content_image/296818.jpg content_image/296817.jpg content_image/296816.jpg content_image/296815.jpg content_image/296814.jpg
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 

<img src='content_image/296818.jpg'>

<img src='content_image/296817.jpg'>

<img src='content_image/296816.jpg'>

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 

<img src='content_image/296815.jpg'>

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 

<img src='content_image/296814.jpg'>

