The inequality (3.4) yields It can be verified by applying the discrete analogue of Gronwall inequality that Thus, for n$_{1}$ − E ≤ n ≤ n$_{2}$ , we have The last inequality implies that One can confirm using (3.3) that ( ‖ A ‖ + L$_{f}$ ) n − n 1 + E +1 ≤ γϵ for n$_{1}$ ≤ n ≤ n$_{2}$ . Therefore, for each k ≥ k$_{0}$ , the inequality is valid for n$_{1}$ ≤ n ≤ n$_{2}$ . Hence, ‖ φ$_{n}$$_{+}$$_{ζ}$$_{k}$ − φ$_{n}$ ‖ → 0 uniformly as k → ∞ for each n with n$_{1}$ ≤ n ≤ n$_{2}$ . Next, we will show the existence of a positive number ϵ$_{0}$ and a sequence $^{{}$˜ η$_{k}$ } with ˜ η$_{k}$ → ∞ as k → ∞ such that ‖ φ$_{ζ}$$_{k}$$_{+}$$_{˜}$ η$_{k}$ − φ$_{˜}$ $_{η$_{k}$}$‖ ≥ ϵ$_{0}$ for each k ∈ N . Using the relations and we obtain for k ∈ N that Therefore, max {‖ φ$_{ζ}$$_{k}$$_{+}$$_{η}$$_{k}$$_{+1}$ − φ$_{η}$$_{k}$$_{+1}$ ‖ , ‖ φ$_{ζ}$$_{k}$$_{+}$$_{η}$$_{k}$ − φ$_{η}$$_{k}$ ‖} ≥ ϵ$_{0}$, (3.5) ‖ φ$_{ζ}$$_{k}$$_{+}$$_{η}$$_{k}$$_{+1}$ − φ$_{η}$$_{k}$$_{+1}$ ‖ ≥ ϵ$_{0}$ − ( ‖ A ‖ + L$_{f}$ ) ‖ φ$_{ζ}$$_{k}$$_{+}$$_{η}$$_{k}$ − φ$_{η}$$_{k}$ ‖ . φ$_{η}$$_{k}$$_{+1}$ = Aφ$_{η}$$_{k}$ + f ( φ$_{η}$$_{k}$ ) + ψ$_{η}$$_{k}$ φ$_{ζ}$$_{k}$$_{+}$$_{η}$$_{k}$$_{+1}$ = Aφ$_{ζ}$$_{k}$$_{+}$$_{η}$$_{k}$ + f ( φ$_{ζ}$$_{k}$$_{+}$$_{η}$$_{k}$ ) + ψ$_{ζ}$$_{k}$$_{+}$$_{η}$$_{k}$ z$_{n}$ < ϑ$_{n}$ + L$_{f}$ ‖ A ‖ n ∑ j = n $_{1}$− E z$_{j}$ − $_{1}$. z$_{n}$ ≤ ϑ$_{n}$ + L$_{f}$ ‖ A ‖ n ∑ j = n $_{1}$− E ϑ$_{j}$ − 1 ( 1 + L$_{f}$ ‖ A ‖ ) n − j . z$_{n}$ ≤ 2( M$_{f}$ + M$_{ψ}$ ) 1 − ‖ A ‖ ‖ A ‖ − $^{n}$( ‖ A ‖ + L$_{f}$ ) n − n 1 + E +1 + γϵ 1 − ‖ A ‖ − L$_{f}$ ‖ A ‖ − n [ 1 − ( ‖ A ‖ + L$_{f}$ ) n − n 1 + E +1 ] . ‖ φ$_{n}$$_{+}$$_{ζ}$$_{k}$ − φ$_{n}$ ‖ < 2( M$_{f}$ + M$_{ψ}$ ) 1 − ‖ A ‖ ( ‖ A ‖ + L$_{f}$ ) n − n 1 + E $^{+1}$+ γϵ 1 − ‖ A ‖ − L$_{f}$ . ‖ φ$_{n}$$_{+}$$_{ζ}$$_{k}$ − φ$_{n}$ ‖ < [ 2( M$_{f}$ + M$_{ψ}$ ) 1 − ‖ A ‖ + 1 1 − ‖ A ‖ − L$_{f}$ ] γϵ ≤ ϵ 5