content_image/319363.jpg content_image/319362.jpg content_image/319361.jpg content_image/319360.jpg content_image/319359.jpg content_image/319358.jpg
˜ 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 

<img src='content_image/319363.jpg'>

<img src='content_image/319362.jpg'>

Thus, the difference w ( t ) = ‖ y ( t ) − ˜ y ( t ) ‖ satisfies the inequality 

<img src='content_image/319361.jpg'>

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 

<img src='content_image/319360.jpg'>

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 

<img src='content_image/319359.jpg'>

<img src='content_image/319358.jpg'>

on an interval J = [ a$_{1}$ + E$_{1}$, a$_{1}$ + 2 E$_{1}$ ] with the length E$_{1}$ ≥ $_{E}$. The proof is finished. □ 

