˜ y ( t ) = φ β ˜ x ( t $_{)}$( t $_{)}$. Then for any t ∈ [ a$_{1}$, a$_{1}$ + 3 E$_{1}$ $_{]}$, it can be verified that Using the estimates from the proof of Lemma 2.1, one can confirm that Thus, the difference w ( t ) = ‖ y ( t ) − ˜ y ( t ) ‖ satisfies the inequality for t ∈ [ a$_{1}$, a$_{1}$ + 3 E$_{1}$ $_{]}$. Since 2 KP$_{2}$r$_{1}$ < α 2 < α, Theorem 5.1 can be applied to derive on [ a$_{1}$, a$_{1}$ + 3 E$_{1}$ $_{]}$, for a positive constant κ independent of y ( t $_{)}$, ˜ y ( t $_{)}$, $_{ϵ$_{1}$}$, $_{a$_{1}$}$, $_{E$_{1}$}$. Hence, on [ a$_{1}$ + E$_{1}$, a$_{1}$ + 2 E$_{1}$ $_{]}$. Taking ϵ$_{1}$ = ϵ 4 κ and E$_{1}$ = max { − ln ϵ 4 κ δ , E } , we obtain on an interval J = [ a$_{1}$ + E$_{1}$, a$_{1}$ + 2 E$_{1}$ ] with the length E$_{1}$ ≥ $_{E}$. The proof is finished. □ 10 ‖ y ( t ) − ˜ y ( t ) ‖ ≤ 3 4 ϵ < ϵ ‖ y ( t ) − ˜ y ( t ) ‖ ≤ κ ( 2 e − δE$_{1}$ + ϵ$_{1}$ ) ‖ y ( t ) − ˜ y ( t ) ‖ ≤ κ ( e − δ ( t − a$_{1}$ ) + e − δ ( a$_{1}$ +3 E$_{1}$ − t ) + ϵ$_{1}$ ) w ( t ) ≤ K α ( 2 P$_{2}$r 2 1 + 2 L$_{2}$H )( e − α ( t − a$_{1}$ ) + e − α ( a$_{1}$ +3 E$_{1}$ − t ) ) + 2 L$_{2}$K α ϵ$_{1}$ + KP$_{2}$r$_{1}$ a$_{1}$ +3 E$_{1}$ ∫ a$_{1}$ e − α | t − s $^{|}$w ( s ) ds (3.14) ‖ y ( t ) − ˜ y ( t ) ‖ ≤ ( 2 P$_{2}$r 2 1 + 2 L$_{2}$H ) K α e − α ( t − a$_{1}$ ) + ( 2 P$_{2}$r 2 1 + 2 L$_{2}$H ) K α e − α ( a$_{1}$ +3 E$_{1}$ − t ) + K a$_{1}$ +3 E$_{1}$ ∫ a$_{1}$ e − α | t − s | ( P$_{2}$r$_{1}$ ‖ y ( s ) − ˜ y ( s ) ‖ + L$_{2}$ϵ$_{1}$ ) ds ≤ ( 2 P$_{2}$r 2 1 + 2 L$_{2}$H ) K α ( e − α ( t − a$_{1}$ ) + e − α ( a$_{1}$ +3 E$_{1}$ − t ) ) + 2 L$_{2}$K α ϵ$_{1}$ + KP$_{2}$r$_{1}$ a$_{1}$ + E$_{1}$ ∫ a$_{1}$ e − α | t − s $^{|}$‖ y ( s ) − ˜ y ( s ) ‖ ds. y ( t ) − ˜ y ( t ) = a$_{1}$ ∫ −∞ G$_{β}$ ( t, s ) ( H$_{2}$ ( y ( s ) , ν$_{β}$ ( s ) , ˜ y ( s ) , s ) + ( h ( x ( s )) − h (˜ x ( s ))) ) ds + a$_{1}$ +3 E$_{1}$ ∫ a$_{1}$ G$_{β}$ ( t, s ) ( H$_{2}$ ( y ( s ) , ν$_{β}$ ( s ) , ˜ y ( s ) , s ) + ( h ( x ( s )) − h (˜ x ( s ))) ) ds + ∞ ∫ a$_{1}$ +3 E$_{1}$ G$_{β}$ ( t, s ) ( H$_{2}$ ( y ( s ) , ν$_{β}$ ( s ) , ˜ y ( s ) , s ) + ( h ( x ( s )) − h (˜ x ( s ))) ) ds.