If we multiply both sides of the last inequality by e − $^{ωt}$, then we obtain that Now, let $^{R > θ$_{k}$$_{0}$}$− 1 be a sufficiently large real number such that For t ≥ R, we have Therefore, lim $^{t}$→∞ ‖ φ$_{η}$ ( t ) − φ$_{ζ}$ ( t ) ‖ = 0 . Consequently, φ$_{η}$ ( t ) ∈ W $^{s}$( φ$_{ζ}$ ( t )) . □ In the next lemma, we reveal the connection between the unstable sets of the solutions of (1.1) and (1.2). Lemma 3.2 Suppose that the conditions ( C 1) − ( C 6) are fulfilled, and let ζ and η be elements of Θ . If η ∈ W $^{u}$( ζ ) , then φ$_{η}$ ( t ) ∈ W $^{u}$( φ$_{ζ}$ ( t )) . Proof. Fix an arbitrary positive number ϵ, and let λ be a real number such that Since η = { η$_{k}$ } , k ∈ Z , is an element of the unstable set W $^{u}$( ζ ) of ζ = { ζ$_{k}$ } , there exists an integer k$_{0}$ such that ‖ η$_{k}$ − ζ$_{k}$ ‖ < ϵ λ for all k ≤ k$_{0}$. In this case, we have that ‖ g ( t, η ) − g ( t, ζ ) ‖ < ϵ λ for t ≤ θ$_{k}$$_{0}$. By using the relation 7 φ$_{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$_{k}$ ] , λ > N ( ω + 1 − e − $^{ωθ}$) ω (1 − e − $^{ωθ}$) − N ( L$_{f}$ (1 − e − $^{ωθ}$) + L$_{J}$ω ) . ∥ φ$_{η}$ ( t ) − φ$_{ζ}$ ( t ) ∥ < ϵ γ [ 1 + N ( 1 ω + 1 1 − e − ωθ )( 1 + NL$_{f}$ (1 + NL$_{J}$ ) α + NL$_{J}$ (1 + NL$_{J}$ ) 1 − e − αθ )] ≤ ϵ. 2 N (1 + NL$_{J}$ ) ( M$_{f}$ + M$_{F}$ ω + M$_{J}$ + M$_{F}$ 1 − e − ωθ ) e − α ( $^{R}$−θ$_{k}$$_{0}$$_{−}$$_{1}$ ) ≤ ϵ γ . ‖ φ$_{η}$ ( t ) − φ$_{ζ}$ ( t ) ‖ ≤ c$_{1}$ (1 + NL$_{J}$ ) e − $^{ωθ$_{k}$$_{0}$$_{−}$$_{1}$}$e − α ( $^{t}$−θ$_{k}$$_{0}$$_{−}$$_{1}$ ) + c$_{2}$ + c$_{2}$NL$_{f}$ (1 + NL$_{J}$ ) α ( 1 − e − α ( $^{t}$−θ$_{k}$$_{0}$$_{−}$$_{1}$ ) ) + c$_{2}$NL$_{J}$ (1 + NL$_{J}$ ) 1 − e − αθ 1 − e − α ( $^{t}$−θ$_{k}$$_{0}$$_{−}$$_{1}$ + θ ) ) < 2 N (1 + NL$_{J}$ ) ( M$_{f}$ + M$_{F}$ ω + M$_{J}$ + M$_{F}$ 1 − e − ωθ ) $_{e}$− α ( $^{t}$−θ$_{k}$$_{0}$$_{−}$$_{1}$ ) + Nϵ γ ( 1 ω + 1 1 − e − ωθ ) ( 1 + NL$_{f}$ (1 + NL$_{J}$ ) α + NL$_{J}$ (1 + NL$_{J}$ ) 1 − e − αθ ) .