content_image/318300.jpg content_image/318299.jpg content_image/318298.jpg content_image/318297.jpg content_image/318296.jpg content_image/318301.jpg content_image/318295.jpg
Applying the Gronwall’s Lemma [14], one can confirm that 

<img src='content_image/318300.jpg'>

Hence, the inequality 

<img src='content_image/318299.jpg'>

is valid. Since the number k$_{0}$ satisfies (4.4), we have e ( KL$_{f}$ − ω ) k 0 ≤ ωαϵ 2 K ( M$_{f}$ + M$_{g}$ ) so that 

Therefore, the inequality 

<img src='content_image/318298.jpg'>

<img src='content_image/318297.jpg'>

holds for each integer k with 1 ≤ k ≤ k$_{0}$. It is clear that ( 1 + K ω − KL$_{f}$ ) αϵ < 1 . Thus, 

For n ≥ n$_{0}$, it can be obtained by using (4.4) one more time that 

Hence, d ( φ$_{t}$$_{n}$, φ ) → 0 as n → ∞ . 

Next, we will verify the presence of a positive number ϵ$_{0}$ and a sequence $^{{}$˜ τ$_{n}$ } , ˜ τ$_{n}$ → ∞ as n → ∞ , such that d ( φ$_{t}$$_{n}$$_{+}$$_{˜}$ $_{τ$_{n}$}$, φ$_{˜}$ $_{τ$_{n}$}$) ≥ ϵ$_{0}$ for all n ∈ N . ∞ 

Let N be a natural number such that ∑ k = N +1 2 − k ≤ ϵ$_{0}$ 2 . One can confirm that 

<img src='content_image/318296.jpg'>

<img src='content_image/318301.jpg'>

<img src='content_image/318295.jpg'>

