a modified mixed ishikawa iteration for common fixed points of two asymptotically quasi pseudocontractive type non self mappings

Motivated and inspired by the above results, in this paper, we introduce a new modified mixed Ishikawa iter-ative sequence with error for common fixed points of two more generalized asym

Research Article

A Modified Mixed Ishikawa Iteration for

Common Fixed Points of Two Asymptotically Quasi

Pseudocontractive Type Non-Self-Mappings

Yuanheng Wang and Huimin Shi

Department of Mathematics, Zhejiang Normal University, Zhejiang 321004, China

Correspondence should be addressed to Yuanheng Wang; wangyuanhengmath@163.com

Received 3 January 2014; Accepted 21 February 2014; Published 26 March 2014

Academic Editor: Rudong Chen

Copyright © 2014 Y Wang and H Shi This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

A new modified mixed Ishikawa iterative sequence with error for common fixed points of two asymptotically quasi pseudocontrac-tive type non-self-mappings is introduced By the flexible use of the iterapseudocontrac-tive scheme and a new lemma, some strong convergence theorems are proved under suitable conditions The results in this paper improve and generalize some existing results

1 Introduction

Let𝐸 be a real Banach space with its dual 𝐸∗and let𝐶 be a

nonempty, closed, and convex subset of𝐸 The mapping 𝐽 :

𝐸 → 2𝐸 ∗

is the normalized duality mapping defined by

𝐽 (𝑥) = {𝑥∗∈ 𝐸∗ : ⟨𝑥, 𝑥∗⟩ = ‖𝑥‖ ⋅ 󵄩󵄩󵄩󵄩𝑥∗󵄩󵄩󵄩󵄩,‖𝑥‖ = 󵄩󵄩󵄩󵄩𝑥∗󵄩󵄩󵄩󵄩},

𝑥 ∈ 𝐸 (1) Let𝑇 : 𝐶 → 𝐸 be a mapping We denote the fixed point

set of𝑇 by 𝐹(𝑇); that is, 𝐹(𝑇) = {𝑥 ∈ 𝐶 : 𝑥 = 𝑇𝑥} Recall that

a mapping𝑇 : 𝐶 → 𝐸 is said to be nonexpansive if, for each

𝑥, 𝑦 ∈ 𝐶,

󵄩󵄩󵄩󵄩𝑇𝑥 − 𝑇𝑦󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩 (2)

𝑇 is said to be asymptotically nonexpansive if there exists

a sequence𝑘𝑛 ⊆ [1, ∞) with 𝑘𝑛 → 1 as 𝑛 → ∞ such that

󵄩󵄩󵄩󵄩𝑇𝑛𝑥 − 𝑇𝑛𝑦󵄩󵄩󵄩󵄩 ≤ 𝑘𝑛󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩, ∀𝑥,𝑦 ∈ 𝐶 (3)

A sequence of self-mappings{𝑇𝑖}∞𝑖=1on𝐶 is said to be uniform

Lipschitzian with the coefficient𝐿 if, for any 𝑖 = 1, 2, , the

following holds:

󵄩󵄩󵄩󵄩𝑇𝑛

𝑖𝑥 − 𝑇𝑖𝑛𝑦󵄩󵄩󵄩󵄩 ≤ 𝐿󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩, ∀𝑥,𝑦 ∈ 𝐶 (4)

𝑇 is said to be asymptotically pseudocontractive if there exist𝑘𝑛 ⊆ [1, ∞) with 𝑘𝑛 → 1 as 𝑛 → ∞ and 𝑗(𝑥 − 𝑦) ∈ 𝐽(𝑥 − 𝑦) such that

⟨𝑇𝑛𝑥 − 𝑇𝑛𝑦, 𝑗 (𝑥 − 𝑦)⟩ ≤ 𝑘𝑛󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩2, ∀𝑥, 𝑦 ∈ 𝐶 (5)

It is obvious to see that every nonexpansive mapping

is asymptotically nonexpansive and every asymptotically nonexpansive mapping is asymptotically pseudocontractive Goebel and Kirk [1] introduced the class of asymptotically nonexpansive mappings in 1972 The class of asymptotically pseudocontractive mappings was introduced by Schu [2] and has been studied by various authors for its generalized mappings in Hilbert spaces, Banach spaces, or generalized topological vector spaces by using the modified Mann or Ishikawa iteration methods (see, e.g.,[3–21])

In 2003, Chidume et al [22] studied fixed points of an asymptotically nonexpansive non-self-mapping𝑇 : 𝐶 →

𝐸 and the strong convergence of an iterative sequence {𝑥𝑛} generated by

𝑥𝑛+1= 𝑃 ((1 − 𝛼𝑛) 𝑥𝑛+ 𝛼𝑛𝑇(𝑃𝑇)𝑛−1𝑥𝑛) , 𝑛 ≥ 1, 𝑥1∈ 𝐶,

(6) where𝑃 : 𝐸 → 𝐶 is a nonexpansive retraction

Abstract and Applied Analysis

Volume 2014, Article ID 129069, 7 pages

http://dx.doi.org/10.1155/2014/129069

In 2011, Zegeye et al [23] proved a strong convergence of

Ishikawa scheme to a uniformly L-Lipschitzian and

asymp-totically pseudocontractive mappings in the intermediate

sense which satisfies the following inequality (see [24]):

lim sup

𝑛 → ∞ sup

𝑥,𝑦∈𝐶(⟨𝑇𝑛𝑥 − 𝑇𝑛𝑦, 𝑥 − 𝑦⟩ − 𝑘𝑛󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩2) ≤ 0,

∀𝑥, 𝑦 ∈ 𝐶,

(7)

where𝑘𝑛⊆ [1, ∞) with 𝑘𝑛 → 1 as 𝑛 → ∞

Motivated and inspired by the above results, in this

paper, we introduce a new modified mixed Ishikawa

iter-ative sequence with error for common fixed points of two

more generalized asymptotically quasi pseudocontractive

type non-self-mappings By the flexible use of the iterative

scheme and a new lemma (i.e., Lemma 6 in this paper),

under suitable conditions, we prove some strong convergence

theorems Our results extend and improve many results of

other authors to a certain extent, such as [6,8,14–23]

2 Preliminaries

Definition 1 Let𝐶 be a nonempty closed convex subset of a

real Banach space𝐸 𝐶 is said to be a nonexpansive retract

(with𝑃) of 𝐸 if there exists a nonexpansive mapping 𝑃 :

𝐸 → 𝐶 such that, for all 𝑥 ∈ 𝐶, 𝑃𝑥 = 𝑥 And 𝑃 is called

a nonexpansive retraction

Let 𝑇 : 𝐶 → 𝐸 be a non-mapping (maybe

self-mapping) 𝑇 is called uniformly L-Lipschitzian (with 𝑃) if

there exists a constant𝐿 > 0 such that

󵄩󵄩󵄩󵄩

󵄩𝑇(𝑃𝑇)𝑛−1𝑥 − 𝑇(𝑃𝑇)𝑛−1𝑦󵄩󵄩󵄩󵄩󵄩 ≤ 𝐿󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩, ∀𝑥,𝑦 ∈ 𝐶, 𝑛 ≥ 1

(8)

𝑇 is said to be asymptotically pseudocontractive (with 𝑃)

if there exist𝑘𝑛 ⊆ [1, ∞) with 𝑘𝑛 → 1 as 𝑛 → ∞ and

∀𝑥, 𝑦 ∈ 𝐶, ∃𝑗(𝑥 − 𝑦) ∈ 𝐽(𝑥 − 𝑦) such that

⟨𝑇(𝑃𝑇)𝑛−1𝑥 − 𝑇(𝑃𝑇)𝑛−1𝑦, 𝑗 (𝑥 − 𝑦)⟩ ≤ 𝑘𝑛󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩2 (9)

𝑇 is said to be an asymptotically pseudocontractive type

(with𝑃) if there exist 𝑘𝑛 ⊆ [1, ∞) with 𝑘𝑛 → 1 as 𝑛 → ∞

and∀𝑥, 𝑦 ∈ 𝐶, 𝑗(𝑥 − 𝑦) ∈ 𝐽(𝑥 − 𝑦) such that

lim sup

𝑛 → ∞ sup

𝑥,𝑦∈𝐶 lim inf

𝑗(𝑥−𝑦)∈𝐽(𝑥−𝑦)(⟨𝑇(𝑃𝑇)𝑛−1𝑥 − 𝑇(𝑃𝑇)𝑛−1𝑦,

𝑗 (𝑥 − 𝑦)⟩ − 𝑘𝑛󵄩󵄩󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩󵄩2) ≤ 0

(10)

𝑇 is said to be an asymptotically quasi pseudocontractive

type (with𝑃) if 𝐹(𝑇) ̸= 0, for 𝑝 ∈ 𝐹(𝑇), there exist 𝑘𝑛⊆ [1, ∞)

with𝑘𝑛 → 1 as 𝑛 → ∞, and, ∀𝑥 ∈ 𝐶, 𝑗(𝑥 − 𝑝) ∈ 𝐽(𝑥 − 𝑝)

such that

lim sup

𝑛 → ∞ sup

𝑥∈𝐶 lim inf

𝑗(𝑥−𝑝)∈𝐽(𝑥−𝑝)(⟨𝑇(𝑃𝑇)𝑛−1𝑥 − 𝑝, 𝑗 (𝑥 − 𝑦)⟩

− 𝑘𝑛󵄩󵄩󵄩󵄩𝑥 − 𝑝󵄩󵄩󵄩󵄩2) ≤ 0

(11)

Remark 2 It is clear that every asymptotically

pseudocon-tractive mapping (with𝑃) is asymptotically pseudocontrac-tive type (with𝑃) and every asymptotically pseudocontrac-tive type (with𝑃) is asymptotically quasi pseudocontractive type (with𝑃) If 𝑇 : 𝐶 → 𝐶 is a self-mapping, then we can choose 𝑃 = 𝐼 as the identical mapping and we can get the usual definition of asymptotically pseudocontractive mapping, and so forth

Definition 3 Let 𝐶 be a nonexpansive retract (with 𝑃) of

𝐸, let 𝑇1, 𝑇2 : 𝐶 → 𝐸 be two uniformly L-Lipschitzian non-self-mappings and let 𝑇1 be an asymptotically quasi pseudocontractive type (with𝑃)

The sequence {𝑥𝑛} is called the new modified mixed Ishikawa iterative sequence with error (with 𝑃), if {𝑥𝑛} is generated by

𝑥𝑛+1= 𝑃 ((1 − 𝛼𝑛− 𝛾𝑛) 𝑥𝑛+ 𝛼𝑛𝑇1(𝑃𝑇1)𝑛−1

× ((1 − 𝛽𝑛) 𝑦𝑛+ 𝛽𝑛𝑇1(𝑃𝑇1)𝑛−1𝑦𝑛) + 𝛾𝑛𝑢𝑛) ,

𝑦𝑛= 𝑃 ((1 − 𝛼󸀠𝑛− 𝛾𝑛󸀠) 𝑥𝑛+ 𝛼󸀠𝑛𝑇2(𝑃𝑇2)𝑛−1

× ((1 − 𝛽𝑛󸀠) 𝑥𝑛+ 𝛽󸀠𝑛𝑇2(𝑃𝑇2)𝑛−1𝑥𝑛) + 𝛾𝑛󸀠V𝑛) ,

(12)

where𝑥1∈ 𝐶 is arbitrary, {𝑢𝑛} and {V𝑛} ⊂ 𝐶 are bounded, and

𝛼𝑛, 𝛽𝑛, 𝛾𝑛, 𝛼󸀠

𝑛, 𝛽󸀠

𝑛, 𝛾󸀠

𝑛∈ [0, 1], 𝑛 = 1, 2,

If𝛼󸀠

𝑛= 𝛽󸀠

𝑛= 𝛾󸀠

𝑛 = 0, (12) turns to

𝑥𝑛+1= 𝑃 ((1 − 𝛼𝑛− 𝛾𝑛) 𝑥𝑛+ 𝛼𝑛𝑇1(𝑃𝑇1)𝑛−1

× ((1 − 𝛽𝑛) 𝑥𝑛+ 𝛽𝑛𝑇1(𝑃𝑇1)𝑛−1𝑥𝑛) + 𝛾𝑛𝑢𝑛) , (13) and it is called the new modified mixed Mann iterative sequence with error (with𝑃)

If𝛾𝑛= 𝛾󸀠

𝑛= 0, (12) becomes

𝑥𝑛+1= 𝑃 ( (1 − 𝛼𝑛) 𝑥𝑛+ 𝛼𝑛𝑇1(𝑃𝑇1)𝑛−1

× ((1 − 𝛽𝑛) 𝑦𝑛+ 𝛽𝑛𝑇1(𝑃𝑇1)𝑛−1𝑦𝑛)) ,

𝑦𝑛= 𝑃 ( (1 − 𝛼𝑛󸀠) 𝑥𝑛+ 𝛼𝑛󸀠𝑇2(𝑃𝑇2)𝑛−1

× ((1 − 𝛽󸀠𝑛) 𝑥𝑛+ 𝛽󸀠𝑛𝑇2(𝑃𝑇2)𝑛−1𝑥𝑛)) ,

(14)

and it is called the new modified mixed Ishikawa iterative sequence (with𝑃)

If𝛽𝑛 = 𝛽󸀠

𝑛= 0, (14) turns to

𝑥𝑛+1= 𝑃 ((1 − 𝛼𝑛) 𝑥𝑛+ 𝛼𝑛𝑇1(𝑃𝑇1)𝑛−1𝑦𝑛) ,

𝑦𝑛= 𝑃 ((1 − 𝛼𝑛󸀠) 𝑥𝑛+ 𝛼𝑛󸀠𝑇2(𝑃𝑇2)𝑛−1𝑥𝑛) , (15) and it is called the new mixed Ishikawa iterative sequence (with𝑃)

If𝑇1 = 𝑇2 = 𝑇 : 𝐶 → 𝐶 is a self-mapping and 𝑃 = 𝐼 is the identical mapping, then (15) is just the modified Ishikawa iterative sequence

𝑥𝑛+1= (1 − 𝛼𝑛) 𝑥𝑛+ 𝛼𝑛𝑇𝑛𝑦𝑛,

𝑦𝑛= (1 − 𝛼𝑛󸀠) 𝑥𝑛+ 𝛼󸀠𝑛𝑇𝑛𝑥𝑛 (16)

𝑛 = 0, (15) becomes (6), obviously So, iterative method

(12) is greatly generalized

The following lemmas will be needed in what follows to

prove our main results

Lemma 4 (see [19]) Let 𝐸 be a real Banach space Then, for

all 𝑥, 𝑦 ∈ 𝐸, 𝑗(𝑥+𝑦) ∈ 𝐽(𝑥+𝑦), the following inequality holds:

󵄩󵄩󵄩󵄩𝑥 + 𝑦󵄩󵄩󵄩󵄩2≤ ‖𝑥‖2+ 2 ⟨𝑥, 𝑗 (𝑥 + 𝑦)⟩ (17)

Lemma 5 (see [6,7]) Let{𝑎𝑛}, {𝑏𝑛}, {𝑐𝑛} be three sequences of

nonnegative numbers satisfying the recursive inequality:

𝑎𝑛+1≤ (1 + 𝑏𝑛) 𝑎𝑛+ 𝑐𝑛, ∀𝑛 ≥ 𝑛0, (18)

where𝑛0is some nonnegative integer IfΣ∞

𝑛=1𝑏𝑛< ∞, Σ∞

𝑛=1𝑐𝑛<

∞, then lim𝑛 → ∞𝑎𝑛exists.

Lemma 6 Suppose that 𝜙 : [0, +∞) → [0, +∞) is a strictly

increasing function with 𝜙(0) = 0 Let {𝑎𝑛}, {𝑏𝑛}, {𝑐𝑛}, {𝜆𝑛} (0 ≤

𝜆𝑛 ≤ 1) be four sequences of nonnegative numbers satisfying

the recursive inequality:

𝑎𝑛+1≤ (1 + 𝑏𝑛) 𝑎𝑛− 𝜆𝑛𝜙 (𝑎𝑛+1) + 𝑐𝑛, ∀𝑛 ≥ 𝑛0, (19)

where𝑛0is some nonnegative integer IfΣ∞

𝑛=1𝑏𝑛< ∞, Σ∞

𝑛=1𝑐𝑛<

∞, Σ∞

𝑛=1𝜆𝑛= ∞, then lim𝑛 → ∞𝑎𝑛= 0.

Proof From (19), we get

𝑎𝑛+1≤ (1 + 𝑏𝑛) 𝑎𝑛+ 𝑐𝑛, ∀𝑛 ≥ 𝑛0 (20)

By Lemma5, we know that lim𝑛 → ∞𝑎𝑛 = 𝑎 ≥ 0 exists Let

𝑀 = sup1≤𝑛≤∞{𝑎𝑛} < ∞ Now we show 𝑎 = 0 Otherwise, if

𝑎 > 0, then ∃𝑛1 ≥ 𝑛0, such that𝑎𝑛+1≥ (1/2)𝑎 > 0 when 𝑛 ≥

𝑛1 Because𝜙 is a strictly increasing function, so 𝜙(𝑎𝑛+1) ≥

𝜙((1/2)𝑎) > 0 From (19) again, we have

0 < 𝜙 (1

2𝑎)

∞

∑

𝑛=1

𝜆𝑛

= 𝜙 (1

2𝑎)

𝑛 1

∑

𝑛=1

𝜆𝑛+ 𝜙 (1

2𝑎)

∞

∑

𝑛=𝑛 1 +1

𝜆𝑛

≤ 𝜙 (12𝑎)∑𝑛1

𝑛=1𝜆𝑛+ ∑∞

𝑛=𝑛1+1𝜆𝑛𝜙 (𝑎𝑛+1)

≤ 𝜙 (12𝑎)∑𝑛1

𝑛=1

𝜆𝑛+ ∑∞

𝑛=𝑛 1 +1

(𝑎𝑛− 𝑎𝑛+1)

+ ∑∞

𝑛=𝑛 1 +1

𝑏𝑛𝑎𝑛+ ∑∞

𝑛=𝑛 1 +1

𝑐𝑛

≤ 𝜙 (1

2𝑎)

𝑛 1

∑

𝑛=1

𝜆𝑛+ 𝑎𝑛1+1+ 𝑀∑∞

𝑛=1

𝑏𝑛+∑∞

𝑛=1

𝑐𝑛< ∞

(21)

This is a contradiction with the given conditionΣ∞𝑛=1𝜆𝑛= ∞

Therefore lim𝑛 → ∞𝑎𝑛 = 0

Lemma 7 Suppose that 𝜙 : [0, +∞) → [0, +∞) is a strictly

increasing function with 𝜙(0) = 0 Let {𝑎𝑛}, {𝑏𝑛}, {𝑐𝑛}, {𝜆𝑛} (0 ≤

𝜆𝑛≤ 1), {𝜀𝑛} be five sequences of nonnegative numbers satisfy-ing the recursive inequality:

𝑎𝑛+1≤ (1 + 𝑏𝑛) 𝑎𝑛− 𝜆𝑛𝜙 (𝑎𝑛+1) + 𝑐𝑛+ 𝜆𝑛𝜀𝑛, ∀𝑛 ≥ 𝑛0,

(22)

where𝑛0is some nonnegative integer IfΣ∞𝑛=1𝑏𝑛< ∞, Σ∞𝑛=1𝑐𝑛<

∞, Σ∞𝑛=1𝜆𝑛= ∞, lim𝑛 → ∞𝜀𝑛= 0, then lim𝑛 → ∞𝑎𝑛= 0 Proof Firstly, we show lim inf𝑛 → ∞𝑎𝑛= 𝑎 = 0 If 𝑎 > 0, then, for arbitrary𝑟 ∈ (0, 𝑎), ∃𝑛1 ≥ 𝑛0, such that𝑎𝑛+1 ≥ 𝑟 > 0 when𝑛 ≥ 𝑛1 Because𝜙 is a strictly increasing function and lim𝑛 → ∞𝜀𝑛 = 0, so 𝜙(𝑎𝑛+1) ≥ 𝜙(𝑟) > 0 and 𝜀𝑛 ≤ (1/2)𝜙(𝑟) when𝑛 ≥ 𝑛1 From (22), we have

𝑎𝑛+1≤ (1 + 𝑏𝑛) 𝑎𝑛− 𝜆𝑛𝜙 (𝑎𝑛+1) + 𝑐𝑛+ 𝜆𝑛1

2𝜙 (𝑎𝑛+1)

= (1 + 𝑏𝑛) 𝑎𝑛−1

2𝜆𝑛𝜙 (𝑎𝑛+1) + 𝑐𝑛, ∀𝑛 ≥ 𝑛1.

(23)

By Lemma6, we get0 = lim𝑛 → ∞𝑎𝑛 = lim inf𝑛 → ∞𝑎𝑛= 𝑎 > 0 This is contradictory So, lim inf𝑛 → ∞𝑎𝑛 = 0

Secondly,∀𝜀 > 0, from the given conditions in Lemma7,

∃𝑛2≥ 𝑛0, when∀𝑛 ≥ 𝑛2, we have

𝜀𝑛≤ 𝜙 (𝜀) , ∑∞

𝑛=𝑛 2

𝑏𝑛≤ ln 2, ∑∞

𝑛=𝑛 2

𝑐𝑛≤ 𝜀 (24)

On the other hand, since lim inf𝑛 → ∞𝑎𝑛 = 0, ∃𝑁 ≥ 𝑛2 such that𝑎𝑁≤ 𝜀 Now we claim

𝑎𝑘≤ (𝜀 + 𝑘−1∑

𝑛=𝑁

𝑐𝑛) exp (𝑘−1∑

𝑛=𝑁

𝑏𝑛) , ∀𝑘 ≥ 𝑁 (25)

In fact, when𝑘 = 𝑁, (25) holds Suppose that (25) holds for

𝑘 dose not for 𝑘 + 1 Then

𝑎𝑘+1> (𝜀 + ∑𝑘

𝑛=𝑁

𝑐𝑛) exp (∑𝑘

𝑛=𝑁

𝑏𝑛) (26)

Furthermore,𝑎𝑘+1> 𝜀, 𝜙(𝑎𝑘+1) > 𝜙(𝜀) But by (22), (24), and the inductive hypothesis, we have

𝑎𝑛+1≤ (1 + 𝑏𝑛) 𝑎𝑛− 𝜆𝑛𝜙 (𝑎𝑛+1) + 𝑐𝑛+ 𝜆𝑛𝜀𝑛

≤ (1 + 𝑏𝑛) 𝑎𝑛− 𝜆𝑛𝜙 (𝜀) + 𝑐𝑛+ 𝜆𝑛𝜙 (𝜀)

≤ (1 + 𝑏𝑛) (𝜀 +𝑘−1∑

𝑛=𝑁

𝑐𝑛) exp (𝑘−1∑

𝑛=𝑁

𝑏𝑛) + 𝑐𝑛

≤ (𝜀 + 𝑘−1∑

𝑛=𝑁

𝑐𝑛) exp (∑𝑘

𝑛=𝑁

𝑏𝑛) + 𝑐𝑛

≤ (𝜀 + ∑𝑘

𝑛=𝑁

𝑐𝑛) exp (∑𝑘

𝑛=𝑁

𝑏𝑛)

(27)

This is a contradiction with (26) So, (25) holds Whereupon,

lim sup

𝑘 → ∞ 𝑎𝑘≤ (𝜀 + ∑∞

𝑛=𝑁

𝑐𝑛) exp (∑∞

𝑛=𝑁

𝑏𝑛)

≤ 2 (𝜀 + 𝜀) = 4𝜀

(28)

Therefore, lim sup𝑘 → ∞𝑎𝑘= 0 = lim𝑛 → ∞𝑎𝑛

3 Main Results

Now, we are in a position to state and prove the main results

of this paper

Theorem 8 Let 𝐶 be nonexpansive retract (with 𝑃) of a

real Banach space 𝐸 Assume that 𝑇1, 𝑇2 : 𝐶 → 𝐸 are

two uniformly L-Lipschitzian non-self-mappings (with 𝑃) and

𝑇1 is an asymptotically quasi pseudocontractive type with

coefficient numbers {𝑘𝑛} ⊂ [1, +∞) : 𝑘𝑛 → 1 satisfying

𝐹 = 𝐹(𝑇1) ∩ 𝐹(𝑇2) ̸= 0 Suppose that {𝑢𝑛}, {V𝑛} ⊂ 𝐶 are two

bounded sequences;{𝛼𝑛}, {𝛽𝑛}, {𝛾𝑛}, {𝛼󸀠𝑛}, {𝛽󸀠𝑛}, {𝛾𝑛󸀠} ⊂ [0, 1] are

six number sequences satisfying the following:

(C1)Σ∞

𝑛=1𝛼𝑛 = +∞, Σ∞

𝑛=1𝛼2

𝑛 < +∞, Σ∞

𝑛=1𝛼𝑛(𝑘𝑛− 1) < +∞;

(C2)𝛼𝑛+ 𝛾𝑛≤ 1, 𝛼󸀠

𝑛+ 𝛾󸀠

𝑛 ≤ 1, Σ∞ 𝑛=1𝛾𝑛< +∞;

(C3)Σ∞

𝑛=1𝛼𝑛𝛽𝑛< +∞, Σ∞

𝑛=1𝛼𝑛𝛼󸀠

𝑛< +∞, Σ∞

𝑛=1𝛼𝑛𝛾󸀠

𝑛< +∞

If𝑥1∈ 𝐶 is arbitrary, then the iterative sequence {𝑥𝑛} generated

by (12) converges strongly to the fixed point𝑥∗ ∈ 𝐹 if and only

if there exists a strictly increasing function𝜙 : [0, +∞) →

[0, +∞) with 𝜙(0) = 0 such that

lim sup

𝑗(𝑥 𝑛+1 −𝑥 ∗)∈𝐽(𝑥 𝑛+1 −𝑥 ∗)[⟨𝑇1(𝑃𝑇1)

𝑛−1𝑥𝑛+1− 𝑥∗,

𝑗 (𝑥𝑛+1− 𝑥∗) ⟩− 𝑘𝑛󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩2

+ 𝜙 (󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩)] ≤ 0

(29)

Proof (Adequacy) Let

𝜀󸀠𝑛= inf

𝑗(𝑥 𝑛+1 −𝑥 ∗)∈𝐽(𝑥 𝑛+1 −𝑥 ∗)[⟨𝑇1(𝑃𝑇1)

𝑛−1𝑥𝑛+1− 𝑥∗,

𝑗 (𝑥𝑛+1− 𝑥∗) ⟩ − 𝑘𝑛󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩2

+ 𝜙 (󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩)],

𝜀𝑛 = max {𝜀𝑛󸀠, 0} +1𝑛

(30) Then there exists𝑗(𝑥𝑛+1− 𝑥∗) ∈ 𝐽(𝑥𝑛+1− 𝑥∗) such that

⟨𝑇1(𝑃𝑇1)𝑛−1𝑥𝑛+1− 𝑥∗, 𝑗 (𝑥𝑛+1− 𝑥∗)⟩

− 𝑘𝑛󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩2

+ 𝜙 (󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩) ≤ 𝜀𝑛 (31)

From (29), we know that lim sup𝑛 → ∞𝜀󸀠

𝑛 ≤ 0 So, lim𝑛 → ∞𝜀𝑛= 0

Now, from the given conditions and (12), we can let

𝜎𝑛= (1 − 𝛽𝑛) 𝑦𝑛+ 𝛽𝑛𝑇1(𝑃𝑇1)𝑛−1𝑦𝑛,

𝛿𝑛= (1 − 𝛽𝑛󸀠) 𝑥𝑛+ 𝛽󸀠𝑛𝑇2(𝑃𝑇2)𝑛−1𝑥𝑛, (32) and𝑀 = sup𝑛≥1{‖𝜇𝑛− 𝑥∗‖, ‖]𝑛− 𝑥∗‖} < ∞ Then

󵄩󵄩󵄩󵄩𝛿𝑛− 𝑥∗󵄩󵄩󵄩󵄩 ≤ 𝛽󸀠

𝑛󵄩󵄩󵄩󵄩𝑇2(𝑃𝑇2) 𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩 + (1 − 𝛽󸀠

𝑛) 󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩

≤ 𝛽󸀠𝑛𝐿 󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩;

󵄩󵄩󵄩󵄩𝑦𝑛− 𝑥∗󵄩󵄩󵄩󵄩 ≤ (1 − 𝛼󸀠

𝑛− 𝛾𝑛󸀠) 󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩

+ 𝛼𝑛󸀠𝐿 󵄩󵄩󵄩󵄩𝛿𝑛− 𝑥∗󵄩󵄩󵄩󵄩 + 𝛾󸀠

𝑛󵄩󵄩󵄩󵄩]𝑛− 𝑥∗󵄩󵄩󵄩󵄩

≤ 󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩 + 𝛼󸀠

𝑛𝛽󸀠

𝑛𝐿2󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩

+ 𝛼𝑛󸀠𝐿 󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩 + 𝛾󸀠

𝑛𝑀

= (1 + 𝛼󸀠

𝑛𝛽󸀠

𝑛𝐿2+ 𝛼󸀠

𝑛𝐿) 󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩 + 𝛾󸀠

𝑛𝑀

≤ (1 + 𝐿 + 𝐿2) 󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩 + 𝑀;

󵄩󵄩󵄩󵄩𝜎𝑛− 𝑥∗󵄩󵄩󵄩󵄩 ≤ 𝛽𝑛󵄩󵄩󵄩󵄩󵄩𝑇1(𝑃𝑇1)𝑛−1𝑦𝑛− 𝑥∗󵄩󵄩󵄩󵄩󵄩

+ (1 − 𝛽𝑛) 󵄩󵄩󵄩󵄩𝑦𝑛− 𝑥∗󵄩󵄩󵄩󵄩

≤ 𝛽𝑛𝐿 󵄩󵄩󵄩󵄩𝑦𝑛− 𝑥∗󵄩󵄩󵄩󵄩 + 󵄩󵄩󵄩󵄩𝑦𝑛− 𝑥∗󵄩󵄩󵄩󵄩

≤ (1 + 𝐿) (1 + 𝐿 + 𝐿2) 󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩 + (1 + 𝐿)𝑀;

󵄩󵄩󵄩󵄩𝑦𝑛− 𝑥𝑛+1󵄩󵄩󵄩󵄩 ≤ 𝛼𝑛𝐿 󵄩󵄩󵄩󵄩𝜎𝑛− 𝑥∗󵄩󵄩󵄩󵄩 + 𝛼𝑛󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩

+ 𝛼𝑛󸀠𝐿 󵄩󵄩󵄩󵄩𝛿𝑛− 𝑥∗󵄩󵄩󵄩󵄩 + 𝛼󸀠

𝑛󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩

+ (𝛾𝑛+ 𝛾𝑛󸀠) 󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩 + (𝛾𝑛+ 𝛾𝑛󸀠) 𝑀

≤ 𝛼𝑛𝐿 [(1 + 𝐿) (1 + 𝐿 + 𝐿2) 󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩

+ (1 + 𝐿) 𝑀]

+ 𝛼𝑛󸀠𝐿 [(1 + 𝛽󸀠𝑛𝐿) 󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩]

+ (𝛼𝑛+ 𝛼𝑛󸀠+ 𝛾𝑛+ 𝛾𝑛󸀠) 󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩 + (𝛾𝑛+ 𝛾𝑛󸀠) 𝑀

≤ [𝛼𝑛𝐿 (1 + 𝐿) (1 + 𝐿 + 𝐿2) + 𝛼󸀠𝑛𝐿 (1 + 𝛽𝑛󸀠𝐿) + 𝛼𝑛+ 𝛼󸀠

𝑛+ 𝛾𝑛+ 𝛾󸀠

𝑛] 󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩

+ (𝛼𝑛𝐿 (1 + 𝐿) + 𝛾𝑛+ 𝛾𝑛󸀠)𝑀;

󵄩󵄩󵄩󵄩𝜎𝑛− 𝑥𝑛+1󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩󵄩𝑦𝑛− 𝑥𝑛+1󵄩󵄩󵄩󵄩 + 𝛽𝑛󵄩󵄩󵄩󵄩󵄩𝑇1(𝑃𝑇1)𝑛−1𝑦𝑛− 𝑦𝑛󵄩󵄩󵄩󵄩󵄩

≤ 𝑠𝑛󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩 + 𝑡𝑛,

(33)

𝑠𝑛= 𝛼𝑛𝐿 (1 + 𝐿) (1 + 𝐿 + 𝐿2) + 𝛼󸀠𝑛𝐿 (1 + 𝛽𝑛󸀠𝐿)+ 𝛼𝑛

+ 𝛼𝑛󸀠+ 𝛾𝑛+ 𝛾󸀠𝑛+ 𝛽𝑛(1 + 𝐿) (1 + 𝐿 + 𝐿2) ;

𝑡𝑛 = [𝛼𝑛𝐿 (1 + 𝐿) + 𝛾𝑛+ 𝛾𝑛󸀠+ 𝛽𝑛(1 + 𝐿)] 𝑀

(34)

So, by Lemma4,

2𝛼𝑛⟨𝑇1(𝑃𝑇1)𝑛−1𝜎𝑛− 𝑇1(𝑃𝑇1)𝑛−1𝑥𝑛+1, 𝑗 (𝑥𝑛+1− 𝑥∗)⟩

≤ 2𝛼𝑛𝐿 󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩𝜎𝑛− 𝑥𝑛+1󵄩󵄩󵄩󵄩

≤ 2𝛼𝑛𝐿 󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩[𝑠𝑛󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩 + 𝑡𝑛] ;

(35)

󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩2

≤ (1 − 𝛼𝑛− 𝛾𝑛)2󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩2

+ 2𝛼𝑛⟨𝑇1(𝑃𝑇1)𝑛−1𝜎𝑛− 𝑥∗, 𝑗 (𝑥𝑛+1− 𝑥∗)⟩

+ 2𝛾𝑛⟨𝜇𝑛− 𝑥∗, 𝑗 (𝑥𝑛+1− 𝑥∗)⟩

≤ (1 − 𝛼𝑛− 𝛾𝑛)2󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩2

+ 2𝛼𝑛⟨𝑇1(𝑃𝑇1)𝑛−1𝜎𝑛− 𝑇1(𝑃𝑇1)𝑛−1𝑥𝑛+1,

𝑗 (𝑥𝑛+1− 𝑥∗) ⟩

+ 2𝛼𝑛⟨𝑇1(𝑃𝑇1)𝑛−1𝑥𝑛+1− 𝑥∗, 𝑗 (𝑥𝑛+1− 𝑥∗)⟩

+ 2𝛾𝑛𝑀 󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩

(36)

For the third in (36), we have

2𝛼𝑛⟨𝑇1(𝑃𝑇1)𝑛−1𝑥𝑛+1− 𝑥∗, 𝑗 (𝑥𝑛+1− 𝑥∗)⟩

= 2𝛼𝑛𝑑𝑛+ 2𝛼𝑛[𝑘𝑛󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩2

− 𝜙 (󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩)]

≤ 2𝛼𝑛𝜀𝑛+ 2𝛼𝑛[𝑘𝑛󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩2

− 𝜙 (󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩)],

(37)

where

𝑑𝑛= ⟨𝑇1(𝑃𝑇1)𝑛−1𝑥𝑛+1− 𝑥∗, 𝑗 (𝑥𝑛+1− 𝑥∗)⟩

− 𝑘𝑛󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩2

+ 𝜙 (󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩) ≤ 𝜀𝑛 (38) Substituting (35) into (36), we get

󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩2≤ (1 − 𝛼𝑛)2󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩2+ 2𝛼𝑛𝜀𝑛

+ 2𝛼𝑛𝑘𝑛󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩2− 2𝛼𝑛𝜙 (󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩) + 2𝛼𝑛𝐿 (𝑠𝑛󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩 + 𝑡𝑛) 󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩

+ 2𝛾𝑛𝑀 󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩

(39)

Let𝑎𝑛 = ‖𝑥𝑛− 𝑥∗‖2,𝜑(𝑡) = 2𝜙(√𝑡), and

𝜉𝑛 = 𝐿𝛼𝑛𝑠𝑛

= 𝐿2𝛼2

𝑛(1 + 𝐿) (1 + 𝐿 + 𝐿2) + 𝛼𝑛𝛼𝑛󸀠𝐿2(1 + 𝛽𝑛󸀠𝐿)+ 𝛼2𝑛𝐿 + 𝛼𝑛𝛼𝑛󸀠𝐿 + 𝐿𝛼𝑛𝛾𝑛 + 𝐿𝛼𝑛𝛾𝑛󸀠+ 𝐿𝛼𝑛𝛽𝑛(1 + 𝐿) (1 + 𝐿 + 𝐿2) ,

(40)

𝜌𝑛 = 𝐿𝛼𝑛𝑡𝑛+ 𝑀𝛾𝑛

= [𝛼2𝑛𝐿2(1 + 𝐿) + 𝐿𝛼𝑛𝛾𝑛+ 𝐿𝛼𝑛𝛾𝑛󸀠+ 𝛼𝑛𝛽𝑛(𝐿 + 𝐿2) ] 𝑀 + 𝛾𝑛𝑀

(41) Then (39) becomes

𝑎𝑛+1≤ (1 − 𝛼𝑛)2𝑎𝑛+ 2𝛼𝑛𝜀𝑛+ 2𝛼𝑛𝑘𝑛𝑎𝑛+1− 𝛼𝑛𝜑 (𝑎𝑛+1) + 2 (𝜉𝑛󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩 + 𝜌𝑛) 󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩 (42)

By using2𝑎𝑏 ≤ 𝑎2+ 𝑏2, we have

𝑎𝑛+1≤ (1 − 𝛼𝑛)2𝑎𝑛+ 2𝛼𝑛𝜀𝑛+ 2𝛼𝑛𝑘𝑛𝑎𝑛+1

− 𝛼𝑛𝜑 (𝑎𝑛+1) + 𝜉𝑛(𝑎𝑛+ 𝑎𝑛+1) + 𝜌𝑛(1 + 𝑎𝑛+1)

= (1 − 2𝛼𝑛+ 𝛼2𝑛+ 𝜉𝑛) 𝑎𝑛+ (2𝛼𝑛𝑘𝑛+ 𝜉𝑛+ 𝜌𝑛) 𝑎𝑛+1

− 𝛼𝑛𝜑 (𝑎𝑛+1) + 2𝛼𝑛𝜀𝑛+ 𝜌𝑛

(43)

From (40), (41), and the given conditions, we know

∞

∑

𝑛=1

𝛼2𝑛< +∞, ∑∞

𝑛=1

𝜉𝑛 < +∞, ∑∞

𝑛=1

𝜌𝑛 < +∞ (44)

Then, lim𝑛 → ∞(2𝛼𝑛𝑘𝑛 + 𝜉𝑛 + 𝜌𝑛) = 0 Therefore ∃𝑛0, when

𝑛 ≥ 𝑛0,2𝛼𝑛𝑘𝑛+ 𝜉𝑛+ 𝜌𝑛≤ 1/2 Let

𝑏𝑛 = 1 − 2𝛼1 − 2𝛼𝑛+ 𝛼2𝑛+ 𝜉𝑛

𝑛𝑘𝑛− 𝜉𝑛− 𝜌𝑛 − 1 =

2𝛼𝑛(𝑘𝑛− 1) + 𝛼𝑛2+ 2𝜉𝑛+ 𝜌𝑛

1 − 2𝛼𝑛𝑘𝑛− 𝜉𝑛− 𝜌𝑛 ;

𝑐𝑛 = 1 − 2𝛼 𝜌𝑛

𝑛𝑘𝑛− 𝜉𝑛− 𝜌𝑛.

(45)

So, when𝑛 ≥ 𝑛0, we get

0 ≤ 𝑏𝑛≤ 2 [2𝛼𝑛(𝑘𝑛− 1) + 𝛼2𝑛+ 2𝜉𝑛+ 𝜌𝑛] , 0 ≤ 𝑐𝑛≤ 2𝜌𝑛

(46) From (44) and the given conditions, we have∑∞𝑛=𝑛0𝑏𝑛 < +∞,

∑∞𝑛=𝑛0𝑐𝑛< +∞ On the other hand, from (43), we have

𝑎𝑛+1≤ (1 + 𝑏𝑛) 𝑎𝑛− 𝛼𝑛𝜑 (𝑎𝑛+1) + 4𝛼𝑛𝜀𝑛+ 𝑐𝑛, ∀𝑛 ≥ 𝑛0

(47)

By Lemma7, we at last get

lim

𝑛 → ∞𝑎𝑛 = lim𝑛 → ∞󵄩󵄩󵄩󵄩𝑥𝑛− 𝑥∗󵄩󵄩󵄩󵄩2= 0; (48)

for example, lim𝑛 → ∞𝑥𝑛= 𝑥∗∈ 𝐹 = 𝐹(𝑇1) ∩ 𝐹(𝑇2)

(Necessity) Suppose that lim𝑛 → ∞𝑥𝑛 = 𝑥∗ ∈ 𝐹 Then we can

choose an arbitrary continuous strictly increasing function

𝜙 : [0, +∞) → [0, +∞) with 𝜙(0) = 0, such as 𝜙(𝑡) = 𝑡

We can get lim𝑛 → ∞𝜙(‖𝑥𝑛+1− 𝑥∗‖) = 0

Because𝑇1is an asymptotically quasi pseudocontractive

type (with𝑃), by (11) in Definition1, for any𝑝 ∈ 𝐹(𝑇1) ⊇ 𝐹,

we have

lim sup

𝑛 → ∞ sup

𝑥∈𝐶 lim inf

𝑗(𝑥−𝑝)∈𝐽(𝑥−𝑝)(⟨𝑇(𝑃𝑇)𝑛−1𝑥 − 𝑝, 𝑗 (𝑥 − 𝑦)⟩

− 𝑘𝑛󵄩󵄩󵄩󵄩𝑥 − 𝑝󵄩󵄩󵄩󵄩2) ≤ 0

(49)

So,

lim sup

𝑛 → ∞ inf

𝑗(𝑥𝑛+1−𝑥 ∗)∈𝐽(𝑥𝑛+1−𝑥 ∗)[⟨𝑇1(𝑃𝑇1)

𝑛−1𝑥𝑛+1− 𝑥∗,

𝑗 (𝑥𝑛+1− 𝑥∗) ⟩− 𝑘𝑛󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩2

+ 𝜙 (󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩)]

= lim sup

𝑛 → ∞ inf

𝑗(𝑥𝑛+1−𝑥 ∗)∈𝐽(𝑥𝑛+1−𝑥 ∗)[⟨𝑇1(𝑃𝑇1)

𝑛−1𝑥𝑛+1− 𝑥∗,

𝑗 (𝑥𝑛+1− 𝑥∗) ⟩

− 𝑘𝑛󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩2] + lim𝑛 → ∞𝜙 (󵄩󵄩󵄩󵄩𝑥𝑛+1− 𝑥∗󵄩󵄩󵄩󵄩) ≤ 0 + 0 = 0;

(50) that is, (29) holds This completes the proof of Theorem8

Combining with Theorem8 and Definition3, we have

some results as follows

Theorem 9 Let 𝐶 be nonexpansive retract (with 𝑃) of a real

Banach space 𝐸 Assume that 𝑇1, 𝑇2 : 𝐶 → 𝐸 are two

uniformly L-Lipschitzian non-self-mappings (with 𝑃) and 𝑇1is

an asymptotically quasi pseudocontractive type with coefficient

numbers{𝑘𝑛} ⊂ [1, +∞) : 𝑘𝑛 → 1 satisfying 𝐹 = 𝐹(𝑇1) ∩

𝐹(𝑇2) ̸= 0 Suppose that {𝛼𝑛}, {𝛽𝑛}, {𝛼󸀠

𝑛}, {𝛽󸀠

𝑛} ⊂ [0, 1] are four number sequences satisfying the following:

(C1)Σ∞

𝑛=1𝛼𝑛 = +∞, Σ∞

𝑛=1𝛼2

𝑛 < +∞, Σ∞

𝑛=1𝛼𝑛(𝑘𝑛− 1) < +∞;

(C2)Σ∞

𝑛=1𝛼𝑛𝛽𝑛< +∞, Σ∞

𝑛=1𝛼𝑛𝛼󸀠

𝑛< +∞.

If𝑥1∈ 𝐶 is arbitrary, then the iterative sequence {𝑥𝑛} generated

by (14) converges strongly to the fixed point𝑥∗ ∈ 𝐹 if and only

if there exists a strictly increasing function𝜙 : [0, +∞) →

[0, +∞) with 𝜙(0) = 0 such that (29) holds.

Theorem 10 Let 𝐶 be nonexpansive retract (with 𝑃) of a real

Banach space 𝐸 Assume that 𝑇1, 𝑇2 : 𝐶 → 𝐸 are two

uniformly L-Lipschitzian non-self-mappings (with 𝑃) and 𝑇1is

an asymptotically quasi pseudocontractive type with coefficient numbers{𝑘𝑛} ⊂ [1, +∞) : 𝑘𝑛 → 1 satisfying 𝐹 = 𝐹(𝑇1) ∩ 𝐹(𝑇2) ̸= 0 Suppose that {𝛼𝑛}, {𝛼󸀠

𝑛} ⊂ [0, 1] are two number sequences satisfying the following:

(C1)Σ∞ 𝑛=1𝛼𝑛 = +∞, Σ∞

𝑛=1𝛼2

𝑛 < +∞, Σ∞

𝑛=1𝛼𝑛(𝑘𝑛− 1) < +∞;

(C2)Σ∞ 𝑛=1𝛼𝑛𝛼󸀠

𝑛< +∞.

If𝑥1∈ 𝐶 is arbitrary, then the iterative sequence {𝑥𝑛} generated

by (15) converges strongly to the fixed point𝑥∗ ∈ 𝐹 if and only

if there exists a strictly increasing function𝜙 : [0, +∞) →

[0, +∞) with 𝜙(0) = 0 such that (29) holds.

Theorem 11 Let 𝐶 be a nonempty closed convex subset of a

real Banach space 𝐸 Assume that 𝑇 : 𝐶 → 𝐶 is uniformly L-Lipschitzian self-mappings and asymptotically quasi pseudo-contractive type with coefficient numbers{𝑘𝑛} ⊂ [1, +∞) :

𝑘𝑛 → 1 satisfying 𝐹 = 𝐹(𝑇) ̸= 0 Suppose that {𝛼𝑛}, {𝛼󸀠

𝑛} ⊂

[0, 1] are two number sequences satisfying the following:

(C1)Σ∞ 𝑛=1𝛼𝑛 = +∞, Σ∞

𝑛=1𝛼2

𝑛 < +∞, Σ∞

𝑛=1𝛼𝑛(𝑘𝑛− 1) < +∞;

(C2)Σ∞ 𝑛=1𝛼𝑛𝛼󸀠

𝑛< +∞.

If𝑥1∈ 𝐶 is arbitrary, then the iterative sequence {𝑥𝑛} generated

by (16) converges strongly to the fixed point𝑥∗∈ 𝐹 if and only

if there exists a strictly increasing function𝜙 : [0, +∞) →

[0, +∞) with 𝜙(0) = 0 such that (29) holds.

Remark 12 Our research and results in this paper have the

following several advantaged characteristics (a) The iterative scheme is the new modified mixed Ishikawa iterative scheme with error on two mappings𝑇1, 𝑇2 (b) The common fixed point 𝑥∗ ∈ 𝐹 = 𝐹(𝑇1) ∩ 𝐹(𝑇2) is studied (c) The research object is the very generalized asymptotically quasi pseudocontractive type (with𝑃) non-self-mapping (d) The tool used by us is the very powerful tool: Lemma 7 So, our results here extend and improve many results of other authors to a certain extent, such as [6,8,14–23], and the proof methods are very different from the previous

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper

Acknowledgments

The authors would like to thank the editors and referees for their many useful comments and suggestions for the improvement of the paper This work was supported by the National Natural Science Foundations of China (Grant no 11271330) and the Natural Science Foundations of Zhejiang Province (Grant no Y6100696)

References

[1] K Goebel and W A Kirk, “A fixed point theorem for

asymp-totically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol 35, pp 171–174, 1972.

[2] J Schu, “Approximation of fixed points of asymptotically

non-expansive mappings,” Proceedings of the American Mathematical

Society, vol 112, pp 143–151, 1991.

[3] W Takahashi, N.-C Wong, and J.-C Yao, “Attractive point

and weak convergence theorems for new generalized hybrid

mappings in Hilbert spaces,” Journal of Nonlinear and Convex

Analysis, vol 13, pp 745–757, 2012.

[4] Y Kurokawa and W Takahashi, “Weak and strong convergence

theorems for nonspreading mappings in Hilbert spaces,”

Non-linear Analysis, Theory, Methods and Applications, vol 73, no 6,

pp 1562–1568, 2010

[5] H K Xu, “Vistosity approximation methods for nonexpansive

mappings,” Journal of Mathematical Analysis and Applications,

vol 298, pp 279–291, 2004

[6] D R Sahu, H.-K Xu, and J.-C Yao, “Asymptotically strict

pseu-docontractive mappings in the intermediate sense,” Nonlinear

Analysis, Theory, Methods and Applications, vol 70, no 10, pp.

3502–3511, 2009

[7] J K Kim, D R Sahu, and Y M Nam, “Convergence theorem for

fixed points of nearly uniformly L-Lipschitzian asymptotically

generalizedΦ-hemicontractive mappings,” Nonlinear Analysis,

Theory, Methods and Applications, vol 71, no 12, pp e2833–

e2838, 2009

[8] S S Chang, “Some results for asymptotically

pseudo-con-tractive mappings and asymptotically nonexpansive mappings,”

Proceedings of the American Mathematical Society, vol 129, no.

3, pp 845–853, 2001

[9] P Cholamjiak and S Suantai, “A new hybrid algorithm for

variational inclusions, generalized equilibrium problems, and

a finite family of quasi-nonexpansive mappings,” Fixed Point

Theory and Applications, vol 2009, Article ID 350979, 7 pages,

2009

[10] C Klin-Eam and S Suantai, “Strong convergence of composite

iterative schemes for a countable family of nonexpansive

map-pings in Banach spaces,” Nonlinear Analysis, Theory, Methods

and Applications, vol 73, no 2, pp 431–439, 2010.

[11] S Plubtieng and K Ungchittrakool, “Approximation of

com-mon fixed points for a countable family of relatively

nonexpan-sive mappings in a Banach space and applications,” Nonlinear

Analysis, Theory, Methods and Applications, vol 72, no 6, pp.

2896–2908, 2009

[12] T Suzuki, “Strong convergence theorems for infinite families of

nonexpansive mappings in general Banach spaces,” Fixed Point

Theory and Applications, vol 2005, no 1, pp 103–123, 2005.

[13] L.-C Zeng and J.-C Yao, “Implicit iteration scheme with

perturbed mapping for common fixed points of a finite family of

nonexpansive mappings,” Nonlinear Analysis, Theory, Methods

and Applications, vol 64, no 11, pp 2507–2515, 2006.

[14] G Marino and H.-K Xu, “A general iterative method for

non-expansive mappings in Hilbert spaces,” Journal of Mathematical

Analysis and Applications, vol 318, no 1, pp 43–52, 2006.

[15] Y.-H Wang and Y.-H Xia, “Strong convergence for

asymptoti-cally pseudocontractions with the demiclosedness principle in

Banach spaces,” Fixed Point Theory and Applications, vol 2012,

8 pages, 2012

[16] Y.-H Wang and W F Xuan, “Convergence theorems for

com-mon fixed points of a finite family of relatively nonexpansive

mappings in Banach spaces,” Abstract and Applied Analysis, vol.

2013, Article ID 259470, 7 pages, 2013

[17] L.-C Zeng and J.-C Yao, “Stability of iterative procedures with

errors for approximating common fixed points of a couple

of q-contractive-like mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol 321, no 2, pp 661–

674, 2006

[18] L.-C Zeng, T Tanaka, and J.-C Yao, “Iterative construction of

fixed points of nonself-mappings in Banach spaces,” Journal of Computational and Applied Mathematics, vol 206, no 2, pp.

814–825, 2007

[19] L C Zeng, N C Wong, and J C Yao, “Convergence analysis

of iterative sequences for a pair of mappings in Banach spaces,”

Acta Mathematica Sinica, vol 24, no 3, pp 463–470, 2008.

[20] X Wu, J C Yao, and L C Zeng, “Uniformly normal structure and strong convergence theorems for asymptotically

pseu-docontractive mappings,” Journal of Nonlinear and Convex Analysis, vol 6, no 3, pp 453–463, 2005.

[21] L.-C Ceng, N.-C Wong, and J.-C Yao, “Fixed point solutions of variational inequalities for a finite family of asymptotically non-expansive mappings without common fixed point assumption,”

Computers and Mathematics with Applications, vol 56, no 9, pp.

2312–2322, 2008

[22] C E Chidume, E U Ofoedu, and H Zegeye, “Strong and weak convergence theorems for asymptotically nonexpansive

mappings,” Journal of Mathematical Analysis and Applications,

vol 280, no 2, pp 364–374, 2003

[23] H Zegeye, M Robdera, and B Choudhary, “Convergence theorems for asymptotically pseudocontractive mappings in

the intermediate sense,” Computers and Mathematics with Applications, vol 62, no 1, pp 326–332, 2011.

[24] S.-S Chang, “Some problems and results in the study of

nonlinear analysis,” Nonlinear Analysis, Theory, Methods and Applications, vol 30, no 7, pp 4197–4208, 1997.

and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission However, users may print, download, or email articles for individual use.