Let us introduce the function w ( t ) = e $^{αt}$‖ φ$_{x,ζ}$ ( t ) − φ$_{˜}$ $_{x,ζ}$( t ) ‖ . Inequality (7.21) implies that Applying Lemma 2 . 2 [40] to the last inequality we obtain that Therefore, we have for t ∈ [ R, R + E$_{0}$ ] that Since the number E is sufficiently large such that E ≥ 2 α − KL$_{f}$ ln ( 1 γϵ ) , if t ∈ [ R + E/ 2 , R + E$_{0}$ ] , then e ( KL$_{f}$ − α )( t − R ) ≤ γϵ . Thus, for t ∈ [ R + E/ 2 , R + E$_{0}$ ] . It is worth noting that the interval [ R + E/ 2 , R + E$_{0}$ ] has a length no less than E/ 2 . Consequently, the couple ( φ$_{x,ζ}$ ( t ) , φ$_{˜}$ $_{x,ζ}$( t )) ∈ B ζ × B ζ is proximal for any sequence ζ ∈ Θ . □ Since the couple ( x ( t ) $^{,}$˜ x ( t )) ∈ A × A is frequently ( ϵ$_{0}$, Δ) − separated, there exist infinitely many disjoint intervals I$_{k}$, k ∈ N , with lengths no less than Δ such that ‖ x ( t ) − ˜ x ( t ) ‖ > ϵ$_{0}$ for each t from these intervals. According to condition ( C 6) the set of functions A is an equicontinuous family on R . Therefore, using the uniform continuity of the function g : Λ × Λ → R n defined as g ( x$_{1}$, x$_{2}$ ) = h ( x$_{1}$ ) − h ( x$_{2}$ ) , one can confirm that the family is also equicontinuous on R . Suppose that h ( x ) = ( h$_{1}$ ( x ) , h$_{2}$ ( x ) , . . . , h$_{n}$ ( x )) , where each h$_{j}$, j = 1 , 2 , . . . , n, is a real valued function. In accordance with the equicontinuity of the family U , there exists a positive number τ < Δ , which does not depend on x ( t ) and ˜ x ( t ) , such that for any t$_{1}$, t$_{2}$ ∈ R with | t$_{1}$ − t$_{2}$ | < τ we have for each j = 1 , 2 , . . . , n . Fix an arbitrary natural number k . Let us denote by s$_{k}$ be the midpoint of the interval I$_{k}$ , and set U = { h ( x ( t )) − h $^{(}$˜ x ( t )) : x ( t ) ∈ A , ˜ x ( t ) ∈ A } | ( h$_{j}$ ( x ( t$_{1}$ )) − h$_{j}$ $^{(}$˜ x ( t$_{1}$ ))) − ( h$_{j}$ ( x ( t$_{2}$ )) − h$_{j}$ $^{(}$˜ x ( t$_{2}$ ))) | < L$_{2}$ϵ$_{0}$ $_{2}$√$_{n}$ (7.22) ‖ φ$_{x,ζ}$ ( t ) − φ$_{˜}$ $_{x,ζ}$( t ) ‖ < [ KL$_{1}$ α − KL$_{f}$ + 2 K ( M$_{f}$ + M$_{h}$ ) α ] γϵ ≤ ϵ ‖ φ$_{x,ζ}$ ( t ) − φ$_{˜}$ $_{x,ζ}$( t ) ‖ < KL$_{1}$γϵ α + 2 K ( M$_{f}$ + M$_{h}$ ) α e ( KL$_{f}$ − α )( t − R $^{)}$. w ( t ) ≤ KL$_{1}$γϵ α − KL$_{f}$ e αt ( 1 − e ( KL$_{f}$ − α )( t − R ) ) + 2 K ( M$_{f}$ + M$_{h}$ ) α e KL$_{f}$$^{t}$e − ( KL$_{f}$ − α ) $^{R}$. w ( t ) ≤ 2 K ( M$_{f}$ + M$_{h}$ ) − KL$_{1}$γϵ α e αR + KL$_{1}$γϵ α e αt + KL$_{f}$ ∫ t R w ( s ) ds. Proof of Lemma 4.2 15