content_image/313314.jpg content_image/313313.jpg content_image/313312.jpg content_image/313311.jpg
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 

<img src='content_image/313314.jpg'>

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 

<img src='content_image/313313.jpg'>

<img src='content_image/313312.jpg'>

Define the functions u ( t ) = e ωt ‖ φ$_{η}$ ( t ) − φ$_{ζ}$ ( t ) ‖ and h ( t ) = c$_{1}$ + c$_{2}$e $^{ωt}$, where 

<img src='content_image/313311.jpg'>

