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

177 U. Inprasit and H. Wattanataweekul / Journal of Nonlinear Analysis and Optimization 1 (2010) 169-182
for all n ≥ n 0 . This implies by (11) that
(18)
η 2 (1 − η ′ )g ( ‖T 1 (PT 1 ) n−1 x n − x n ‖ ) ≤ ‖x n − q‖ 2 − ‖x n+1 − q‖ 2
+ K 0
(
(k
2
n − 1) + (l 2 n − 1) + (m 2 n − 1) )
+ K 0 (δ n + σ n + ρ n )
for all n ≥ n 0 . It follows from (18) that for r ≥ n 0 ,
(19)
r (
∑ g ‖T 1 (PT 1 ) n−1 x n − x n ‖ ) ≤
n=n 0
≤
1 ( r
η 2 (1 − η ′ )
∑ (‖x n − q‖ 2 − ‖x n+1 − q‖ 2 )
n=n 0
+ K 0
r
∑
n=n 0
(
(k
2
n − 1) + (l 2 n − 1) + (m 2 n − 1)
+ δ n + σ n + ρ n
) )
1
(‖x
η 2 (1 − η ′ n0 − q‖ 2 r (
+ K 0
)
∑ (k
2
n − 1)
n=n 0
+ (l 2 n − 1) + (m 2 n − 1) )) .
Since 0 ≤ t 2 − 1 ≤ 2t(t − 1) for all t ≥ 1 and ∑
∞ (k n − 1) < ∞,
∞, we get
∞
∑
n=1
∞
∑
(k 2 n − 1) < ∞,
∞
∑
n=1
(l 2 n − 1) < ∞,
n=1
∞
∑
n=1
(l n − 1) < ∞,
∞
∑ (m n − 1) <
n=1
∞
∑ (m 2 n − 1) < ∞. By inequality (19), let r → ∞.
n=1
We get g ( ‖T 1 (PT 1 ) n−1 x n − x n ‖ ) < ∞. Thus lim g ( ‖T 1 (PT 1 ) n−1 x n − x n ‖ ) = 0. Since g is
n=n 0 n→∞
strictly increasing and continuous at 0 with g(0) = 0, it follows that lim ‖T 1 (PT 1 ) n−1 x n −
n→∞
x n ‖ = 0.
By using a similar method as in (ii) part (a) together with (10), (17), (15), (16), (12) and
(13), the results in (ii) (b,c,d), (iii) (a,b) and (iv), respectively, can be proved. □
Lemma 2.2. Let X be a uniformly convex Banach space and C a nonempty closed convex nonexpansive
retract of X with P as a nonexpansive retraction. Let T 1 , T 2 , T 3 : C → X be asymptotically
quasi-nonexpansive mappings with respect to sequences {k n }, {l n }, {m n }, respectively, such that
∞
∞
∞
F ̸= ∅, k n ≥ 1, l n ≥ 1, m n ≥ 1, ∑ (k n − 1) < ∞, ∑ (l n − 1) < ∞, ∑ (m n − 1) < ∞. Let
n=1
n=1
n=1
{a n }, {b n }, {c n }, {α n }, {β n }, {γ n }, {δ n }, {σ n }, {ρ n } be real sequences in [0, 1] such that a n + δ n ,
b n + c n + σ n , α n + β n + γ n + ρ n are in [0, 1] for all n ≥ 1,
∞
∑ δ n < ∞,
n=1
∞
∑
n=1
σ n < ∞,
∞
∑
n=1
ρ n < ∞
and let {u n }, {v n }, {w n } be bounded sequences in C. For a given x 1 ∈ C, let {x n }, {y n }, {z n } be the
sequences defined as in (1). Suppose T 1 , T 2 , T 3 are uniformly L-Lipschitzian. If lim ‖T 1 (PT 1 ) n−1 x n −
n→∞
x n ‖ = 0, lim ‖T 2 (PT 2 ) n−1 z n − x n ‖ = 0, lim ‖T 3 (PT 3 ) n−1 y n − x n ‖ = 0, then
n→∞ n→∞
(i)
(ii)
(iii)
lim ‖T 1 x n − x n ‖ = 0,
n→∞
lim ‖T 2 x n − x n ‖ = 0, and
n→∞
lim ‖T 3 x n − x n ‖ = 0.
n→∞
Proof. Since
‖x n+1 − x n ‖ ≤ α n ‖T 3 (PT 3 ) n−1 y n − x n ‖ + β n ‖T 2 (PT 2 ) n−1 z n − x n ‖
+ γ n ‖T 1 (PT 1 ) n−1 x n − x n ‖ + ρ n ‖w n − x n ‖
−→ 0 as n → ∞,