Common Fixed Points of a New Three-Step Iteration with Errors of ...

173 U. Inprasit and H. Wattanataweekul / Journal of Nonlinear Analysis and Optimization 1 (2010) 169-182
Since {l n }, {m n }, {u n }, {v n }, {w n } are bounded, there exists a constant K > 0 such that α n m n b n +
β n ≤ K, γ n + α n m n c n + (α n m n b n l n + β n l n )a n ≤ K, (m n l n + l n )‖u n − q‖ ≤ K, m n ‖v n − q‖ ≤
K, ‖w n − q‖ ≤ K and α n ≤ K for all n ≥ 1. Then
(6)
‖x n+1 − q‖ ≤
(
1 + K ( (k n − 1) + (l n − 1) + (m n − 1) )) ‖x n − q‖
+ K(δ n + σ n + ρ n )
By Lemma 1.1, we obtain lim
n→∞
‖x n − q‖ exists.
Next, we want to prove (ii), (iii) and (iv). It follows from (i) that {x n − q}, {T 1 (PT 1 ) n−1 x n −
q}, {y n − q}, {T 3 (PT 3 ) n−1 y n − q}, {z n − q} and {T 2 (PT 2 ) n−1 z n − q} are all bounded. Let
{
M = max
sup
n≥1
sup
n≥1
sup
n≥1
‖x n − q‖, sup
n≥1
‖T 3 (PT 3 ) n−1 y n − q‖, sup
n≥1
‖T 2 (PT 2 ) n−1 z n − q‖, sup
n≥1
‖T 1 (PT 1 ) n−1 x n − q‖, sup ‖y n − q‖,
n≥1
‖z n − q‖, sup ‖u n − q‖,
n≥1
}
‖v n − q‖, sup ‖w n − q‖
n≥1
By Lemma 1.3, there exists a continuous, strictly increasing convex function g : [0, ∞) → [0, ∞)
with g(0) = 0 such that
.
(7)
‖λx + µy + ξz + ϑw + ζs‖ 2 ≤ λ‖x‖ 2 + µ‖y‖ 2 + ξ‖z‖ 2 + ϑ‖w‖ 2 + ζ‖s‖ 2
− λµg ( ‖x − y‖ )
for all x, y, z, w, s ∈ B r and all λ, µ, ξ, ϑ, ζ ∈ [0, 1] with λ + µ + ξ + ϑ + ζ = 1. By (7), we have
(8)
‖z n − q‖ 2 = ‖P[a n T 1 (PT 1 ) n−1 x n + (1 − a n − δ n )x n + δ n u n ] − P(q)‖ 2
≤ ‖a n (T 1 (PT 1 ) n−1 x n − q) + (1 − a n − δ n )(x n − q) + δ n (u n − q)‖ 2
≤ a n ‖T 1 (PT 1 ) n−1 x n − q‖ 2 + (1 − a n − δ n )‖x n − q‖ 2
+ δ n ‖u n − q‖ 2 − a n (1 − a n − δ n )g ( ‖T 1 (PT 1 ) n−1 x n − x n ‖ )
≤ a n k 2 n‖x n − q‖ 2 + (1 − a n − δ n )‖x n − q‖ 2 + δ n ‖u n − q‖ 2
− a n (1 − a n − δ n )g ( ‖T 1 (PT 1 ) n−1 x n − x n ‖ )
≤ (1 + a n (k 2 n − 1) − δ n )‖x n − q‖ 2 + δ n ‖u n − q‖ 2
and
(9)
‖y n − q‖ 2 = ‖P[b n T 2 (PT 2 ) n−1 z n + c n T 1 (PT 1 ) n−1 x n + (1 − b n − c n − σ n )
x n + σ n v n ] − P(q)‖ 2
≤ ‖b n (T 2 (PT 2 ) n−1 z n − q) + c n (T 1 (PT 1 ) n−1 x n − q)
+ (1 − b n − c n − σ n )(x n − q) + σ n (x n − q)‖ 2
≤ b n ‖T 2 (PT 2 ) n−1 z n − q‖ 2 + (1 − b n − c n − σ n )‖x n − q‖ 2
+ c n ‖T 1 (PT 1 ) n−1 x n − q‖ 2 + σ n ‖v n − q‖ 2
− b n (1 − b n − c n − σ n )g ( ‖T 2 (PT 2 ) n−1 z n − x n ‖ )
≤ b n l 2 n‖z n − q‖ 2 + (1 − b n − c n − σ n )‖x n − q‖ 2 + c n k 2 n‖x n − q‖ 2
+ σ n ‖v n − q‖ 2 − b n (1 − b n − c n − σ n )g ( ‖T 2 (PT 2 ) n−1 z n − x n ‖ )