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

Journal of Nonlinear Analysis and Optimization 

Vol. 1 No. 1 (2010), 169-182 

ISSN : 1906-9605 

http://www.sci.nu.ac.th/jnao 

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

Asymptotically Quasi-Nonexpansive Nonself-Mappings in Banach 

spaces 

Utith Inprasit 1,2 and Hathaikarn Wattanataweekul 1,2 

ABSTRACT: In this paper, we introduce an iterative method for finding a common element of 

the set of solutions of a generalized mixed equilibrium problem and the set of common fixed 

points of a finite family of nonexpansive mappings in a real Hilbert space. Then, we prove that 

the sequence converges strongly to a common element of the above two sets. Furthermore, 

we apply our result to prove three new strong convergence theorems in fixed point problems, 

mixed equilibrium problems, generalized equilibrium problems and equilibrium problems. 

1. Introduction 

We assume that X is a normed space and C is a nonempty subset of X. A mapping T : C → 

C is said to be asymptotically nonexpansive [3] if there exists a sequence {k n } of real numbers 

with k n ≥ 1 and lim k n = 1 such that ‖T n x − T n y‖ ≤ k n ‖x − y‖ for all x, y ∈ C and each 

n→∞ 

n ≥ 1. The class of asymptotically nonexpansive mappings is a natural generalization of the 

important class of nonexpansive mappings. Goebel and Kirk [3] proved that if C is a nonempty 

closed and bounded subset of a uniformly convex Banach space, then every asymptotically 

nonexpansive self-mapping has a fixed point. A mapping T : C → C is called asymptotically 

quasi-nonexpansive if F(T) ̸= ∅ and there exists a sequence {k n } of real numbers with k n ≥ 1 

and lim k n = 1 such that ‖T n x − q‖ ≤ k n ‖x − q‖ for all x ∈ C, q ∈ F(T), n ≥ 1, where F(T) 

n→∞ 

is the set of fixed points of T. The mapping T is called uniformly L-Lipschitzian if there exists a 

positive constant L such that ‖T n x − T n y‖ ≤ L‖x − y‖ for all x, y ∈ C and each n ≥ 1. It is easy 

to see that an asymptotically nonexpansive mapping must be uniformly L-Lipschitzian as well 

as asymptotically quasi-nonexpansive but the converse does not hold. 

In 2000, Noor [9] introduced a three-step iterative sequence and studied the approximate 

solutions of variational inclusions in Hilbert spaces. Glowinski and Le Tallec [4] applied 

three-step iterative sequences for finding the approximate solutions of the elastoviscoplasticity 

problem, eigenvalue problems and in the liquid crystal theory. It has been shown in [1], 

that three-step method performs better than two-step and one-step methods for solving variational 

inequalities. The three-step schemes are natural generalization of the splitting methods 

to solve partial differential equations; see, Noor [9, 10, 11]. This signifies that Noor three-step 

Corresponding author: utith@sci.ubu.ac.th(U. Inprasit) 

Manuscript received March 16, 2010. Accepted July 31, 2010.

170 U. Inprasit and H. Wattanataweekul / Journal of Nonlinear Analysis and Optimization 1 (2010) 169-182 

methods are robust and more efficient than the Mann (one-step) and Ishikawa (two-step) type 

iterative methods to solve problems of pure and applied sciences. 

In 2001, Khan and Takahashi [5] have approximated common fixed points of two asymptotically 

nonexpansive mappings by the modified Ishikawa iteration. Recently Shahzad and 

Udomene [15] established convergence theorems for the modified Ishikawa iteration process 

of two asymptotically quasi-non expansive mappings to a common fixed point of the mappings. 

For related results with error terms, we refer to [2, 6, 13] and [15]. 

The purpose of this paper is to establish strong and weak convergence theorems of a 

new three-step iteration for three asymptotically quasi-nonexpan sive non-self mappings in a 

uniformly convex Banach space. This scheme can be viewed as an extension of Xu and Noor 

[18], Suantai [16] and Nilsrakoo and Saejung [8]. 

Let X be a normed space. A subset C of X is said to be a retract of X if there exists a continuous 

map P : X → C such that Px = x for all x ∈ C. Every closed convex set of a uniformly 

convex Banach space is a retract. A map P : X → C is said to be a retraction if P 2 = P. It follows 

that if a map P is a retraction, then Py = y for all y in the range of P. A mapping T : C → X is 

said to be asymptotically quasi-nonexpansive if F(T) ̸= ∅ and there exists a sequence {k n } of real 

numbers with k n ≥ 1 and lim 

n→∞ 

k n = 1 such that 

‖T(PT) n−1 x − q‖ ≤ k n ‖x − q‖ 

for all x ∈ C, q ∈ F(T), n ≥ 1, where F(T) is the set of fixed points of T and (PT) 0 = I, the 

identity operator on C. 

The mapping T : C → X is called uniformly L-Lipschitzian if there exists a positive constant 

L such that 

‖T(PT) n−1 x − T(PT) n−1 y‖ ≤ L‖x − y‖ 

for all x, y ∈ C and n ∈ N. 

Let C be a nonempty closed convex subset of X and P : X → C a nonexpansive retraction 

of X onto C, and let T 1 , T 2 , T 3 : C → X be asymptotically quasi-nonexpansive mappings and F 

⋂ 

is the set of all common fixed points of T i i.e., F = 3 F(T i ), where F(T i ) = {x ∈ C : T i x = x} 

for all i = 1, 2, 3. Then, for arbitrary x 1 ∈ C, compute the sequences {x n }, {y n }, {z n } by the 

iterative scheme 

i=1 

z n = P[a n T 1 (PT 1 ) n−1 x n + (1 − a n − δ n )x n + δ n u n ], 

y n = 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 ], 

(1) 

x n+1 = P[α n T 3 (PT 3 ) n−1 y n + β n T 2 (PT 2 ) n−1 z n + γ n T 1 (PT 1 ) n−1 x n 

+ (1 − α n − β n − γ n − ρ n )x n + ρ n w n ] 

for all n ≥ 1, where {a n }, {b n }, {c n }, {α n }, {β n }, {γ n }, {δ n }, {σ n }, {ρ n } are appropriate sequences 

in [0, 1] and {u n }, {v n }, {w n } are bounded sequences in C. 

Without errors (δ n = σ n = ρ n ≡ 0), and T 1 , T 2 , T 3 are self-maps of C, the iterative scheme 

(1) reduces to the following iterative scheme: 

(2) 

z n = a n T n 1 x n + (1 − a n )x n , 

y n = b n T n 2 z n + c n T n 1 x n + (1 − b n − c n )x n , 

x n+1 = α n T n 3 y n + β n T n 2 z n + γ n T n 1 x n + (1 − α n − β n − γ n )x n , n ≥ 1, 

where {a n }, {b n }, {c n }, {α n }, {β n }, {γ n } are appropriate sequences in [0, 1]. 

If T := T 1 = T 2 = T 3 , then (2) reduces to the iterative scheme defined by Nilsrakoo and 

Saejung [8]. 

If γ n ≡ 0 and T := T 1 = T 2 = T 3 , then (2) reduces to the iterative scheme defined by 

Suantai [16]. 

If c n = β n = γ n ≡ 0 and T := T 1 = T 2 = T 3 , then (2) reduces to the iterative scheme 

defined by Xu and Noor [18].


(3) 

If a n = b n = c n ≡ 0, then (2) reduces to the following iterative scheme: 

x n+1 = α n T n 3 x n + β n T n 2 x n + γ n T n 1 x n + (1 − α n − β n − γ n )x n 

for all n ≥ 1, where {α n }, {β n }, {γ n } are appropriate sequences in [0, 1]. 

To study strong and weak convergence theorems of the iterative scheme 1, we recall 

some useful well-known concepts and results. 

Recall that a Banach space X is said to satisfy Opial’s condition [12] if for each sequence 

{x n } and x, y ∈ X with x n → x weakly as n → ∞ and x ̸= y imply that 

lim sup ‖x n − x‖ < lim sup ‖x n − y‖. 

n→∞ n→∞ 

In what follows, we shall make use of the following lemmas. 

Lemma 1.1. [17, Lemma 1]. Let {a n }, {b n }, {δ n } be sequences of nonnegative real numbers satisfying 

the inequality 

a n+1 ≤ (1 + δ n )a n + b n for all n = 1, 2, . . . . 

If 

∞ 

∑ δ n < ∞ and ∑ 

∞ b n < ∞, then 

n=1 

n=1 

(i) lim a n exists, and 

n→∞ 

(ii) lim a n = 0 whenever lim inf a n = 0. 

n→∞ n→∞ 

Lemma 1.2. [7, Lemma 1.4]. Let X be a uniformly convex Banach space and let B r = {x ∈ X : 

‖x‖ ≤ r}, r > 0 be a closed ball of X. Then there exists a continuous, strictly increasing convex 

function g : [0, ∞) → [0, ∞), g(0) = 0 such that 

‖λx + µy + ξz + ϑw‖ 2 ≤ λ‖x‖ 2 + µ‖y‖ 2 + ξ‖z‖ 2 + ϑ‖w‖ 2 − λµg ( ‖x − y‖ ) 

for all x, y, z, w ∈ B r and all λ, µ, ξ, ϑ ∈ [0, 1] with λ + µ + ξ + ϑ = 1. 

Similar to Lemma 1.2, we can prove the next lemma. 

Lemma 1.3. Let X be a uniformly convex Banach space and let B r be a closed ball of X. Then there 

exists a continuous, strictly increasing convex function g : [0, ∞) → [0, ∞), g(0) = 0 such that 

‖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. 

Lemma 1.4. [16, Lemma 2.7]. Let X be a Banach space which satisfies Opial’s condition and let {x n } 

be a sequence in X. Let u, v ∈ X be so that lim ‖x n − u‖ and lim ‖x n − v‖ exist. If {x nk } and {x mk } 

n→∞ n→∞ 

are subsequences of {x n } which converge weakly to u and v, respectively, then u = v. 

2. Main Results 

In this section, we prove strong and weak convergence theorems for the iterative scheme 

(1) for asymptotically quasi-nonexpansive nonself-mappings in a Banach space. In order to 

prove our main results, the following lemmas are needed. 

Lemma 2.1. 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, 

∞ 

∑ 

n=1 

(k n − 1) < ∞, 

∞ 

∑ 

n=1 

(l n − 1) < ∞ ∑ 

∞ (m n − 1) < ∞. Let 

{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 and α n + β n + γ n + ρ n are in [0, 1] for all n ≥ 1, 

∞ 

∑ 

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). Then 

(i) lim ‖x n − q‖ exists for all q ∈ F. 

n→∞ 

(ii) If one of the following conditions (a), (b), (c) and (d) holds, then 

lim ‖T 1(PT 1 ) n−1 x n − x n ‖ = 0. 

n→∞ 

(a) lim inf β n > 0 and 0 < lim inf a n ≤ lim sup(a n + δ n ) < 1. 

n→∞ n→∞ n→∞ 

(b) lim inf α n , lim inf b n > 0 and 0 < lim inf a n ≤ lim sup(a n + δ n ) < 1. 

n→∞ n→∞ n→∞ n→∞ 

(c) 0 < lim inf γ n ≤ lim sup(α n + β n + γ n + ρ n ) < 1. 

n→∞ n→∞ 

(d) 0 < lim inf α n and 0 < lim inf c n ≤ lim sup(b n + c n + σ n ) < 1. 

n→∞ n→∞ n→∞ 

(iii) If either (a) 0 < lim inf β n ≤ lim sup(α n + β n + γ n + ρ n ) < 1 

n→∞ n→∞ 

or (b) lim inf α n > 0 and 0 < lim inf b n ≤ lim sup(b n + c n + σ n ) < 1, 

n→∞ n→∞ n→∞ 

then lim ‖T 2 (PT 2 ) n−1 z n − x n ‖ = 0. 

n→∞ 

(iv) If 0 < lim inf α n ≤ lim sup(α n + β n + γ n + ρ n ) < 1, 

n→∞ n→∞ 

then lim ‖T 3 (PT 3 ) n−1 y n − x n ‖ = 0. 

n→∞ 

Proof. (i) Let q ∈ F. By (1), we obtain 

(4) 

‖z n − q‖ = ‖P[a n T 1 (PT 1 ) n−1 x n + (1 − a n − δ n )x n + δ n u n ] − P(q)‖ 

≤ a n ‖T 1 (PT 1 ) n−1 x n − q‖ + (1 − a n − δ n )‖x n − q‖ + δ n ‖u n − q‖ 

≤ (1 + a n (k n − 1) − δ n )‖x n − q‖ + δ n ‖u n − q‖ 

and 

(5) 

‖y n − q‖ = ‖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)‖ 

≤ 

≤ 

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 ‖v n − q‖ 

b n l n ‖z n − q‖ + c n k n ‖x n − q‖ + (1 − b n − c n − σ n )‖x n − q‖ 

+ σ n ‖v n − q‖. 

By (4) and (5), we obtain 

‖x n+1 − q‖ = ‖P[α n T 3 (PT 3 ) n−1 y n + β n T 2 (PT 2 ) n−1 z n + γ n T 1 (PT 1 ) n−1 x n 

+ (1 − α n − β n − γ n − ρ n )x n + ρ n w n ] − P(q)‖ 

≤ α n ‖T 3 (PT 3 ) n−1 y n − q‖ + β n ‖T 2 (PT 2 ) n−1 z n − q‖ 

+ γ n ‖T 1 (PT 1 ) n−1 x n − q‖ + (1 − α n − β n − γ n − ρ n )‖x n − q‖ 

+ ρ n ‖w n − q‖ 

≤ α n m n ‖y n − q‖ + β n l n ‖z n − q‖ + γ n k n ‖x n − q‖ 

+ (1 − α n − β n − γ n − ρ n )‖x n − q‖ + ρ n ‖w n − q‖ 

≤ (α n m n b n l n + β n l n )‖z n − q‖ + α n m n c n k n ‖x n − q‖ 

+ (α n m n − α n m n b n − α n m n c n − α n m n σ n )‖x n − q‖ 

+ α n m n σ n ‖v n − q‖ + γ n k n ‖x n − q‖ 

+ (1 − α n − β n − γ n − ρ n )‖x n − q‖ + ρ n ‖w n − q‖ 

≤ ‖x n − q‖ + ((l n − 1)(α n m n b n + β n ) + (k n − 1)(γ n + α n m n c n 

+ (α n m n b n l n + β n l n )a n ) + α n (m n − 1))‖x n − q‖ 

+ (m n l n + l n )δ n ‖u n − q‖ + m n σ n ‖v n − q‖ + ρ n ‖w n − q‖.


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 ‖ )


By (7) , (8) and (9), we obtain 

‖x n+1 − q‖ 2 = ‖P[α n T 3 (PT 3 ) n−1 y n + β n T 2 (PT 2 ) n−1 z n + γ n T 1 (PT 1 ) n−1 x n 

+ (1 − α n − β n − γ n − ρ n )x n + ρ n w n ] − P(q)‖ 2 

≤ α n ‖T 3 (PT 3 ) n−1 y n − q‖ 2 + β n ‖T 2 (PT 2 ) n−1 z n − q‖ 2 

+ γ n ‖T 1 (PT 1 ) n−1 x n − q‖ 2 + (1 − α n − β n − γ n − ρ n )‖x n − q‖ 2 

− α n (1 − α n − β n − γ n − ρ n )g ( ‖T 3 (PT 3 ) n−1 y n − x n ‖ ) 

+ ρ n ‖w n − q‖ 2 

≤ α n m 2 n‖y n − q‖ 2 + β n l 2 n‖z n − q‖ 2 + γ n k 2 n‖x n − q‖ 2 + ρ n ‖w n − q‖ 2 

+ (1 − α n − β n − γ n − ρ n )‖x n − q‖ 2 


≤ α n m 2 nc n k 2 n‖x n − q‖ 2 + α n m 2 n(1 − b n − c n − σ n )‖x n − q‖ 2 

+ γ n k 2 n‖x n − q‖ 2 + (1 − α n − β n − γ n − ρ n )‖x n − q‖ 2 

+ (α n m 2 nb n l 2 n + β n l 2 n)‖z n − q‖ 2 + α n m 2 nσ n ‖v n − q‖ 2 + ρ n ‖w n − q‖ 2 

− α n m 2 nb n (1 − b n − c n − σ n )g ( ‖T 2 (PT 2 ) n−1 z n − x n ‖ ) 


≤ α n m 2 nc n k 2 n‖x n − q‖ 2 + α n m 2 n(1 − b n − c n )‖x n − q‖ 2 

+ γ n k 2 n‖x n − q‖ 2 + (1 − α n − β n − γ n )‖x n − q‖ 2 + (α n m 2 nb n l 2 n 

+ β n l 2 n)‖x n − q‖ 2 + (α n m 2 nb n l 2 n + β n l 2 n)(a n (k 2 n − 1))‖x n − q‖ 2 

− (α n m 2 nb n l 2 n + β n l 2 n)a n (1 − a n − δ n )g ( ‖T 1 (PT 1 ) n−1 x n − x n ‖ ) 

+ (m 2 nl 2 n + l 2 n)δ n ‖u n − q‖ 2 + m 2 nσ n ‖v n − q‖ 2 + ρ n ‖w n − q‖ 2 



= ‖x n − q‖ 2 + ( (k 2 n − 1)(α n m 2 nc n + γ n + (α n m 2 nb n l 2 n + β n l 2 n)a n ) 

+ (l 2 n − 1)(α n m 2 nb n + β n ) + α n (m 2 n − 1) ) ‖x n − q‖ 2 


− α n m 2 nb n l 2 na n (1 − a n − δ n )g ( ‖T 1 (PT 1 ) n−1 x n − x n ‖ ) 

− β n l 2 na n (1 − a n − δ n )g ( ‖T 1 (PT 1 ) n−1 x n − x n ‖ ) 



Since {k n }, {l n }, {m n }, {u n }, {v n }, {w n } are bounded and {x n } is bounded, there exist constants 

K 0 > 0 such that 

(α n m 2 nc n + γ n + (α n m 2 nb n l 2 n + β n l 2 n)a n )‖x n − q‖ 2 ≤ K 0 , 

(α n m 2 nb n + β n )‖x n − q‖ 2 ≤ K 0 , α n ‖x n − q‖ 2 ≤ K 0 , (m 2 nl 2 n + l 2 n)‖u n − q‖ 2 ≤ K 0 , 

m 2 n‖v n − q‖ 2 ≤ K 0 and ‖w n − q‖ 2 ≤ K 0 for all n ≥ 1. Thus 

α n m 2 nb n l 2 na n (1 − a n − δ n )g ( ||T 1 (PT 1 ) n−1 x n − x n || ) ≤ ||x n − q|| 2 − ||x n+1 − q|| 2 

(10) 

(11) 

+ K 0 

( 

(k 

2 

n − 1) + (l 2 n − 1) + (m 2 n − 1) ) 

+ K 0 (δ n + σ n + ρ n ). 

β n l 2 na n (1 − a n − δ n )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 ).


α n m 2 nb n (1 − b n − c n − σ n )g ( ‖T 2 (PT 2 ) n−1 z n − x n ‖ ) ≤ ‖x n − q‖ 2 − ‖x n+1 − q‖ 2 

(12) 

+ K 0 

( 

(k 

2 

n − 1) + (l 2 n − 1) + (m 2 n − 1) ) 

+ K 0 (δ n + σ n + ρ n ). 

α n (1 − α n − β n − γ n − ρ n )g ( ‖T 3 (PT 3 ) n−1 y n − x n ‖ ) ≤ ‖x n − q‖ 2 − ‖x n+1 − q‖ 2 

(13) 

Again (7), we obtain 

(14) 

+ K 0 

( 

(k 

2 

n − 1) + (l 2 n − 1) + (m 2 n − 1) ) 

+ K 0 (δ n + σ n + ρ n ). 

‖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 

By (7), (8) and (14), we have 

≤ c n ‖T 1 (PT 1 ) n−1 x n − q‖ 2 + (1 − b n − c n − σ n )‖x n − q‖ 2 

+ b n ‖T 2 (PT 2 ) n−1 z n − q‖ 2 + σ n ‖v n − q‖ 2 

− c n (1 − b n − c n − σ n )g ( ‖T 1 (PT 1 ) n−1 x n − x n ‖ ) 

≤ c n k 2 n‖x n − q‖ 2 + (1 − b n − c n − σ n )‖x n − q‖ 2 + b n l 2 n‖z n − q‖ 2 

+ σ n ‖v n − q‖ 2 − c n (1 − b n − c n − σ n )g ( ‖T 1 (PT 1 ) n−1 x n − x n ‖ ) 



≤ α n ‖T 3 (PT 3 ) n−1 y n − q‖ 2 + β n ‖T 2 (PT 2 ) n−1 z n − q‖ 2 + ρ n ‖w n − q‖ 2 

+ γ n ‖T 1 (PT 1 ) n−1 x n − q‖ 2 + (1 − α n − β n − γ n − ρ n )‖x n − q‖ 2 

− β n (1 − α n − β n − γ n − ρ n )g ( ‖T 2 (PT 2 ) n−1 z n − x n ‖ ) 

≤ α n m 2 n‖y n − q‖ 2 + β n l 2 n‖z n − q‖ 2 + γ n k 2 n‖x n − q‖ 2 

+ ρ n ‖w n − q‖ 2 + (1 − α n − β n − γ n − ρ n )‖x n − q‖ 2 

+ β n (1 − α n − β n − γ n − ρ n )g ( ‖T 2 (PT 2 ) n−1 z n − x n ‖ ) 

≤ α n m 2 nc n k 2 n‖x n − q‖ 2 + α n m 2 n(1 − b n − c n − α n )‖x n − q‖ 2 

+ γ n k 2 n‖x n − q‖ 2 + (1 − α n − β n − γ n − ρ n )‖x n − q‖ 2 

+ (α n m 2 nb n l 2 n + β n l 2 n)‖z n − q‖ 2 + α n m 2 nσ n ‖v n − q‖ 2 + ρ n ‖w n − q‖ 2 

− α n m 2 nc n (1 − b n − c n − σ n )g ( ‖T 1 (PT 1 ) n−1 x n − x n ‖ ) 


≤ α n m 2 nc n k 2 n‖x n − q‖ 2 + α n m 2 n(1 − b n − c n )‖x n − q‖ 2 

+ γ n k 2 n‖x n − q‖ 2 + (1 − α n − β n − γ n )‖x n − q‖ 2 + (α n m 2 nb n l 2 n 

+ β n l 2 n)‖x n − q‖ 2 + (α n m 2 nb n l 2 n + β n l 2 n)(a n (k 2 n − 1))‖x n − q‖ 2 








− β n (1 − α n − β n − γ n − ρ n )g ( ‖T 2 (PT 2 ) n−1 z n − x n ‖ )


Thus 

α n m 2 nc n (1 − b n − c n − σ n )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) + (ln 2 − 1) + (m 2 n − 1) ) 

(15) 

+ K 0 (δ n + σ n + ρ n ). 

β n (1 − α n − β n − γ n − ρ n )g ( ‖T 2 (PT 2 ) n−1 z n − x n ‖ ) 

(16) 

By (7) , (8) and (9), we obtain 

≤ ‖x n − q‖ 2 − ‖x n+1 − q‖ 2 

( 

+ K 0 (k 

2 

n − 1) + (ln 2 − 1) + (m 2 n − 1) ) 

+ K 0 (δ n + σ n + ρ n ). 

Thus 



≤ γ n ‖T 1 (PT 1 ) n−1 x n − q‖ 2 + α n ‖T 3 (PT 3 ) n−1 y n − q‖ 2 + ρ n ‖w n − q‖ 2 

+ (1 − α n − β n − γ n − ρ n )‖x n − q‖ 2 + β n ‖T 2 (PT 2 ) n−1 z n − q‖ 2 

− γ n (1 − α n − β n − γ n − ρ n )g ( ‖T 1 (PT 1 ) n−1 x n − x n ‖ ) 

≤ γ n k 2 n‖x n − q‖ 2 + α n m 2 n‖y n − q‖ 2 + β n l 2 n‖z n − q‖ 2 

+ ρ n ‖w n − q‖ 2 + (1 − α n − β n − γ n − ρ n )‖x n − q‖ 2 


≤ γ n k 2 n‖x n − q‖ 2 + β n l 2 n‖z n − q‖ 2 + α n m 2 nσ n ‖v n − q‖ 2 + ρ n ‖w n − q‖ 2 

+ α n m 2 nb n l 2 n‖z n − q‖ 2 + α n m 2 n(1 − b n − c n − σ n )‖x n − q‖ 2 

+ α n m 2 nc n k 2 n‖x n − q‖ 2 + (1 − α n − β n − γ n − ρ n )‖x n − q‖ 2 


≤ γ n k 2 n‖x n − q‖ 2 + m 2 nσ n ‖v n − q‖ 2 + ρ n ‖w n − q‖ 2 

+ (α n m 2 nb n l 2 n + β n l 2 n)‖x n − q‖ 2 + (m 2 nl 2 n + l 2 n)δ n ‖u n − q‖ 2 

+ ( (α n m 2 nb n l 2 n + β n l 2 n)a n (k 2 n − 1) ) ‖x n − q‖ 2 + α n m 2 nc n k 2 n‖x n − q‖ 2 

+ (1 − α n − β n − γ n )‖x n − q‖ 2 + α n m 2 n(1 − b n − c n )‖x n − q‖ 2 





− γ n (1 − α n − β n − γ n − ρ n )g ( ‖T 1 (PT 1 ) n−1 x n − x n ‖ ) . 

γ n (1 − α n − β n − γ n − ρ n )g ( ‖T 1 (PT 1 ) n−1 x n − x n ‖ ) ≤ ‖x n − q‖ 2 − ‖x n+1 − q‖ 2 

(17) 

+ K 0 

( 

(k 

2 

n − 1) + (l 2 n − 1) + (m 2 n − 1) ) 

+ K 0 (δ n + σ n + ρ n ). 

(ii) (a) Let lim inf β n > 0 and 0 < lim inf a n ≤ lim sup(a n + δ n ) < 1. Then there exists 

n→∞ n→∞ n→∞ 

a positive integer n 0 and η, η ′ ∈ (0, 1) such that 0 < η < β n , 0 < η < a n and a n + δ n < η ′ < 1


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 



∞ 

∞ 

∞ 

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 → ∞,


It follows that, 

Hence (iii) is satisfied. 

‖T 3 x n − x n ‖ ≤ ‖T 3 (PT 3 ) n−1 x n − x n ‖ + ‖T 3 (PT 3 ) n−1 x n − T 3 x n ‖ 

≤ ‖T 3 (PT 3 ) n−1 x n − x n ‖ + L‖T 3 (PT 3 ) n−2 x n − x n ‖ 

−→ 0 as n → ∞. 

Theorem 2.3. Let X be a uniformly convex Banach space and C a nonempty closed convex nonexpansive 



F ̸= ∅, k n ≥ 1, l n ≥ 1, m n ≥ 1, 

∞ 

∑ 

n=1 

(k n − 1) < ∞, 

∞ 

∑ 

n=1 

(l n − 1) < ∞ and ∑ 

∞ (m n − 1) < ∞. 

Let {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 and α n + β n + γ n + ρ n are in [0, 1] for all n ≥ 1, and 

∞ ∑ 

n=1 

n=1 

δ n < ∞, 

∞ 

∑ 

n=1 

□ 

σ n < ∞, 

∞ 

∑ ρ n < ∞ and let {u n }, {v n }, {w n } be bounded sequences in C. Assume that T 1 , T 2 , T 3 are uniformly 

L-Lipschitzian. If one of T i (i = 1, 2, 3) is a completely continuous and one of the 

n=1 

following 

conditions (C1)-(C5) is satisfied: 

(C1) 0 < lim inf a n ≤ lim sup(a n + δ n ) < 1, 

n→∞ n→∞ 

0 < lim inf b n ≤ lim sup(b n + c n + σ n ) < 1, and 

n→∞ n→∞ 

0 < lim inf α n ≤ lim sup(α n + β n + γ n + ρ n ) < 1. 

n→∞ n→∞ 

(C2) 0 < lim inf b n , lim inf c n ≤ lim sup(b n + c n + σ n ) < 1, and 

n→∞ n→∞ n→∞ 

0 < lim inf α n ≤ lim sup(α n + β n + γ n + ρ n ) < 1. 

n→∞ n→∞ 

(C3) 0 < lim inf b n ≤ lim sup(b n + c n + σ n ) < 1, and 

n→∞ n→∞ 

0 < lim inf α n , lim inf γ n ≤ lim sup(α n + β n + γ n + ρ n ) < 1. 

n→∞ n→∞ n→∞ 

(C4) lim inf b n > 0, and 0 < lim inf a n ≤ lim sup(a n + δ n ) < 1, and 

n→∞ n→∞ n→∞ 

0 < lim inf α n , lim inf β n ≤ lim sup(α n + β n + γ n + ρ n ) < 1. 

n→∞ n→∞ n→∞ 

(C5) 0 < lim inf α n , lim inf β n , lim inf γ n ≤ lim sup(α n + β n + γ n + ρ n ) < 1. 

n→∞ n→∞ n→∞ n→∞ 

Then the sequences {x n }, {y n }, {z n } defined as in (1) converge strongly to a common fixed point of T 1 , 

T 2 and T 3 . 

Proof. Suppose one of the conditions (C1)-(C5) is satisfied. By Lemma 2.2, we obtain lim 

n→∞ 

‖T i x n − 

x n ‖ = 0 for i = 1, 2, 3. Assume one of T 1 , T 2 and T 3 says T 1 is completely continuous. Since 

{x n } is a bounded sequence in C, there exists a subsequence {x nk } of {x n } such that {T 1 x nk } 

converges to q ∈ C. Since ‖x nk − q‖ ≤ ‖T 1 x nk − x nk ‖ + ‖T 1 x nk − q‖, we get lim 

k→∞ 

‖x nk − q‖ = 0. 

Thus {x nk } converges to q ∈ C. By continuity of T i , we have T i x nk → T i q as k → ∞. Since 

‖T i q − q‖ ≤ ‖T i x nk − T i q‖ + ‖T i x nk − x nk ‖ + ‖x nk − q‖ → 0 as k → ∞, 

we obtain T i q = q (i = 1, 2, 3). Thus q ∈ F. By Lemma 2.1 (i), lim 

n→∞ 

‖x n − q‖ exists. This implies 

lim 

n→∞ ‖x n − q‖ = 0. By Lemma 2.1, we have 

It follows that 

‖T 1 (PT 1 ) n−1 x n − x n ‖ → 0 and ‖T 2 (PT 2 ) n−1 z n − x n ‖ → 0 as n → ∞. 

‖y n − x n ‖ ≤ b n ‖T 2 (PT 2 ) n−1 z n − x n ‖ + c n ‖T 1 (PT 1 ) n−1 x n − x n ‖ + σ n ‖v n − x n ‖ 

−→ 0 and ‖z n − x n ‖ ≤ a n ‖T 1 (PT 1 ) n−1 x n − x n ‖ + δ n ‖u n − x n ‖ → 0 as n → ∞. These imply 

lim y n = q and lim z n = q. 

□ 

n→∞ n→∞ 

Remark 2.4. In Theorem 2.3, assume that T 1 , T 2 and T 3 are asymptotically nonexpansive selfmappings 

of C such that one of them is completely continuous and thus T 1 , T 2 , T 3 are uniformly


L-Lipschitzian and δ n = σ n = ρ n ≡ 0. We obtain the following results. 

(1) If one of the conditions (C1) − (C5) is satisfied, then the sequences {x n }, 

{y n }, {z n } defined as in (2) converge strongly to a common fixed 

point of T 1 , T 2 and T 3 . 

(2) If T := T 1 = T 2 = T 3 and one of the conditions (C1) − (C5) is satisfied, 

then we obtain the results of Nilsrakoo and Saejung [8]. 

(3) If T := T 1 = T 2 = T 3 and one of the conditions (C1), (C2), (C4) is 

satisfied and γ n ≡ 0, then we obtain the results of Suantai [16]. 

(4) If T := T 1 = T 2 = T 3 with condition (C1) is satisfied and c n = β n = γ n 

≡ 0, then we obtain the results of Xu and Noor [18]. 

(5) If the condition (C5) is satisfied and a n = b n = c n ≡ 0, then the sequence 

{x n } defined as in (3) converges strongly to a common fixed point of 

T 1 , T 2 and T 3 . 

The mapping T : C → X with F(T) ̸= ∅ is said to satisfy Condition A [14] if there is a 

nondecreasing function f : [0, ∞) → [0, ∞) with f (0) = 0, f (r) > 0 for all r ∈ (0, ∞) such that 

‖x − Tx‖ ≥ f (d(x, F(T))) for all x ∈ C, where d(x, F(T)) = inf { ‖x − q‖ : q ∈ F(T) } . As Tan 

and Xu [17] pointed out, the Condition A is weaker than the compactness of C. 

The following result gives a strong convergence theorem for asymptotically quasi-nonexpansive 

nonself-mappings in a uniformly convex Banach space satisfying Condition A. 

Theorem 2.5. Let X be a uniformly convex Banach space and C a nonempty closed convex nonexpansive 



F ̸= ∅, k n ≥ 1, l n ≥ 1, m n ≥ 1, 

∞ 

∑ 

n=1 

(k n − 1) < ∞, 

∞ 

∑ 

n=1 

(l n − 1) < ∞ and ∑ 

∞ (m n − 1) < ∞. 

Let {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 and α n + β n + γ n + ρ n are in [0, 1] for all n ≥ 1 and 

∞ 

∑ 

n=1 

∞ ∑ 

n=1 

n=1 

δ n < ∞, 

∞ 

∑ 

n=1 

σ n < ∞, 

ρ n < ∞ and let {u n }, {v n }, {w n } be bounded sequences in C. Suppose T 1 satisfies Condition A 

and T 2 , T 3 are uniformly L-Lipschitzian and one of the conditions (C1)-(C5) in Theorem 2.3 is satisfied. 

Then the sequence {x n } defined as in (1) converges strongly to a common fixed point of T 1 , T 2 and T 3 . 

Proof. Let q ∈ F. By Lemma 2.1, lim 

n→∞ 

‖x n − q‖ exists. Thus {x n − q} is bounded. Then there 

is a constant H such that ‖x n − q‖ ≤ H for all n ≥ 1. This together with (6), we have 

(21) 

‖x n+1 − q‖ ≤ ‖x n − q‖ + D n , 

where D n = KH ( (k n − 1)+ (l n − 1) +(m n − 1) ) + K(δ n + σ n + ρ n ) < ∞ for all n ≥ 1. By 

Lemma 2.2, we have lim ‖x n − T i x n ‖ = 0 (i = 1, 2, 3). Since T 1 satisfies Condition A, we obtain 

n→∞ 

lim d(x n, F(T 1 )) = 0. Next, we want to show {x n } is a Cauchy sequence. Since lim d(x n , F(T 1 )) = 

n→∞ n→∞ 

0 and ∞ ∑ 

n=1 

D n < ∞, for any ɛ > 0, there exists a positive integer n 0 such that d(x n , F(T 1 )) < ɛ/4 

and ∑ 

n D k < ɛ/2 for all n ≥ n 0 . Now, let n ∈ N be such that n ≥ n 0 . Then we can find q ∗ ∈ F 

k=n 0 

such that ‖x n − q ∗ ‖ < ɛ/4. This implies by (21) that for m ≥ 1, 

‖x n+m − x n ‖ ≤ ‖x n+m − q ∗ ‖ + ‖x n − q ∗ ‖ 

≤ 2‖x n − q ∗ n+m−1 

‖ + 

∑ 

k=n 

= 2‖x n − q ∗ n+m−1 

‖ + 

∑ 

k=n 0 

D k 

D k < 2 ( ɛ ) ɛ + 

4 2 = ɛ.


This shows that {x n } is a Cauchy sequence and so it is convergent. Let lim 

n→∞ 

x n = p. Since 

d(x n , F(T 1 )) → 0 as n → ∞, it follows that d(p, F(T 1 )) = 0 and hence p ∈ F(T 1 ). Next, we 

want to show p ∈ F(T 2 ) ∩ F(T 3 ). Since T 2 , T 3 are uniformly L-Lipschitzian and by Lemma 2.2, 

we obtain 

‖T i p − p‖ 

≤ ‖T i x n − T i p‖ + ‖T i x n − x n ‖ + ‖x n − p‖ 

≤ L‖x n − p‖ + ‖T i x n − x n ‖ + ‖x n − p‖ → 0 as n → ∞. 

Thus T i p = p (i = 2, 3). Therefore p ∈ F. 

□ 

In the next result, we prove weak convergence for the iterative scheme (1) for asymptotically 

quasi-nonexpansive nonself-mappings in a uniformly convex Banach space satisfying 

Opial’s condition. 

Theorem 2.6. Let X be a uniformly convex Banach space which satisfies Opial’s condition and let C be 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, 

∞ 

∑ 

n=1 

(k n − 1) < ∞, 

∞ 

∑ (l n − 1) < ∞ and 

n=1 

∞ 

∑ (m n − 1) < ∞. Let {a n }, {b n }, {c n }, {α n }, {β n }, {γ n }, {δ n }, {σ n }, {ρ n } be real sequences in 

n=1 

[0, 1] such that a n + δ n , b n + c n + σ n and α n + β n + γ n + ρ n are in [0, 1] for all n ≥ 1 and ∞ ∑ 

∞, 

∞ 

∑ 

n=1 

σ n < ∞, 

n=1 

δ n < 

∞ 

∑ ρ n < ∞ and let {u n }, {v n }, {w n } be bounded sequences in C. Suppose T 1 , T 2 , T 3 

n=1 

are uniformly L-Lipschitzian and I − T i (i = 1, 2, 3) is demiclosed at 0. If one of the following conditions 

(C1)-(C5) in Theorem 2.3 is satisfied, then the sequence {x n } defined as in (1) converges weakly to a 

common fixed point of T 1 , T 2 and T 3 . 

Proof. Assume one of the conditions (C1)-(C5) is satisfied. By Lemma 2.1 and Lemma 2.2, 

we have lim ‖T i x n − x n ‖ = 0 (i = 1, 2, 3). Since X is uniformly convex and {x n } is bounded, 

n→∞ 

without loss of generality, we may assume that x n → u weakly as n → ∞. Since I − T i is 

demiclosed at 0, we obtain u ∈ F. Suppose subsequences {x nk } and {x mk } of {x n } converge 

weakly to u and v, respectively. Also, since I − T i (i = 1, 2, 3) is demiclosed at 0, we have u and 

v ∈ F. By Lemma 2.1, we obtain lim ‖x n − u‖ and lim ‖x n − v‖ exist. It follows from Lemma 

n→∞ n→∞ 

1.4 that u = v. Therefore {x n } converges weakly to a common fixed point of T 1 , T 2 and T 3 . □ 

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1,2 DEPARTMENT OF MATHEMATICS, STATISTICS AND COMPUTER, FACULTY OF SCIENCE,, UBON RATCHATHANI 

UNIVERSITY, UBON RATCHATHANI 34190, THAILAND. 

1,2 CENTRE OF EXCELLENCE IN MATHEMATICS, CHE, SI AYUTTHAYA RD., BANGKOK 10400, THAILAND. 

Email address: utith@sci.ubu.ac.th(U. Inprasit) and hathaikarn@sci.ubu.ac.th ( H. Wattanataweekul)

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

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