , there exists an integer k$_{0}$ such that ‖ η$_{k}$ − ζ$_{k}$ ‖ < ϵ γ for all k ≥ k$_{0}$. One can confirm that ‖ g ( t, η ) − g ( t, ζ ) ‖ < ϵ γ for $^{t > θ$_{k}$$_{0}$}$− $_{1}$. Making use of the relation we obtain for $^{t > θ$_{k}$$_{0}$}$− 1 that Define the functions u ( t ) = e ωt ‖ φ$_{η}$ ( t ) − φ$_{ζ}$ ( t ) ‖ and h ( t ) = c$_{1}$ + c$_{2}$e $^{ωt}$, where 5 c$_{1}$ = 2 N ( M$_{f}$ + M$_{F}$ ω + M$_{J}$ + M$_{F}$ 1 − e − ωθ ) e ωθ$_{k}$$_{0}$$_{−}$$_{1}$ − Nϵ γ ( e ωθ$_{k}$$_{0}$$_{−}$$_{1}$ ω + e ω ( $^{θ$_{k}$$_{0}$$_{−}$$_{1}$}$−θ ) 1 − e − ωθ ) ‖ φ$_{η}$ ( t ) − φ$_{ζ}$ ( t ) ‖ ≤ ∫ θ$_{k}$$_{0}$$_{−}$$_{1}$ −∞ 2 N ( M$_{f}$ + M$_{F}$ ) $_{e}$− ω ( $^{t}$−s $^{)}$ds + ∑ −∞ $^{<θ$_{k}$}$≤θ$_{k}$$_{0}$$_{−}$$_{1}$ 2 N ( M$_{J}$ + M$_{F}$ ) e − ω ( $^{t}$−θ$_{k}$ ) + ∫ t θ$_{k}$$_{0}$$_{−}$$_{1}$ Nϵ γ e − ω ( $^{t}$−s $^{)}$ds + θ$_{k}$$_{0}$$_{−}$$_{1}$<θ$_{k}$<t Nϵ γ e − ω ( $^{t}$−θ$_{k}$ ) + ∫ t θ$_{k}$$_{0}$$_{−}$$_{1}$ NL$_{f}$ e − ω ( $^{t}$−s ) ‖ φ$_{η}$ ( s ) − φ$_{ζ}$ ( s ) ‖ ds + ∑ θ$_{k}$$_{0}$$_{−}$$_{1}$<θ$_{k}$<t NL$_{J}$e − ω ( $^{t}$−θ$_{k}$ ) ‖ φ$_{η}$ ( θ$_{k}$ ) − φ$_{ζ}$ ( θ$_{k}$ ) ‖ ≤ 2 N ( M$_{f}$ + M$_{F}$ ω + M$_{J}$ + M$_{F}$ 1 − e − ωθ ) $_{e}$− ω ( $^{t}$−θ$_{k}$$_{0}$$_{−}$$_{1}$ ) + Nϵ γω 1 − $_{e}$− ω ( $^{t}$−θ$_{k}$$_{0}$$_{−}$$_{1}$ ) ) + Nϵ γ (1 − e − $^{ωθ}$) ( 1 − $_{e}$− ω ( $^{t}$−θ$_{k}$$_{0}$$_{−}$$_{1}$ + θ ) ) + ∫ t θ$_{k}$$_{0}$$_{−}$$_{1}$ NL$_{f}$$_{e}$− ω ( $^{t}$−s ) ‖ φ$_{η}$ ( s ) − φ$_{ζ}$ ( s ) ‖ ds + ∑ θ$_{k}$$_{0}$$_{−}$$_{1}$<θ$_{k}$<t NL$_{J}$e − ω ( $^{t}$−θ$_{k}$ ) ‖ φ$_{η}$ ( θ$_{k}$ ) − φ$_{ζ}$ ( θ$_{k}$ ) ‖ . (3.5) φ$_{η}$ ( t ) − φ$_{ζ}$ ( t ) = ∫ t −∞ U ( t, s ) [ f ( s, φ$_{η}$ ( s )) − f ( s, φ$_{ζ}$ ( s )) + g ( s, η ) − g ( s, ζ )] ds + ∑ −∞ <θ$_{k}$<t U ( t, θ$_{k}$ +) [ J$_{k}$ ( φ$_{η}$ ( θ$_{k}$ )) − J$_{k}$ ( φ$_{ζ}$ ( θ$_{k}$ )) + η$_{k}$ − ζ$_{k}$ ] , γ ≥ 1 + N ( 1 ω + 1 1 − e − ωθ ) ( 1 + NL$_{f}$ (1 + NL$_{J}$ ) α + NL$_{J}$ (1 + NL$_{J}$ ) 1 − e − αθ ) .