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}$$_{0}$}$− 1 > Eκ, we have Hence, The last inequality yields ∥ ϕ$_{ζ}$ ( t ) − ϕ$_{˜}$ $_{ζ}$( t ) ∥ < ϵ for t ∈ [ $^{θ$_{2(}$$_{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}$$_{q}$ ] , the solutions φ$_{ζ}$ ( s ) = ϕ$_{ζ}$ ( ψ − $^{1}$( s )) and φ$_{˜}$ $_{ζ}$( s ) = ϕ$_{˜}$ $_{ζ}$( ψ − $^{1}$( s )) of (2.6) satisfy the relations and respectively. Therefore, one can obtain that ∥ φ$_{ζ}$ ( s$_{k}$$_{q}$ ) − φ$_{˜}$ $_{ζ}$( s$_{k}$$_{q}$ ) ∥ > ¯ ϵ$_{0}$κ − ∥ φ$_{ζ}$ ( $^{s$_{k}$$_{q}$}$− $_{1}$+) − φ$_{˜}$ $_{ζ}$( $^{s$_{k}$$_{q}$}$− $_{1}$+) ∥ − ∫ s$_{k}$$_{q}$ s$_{k}$$_{q}$$_{−}$$_{1}$ ( ‖ A ‖ + L$_{f}$ ) ∥ φ$_{ζ}$ ( r ) − φ$_{˜}$ $_{ζ}$( r ) ∥ dr ≥ ¯ ϵ$_{0}$κ − [1 + ( ‖ A ‖ + L$_{f}$ ) κ ] sup $^{s}$∈ ( s$_{k}$$_{q}$$_{−}$$_{1}$,s$_{k}$$_{q}$ ] ∥ φ$_{ζ}$ ( s ) − φ$_{˜}$ $_{ζ}$( s ) ∥ . φ$_{ζ}$ ( s ) = φ$_{ζ}$ ( $^{s$_{k}$$_{q}$}$− $_{1}$+) + ∫ s s$_{k}$$_{q}$$_{−}$$_{1}$ [ Aφ$_{ζ}$ ( r ) + f ( ψ − $^{1}$( r ) , φ$_{ζ}$ ( r )) + ζ$_{k}$$_{q}$ ] dr φ$_{˜}$ $_{ζ}$( s ) = φ$_{˜}$ $_{ζ}$( $^{s$_{k}$$_{q}$}$− $_{1}$+) + ∫ s s$_{k}$$_{q}$$_{−}$$_{1}$ [ Aφ$_{˜}$ $_{ζ}$( r ) + f ( ψ − $^{1}$( r ) , φ$_{˜}$ $_{ζ}$( r )) + ˜ ζ$_{k}$$_{q}$ ] dr, R$_{1}$ (1 + N L$_{f}$ ¯ δ ) $^{p}$$_{e}$−α ( $^{s}$−s$_{k}$$_{0}$$_{−}$$_{1}$ ) < ϵ γ . ∥ φ$_{ζ}$ ( s ) − φ$_{˜}$ $_{ζ}$( s ) ∥ < ϵ γ + N ϵ γ ( 1 λ + ¯ δp 1 − e − λψ ( ω ) ) ( 1 + N L$_{f}$ (1 + N L$_{f}$ ¯ δ ) p α + N L$_{f}$ ¯ δp (1 + N L$_{f}$ ¯ δ ) p 1 − e − αψ ( ω ) ) ≤ ϵ. 9