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for s ∈ ( $^{s$_{k}$$_{0}$}$− $_{1}$, s$_{k}$$_{0}$$_{+2}$$_{E}$ ] . 

Suppose that s belongs to the interval ( $^{s$_{k}$$_{0}$}$− 1+ $_{E}$, s$_{k}$$_{0}$$_{+2}$$_{E}$ ] . Because the number E is sufficiently large such that E ≥ 1 ακ ln ( γR$_{1}$ (1 + N L$_{f}$ ¯ δ ) p ϵ ) and s − $^{s$_{k}$$_{0}$}$− 1 > Eκ, we have 

Hence, 

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The last inequality yields ∥ ϕ$_{ζ}$ ( t ) − ϕ$_{˜}$ $_{ζ}$( t ) ∥ < ϵ for t ∈ [ $^{θ$_{2(}$$_{k}$$_{0}$$_{+}$$_{E}$$_{)}$}$− $_{1}$, θ$_{2(}$$_{k}$$_{0}$$_{+2}$$_{E}$$_{)}$ ] ∩ T $_{0}$. Consequently, the couple ϕ$_{ζ}$ ( t ) , ϕ$_{˜}$ $_{ζ}$( t ) is proximal. □ 

The frequent separation feature of the bounded solutions of (1.1) is presented in the next lemma. 

Lemma 4.2 Under the conditions ( C 1) − ( C 5) , if a pair of sequences ζ, ˜ ζ ∈ Θ is frequently separated, then the pair of solutions ϕ$_{ζ}$ ( t ) , ϕ$_{˜}$ $_{ζ}$( t ) ∈ A is frequently ( ϵ$_{0}$, Δ) -separated for some positive numbers ϵ$_{0}$ and Δ . 

Proof. Because the pair of sequences ζ, ˜ ζ is frequently separated, there exists a positive number ¯ ϵ$_{0}$ and a sequence { k$_{q}$ } of integers satisfying k$_{q}$ → ∞ as q → ∞ such that ∥ ζ$_{k}$$_{q}$ − ˜ ζ$_{k}$$_{q}$ ∥ > ¯ ϵ$_{0}$ for each q ∈ N . 

Let us fix a natural number q. For s ∈ ( $^{s$_{k}$$_{q}$}$− $_{1}$, s$_{k}$$_{q}$ ] , the solutions φ$_{ζ}$ ( s ) = ϕ$_{ζ}$ ( ψ − $^{1}$( s )) and φ$_{˜}$ $_{ζ}$( s ) = ϕ$_{˜}$ $_{ζ}$( ψ − $^{1}$( s )) of (2.6) satisfy the relations 

and 

respectively. Therefore, one can obtain that 

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