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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 395098, 11 pages
http://dx.doi.org/10.1155/2012/395098
Research Article

The Equivalence of Convergence Results between Ishikawa and Mann Iterations with Errors for Uniformly Continuous Generalized Φ-Pseudocontractive Mappings in Normed Linear Spaces

Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China

Received 16 February 2012; Accepted 31 March 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Guiwen Lv and Zhiqun Xue. 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.

Abstract

We prove the equivalence of the convergence of the Mann and Ishikawa iterations with errors for uniformly continuous generalized Φ-pseudocontractive mappings in normed linear spaces. Our results extend and improve the corresponding results of Xu, 1998, Kim et al., 2009, Ofoedu, 2006, Chidume and Zegeye, 2004, Chidume, 2001, Chang et al. 2002, Liu, 1995, Hirano and huang, 2003, C. E. Chidume and C. O. Chidume, 2005, and huang, 2007.

1. Introduction

Let 𝐸 be a real normed linear space, 𝐸 its dual space, and 𝐽𝐸2𝐸 the normalized duality mapping defined by𝐽(𝑥)=𝑓𝐸𝑥,𝑓=𝑥𝑓=𝑓2,(1.1) where , denotes the generalized duality pairing. The single-valued normalized duality mapping is denoted by 𝑗.

Definition 1.1. A mapping 𝑇𝐸𝐸 is said to be(1)strongly accretive if, for all 𝑥,𝑦𝐸, there exist a constant 𝑘(0,1) and 𝑗(𝑥𝑦)𝐽(𝑥𝑦) such that 𝑇𝑥𝑇𝑦,𝑗(𝑥𝑦)𝑘𝑥𝑦2,(1.2)(2)𝜙-strongly accretive if there exist 𝑗(𝑥𝑦)𝐽(𝑥𝑦) and a strictly increasing function 𝜙[0,+)[0,+) with 𝜙(0)=0 such that )𝑇𝑥𝑇𝑦,𝑗(𝑥𝑦)𝜙(𝑥𝑦𝑥𝑦,𝑥,𝑦𝐸,(1.3)(3)generalized Φ-accretive if, for all 𝑥,𝑦𝐸, there exist 𝑗(𝑥𝑦)𝐽(𝑥𝑦) and a strictly increasing function Φ[0,+)[0,+) with Φ(0)=0 such that 𝑇𝑥𝑇𝑦,𝑗(𝑥𝑦)Φ(𝑥𝑦).(1.4)

Definition 1.2. Let 𝑁(𝑇)={𝑥𝐸𝑇𝑥=0}. The mapping 𝑇 is called strongly quasi-accretive if, for all 𝑥𝐸,𝑞𝑁(𝑇), there exist a constant 𝑘(0,1) and 𝑗(𝑥𝑦)𝐽(𝑥𝑦) such that 𝑇𝑥𝑇𝑞,𝑗(𝑥𝑞)𝑘𝑥𝑞2;𝑇 is called 𝜙-strongly quasi-accretive if, for all 𝑥𝐸,𝑞𝑁(𝑇), there exists a function 𝜙 such that 𝑇𝑥𝑇𝑞,𝑗(𝑥𝑞)𝜙(𝑥𝑞)𝑥𝑞, where 𝜙 is as in Definition 1.1. Finally, 𝑇 is called generalized Φ-quasi-accretive if, for each 𝑥𝐸,𝑞𝑁(𝑇), there exist 𝑗(𝑥𝑞)𝐽(𝑥𝑞) and a strictly increasing function Φ[0,+)[0,+) with Φ(0)=0 such that 𝑇𝑥𝑇𝑞,𝑗(𝑥𝑞)Φ(𝑥𝑞).

Closely related to the class of accretive-type mappings are those of pseudocontractive types.

Definition 1.3. A mapping 𝑇 with domain 𝐷(𝑇) and range 𝑅(𝑇) is said to be(1)strongly pseudocontractive if there exist a constant 𝑘(0,1) and 𝑗(𝑥𝑦)𝐽(𝑥𝑦) such that, for each 𝑥,𝑦𝐷(𝑇), 𝑇𝑥𝑇𝑦,𝑗(𝑥𝑦)𝑘𝑥𝑦2,(1.5)(2)𝜙-strongly pseudocontractive if there exist 𝑗(𝑥𝑦)𝐽(𝑥𝑦) and a strictly increasing function 𝜙[0,+)[0,+) with 𝜙(0)=0 such that 𝑇𝑥𝑇𝑦,𝑗(𝑥𝑦)𝑥𝑦2()𝜙𝑥𝑦𝑥𝑦,𝑥,𝑦𝐷(𝑇),(1.6)(3)generalized Φ-pseudocontractive if, for all 𝑥,𝑦𝐷(𝑇), there exist 𝑗(𝑥𝑦)𝐽(𝑥𝑦) and a strictly increasing function Φ[0,+)[0,+) with Φ(0)=0 such that 𝑇𝑥𝑇𝑦,𝑗(𝑥𝑦)𝑥𝑦2(Φ𝑥𝑦).(1.7)

Definition 1.4. Let 𝐹(𝑇)={𝑥𝐸𝑇𝑥=𝑥}. The mapping 𝑇 is called generalized Φ-hemi-pseudocontractive if, for all 𝑥𝐷(𝑇),𝑞𝐹(𝑇), there exist 𝑗(𝑥𝑞)𝐽(𝑥𝑞) and a strictly increasing function Φ[0,+)[0,+) with Φ(0)=0 such that 𝑇𝑥𝑇𝑞,𝑗(𝑥𝑞)𝑥𝑞2(Φ𝑥𝑞).(1.8)

Obviously, a mapping 𝑇 is strongly pseudocontractive, 𝜙-strongly pseudocontractive, generalized Φ-pseudocontractive, and generalized Φ-hemicontractive if and only if (𝐼𝑇) is strongly accretive, 𝜙-strongly accretive, generalized Φ-accretive, and generalized Φ-quasi-accretive, respectively.

It is shown in [1] that the class of strongly pseudocontractive mappings is a proper subclass of 𝜙-strongly pseudocontractive mappings. Furthermore, an example in [2] shows that the class of 𝜙-strongly pseudocontractive mappings with the nonempty fixed point set is a proper subclass of Φ-hemicontractive mappings. Hence, the class of generalized Φ-hemicontractive mappings is the most general among those defined above.

Definition 1.5. The mapping 𝑇𝐸𝐸 is called Lipschitz if exists a constant 𝐿>0 such that 𝑇𝑥𝑇𝑦𝐿𝑥𝑦,𝑥,𝑦𝐸.(1.9) It is clear that if 𝑇 is Lipschitz then it must be uniformly continuous. Otherwise, it is not true.

The following iteration schemes were introduced by Xu [3] in 1998. Let 𝐾 be a nonempty convex subset of 𝐸. For any given 𝑥0𝐾, the sequence {𝑥𝑛}𝑛=0 is defined by𝑥𝑛+1=𝛼𝑛𝑥𝑛+𝛽𝑛𝑇𝑦𝑛+𝛾𝑛𝛿𝑛,𝑦𝑛=𝛼𝑛𝑥𝑛+𝛽𝑛𝑇𝑥𝑛+𝛾𝑛𝜂𝑛,𝑛0,(1.10) is called the Ishikawa iteration sequence with errors, where {𝛿𝑛}𝑛=0,{𝜂𝑛}𝑛=0 are arbitrary bounded sequences in 𝐾 and {𝛼𝑛}𝑛=0,{𝛽𝑛}𝑛=0,{𝛾𝑛}𝑛=0,{𝛼𝑛}𝑛=0,{𝛽𝑛}𝑛=0, and {𝛾𝑛}𝑛=0 are six real sequences in [0,1] such that 𝛼𝑛+𝛽𝑛+𝛾𝑛=𝛼𝑛+𝛽𝑛+𝛾𝑛=1 for all 𝑛0 and satisfy certain conditions.

If 𝛽𝑛=𝛾𝑛=0, for all 𝑛0, then, from (1.10), we get the Mann iteration sequence with errors {𝑢𝑛}𝑛=0 defined by𝑢0𝐾,𝑢𝑛+1=𝛼𝑛𝑢𝑛+𝛽𝑛𝑇𝑢𝑛+𝛾𝑛𝜀𝑛,𝑛0,(1.11) where {𝜀𝑛}𝑛=0 is an arbitrary bounded sequence in 𝐾.

Numerous convergence results have been proved through iterative methods of approximating fixed points of Lipschitz pseudocontractive- (accretive-) type nonlinear mappings [310]. Most of these results have been extended to uniformly continuous mappings by some authors. Recently, C. E. Chidume and C. O. Chidume in [11] gave the most general result for uniformly continuous generalized Φ-hemicontractive mappings in normed linear. Their results are as follows.

Theorem 1.6 (see [11, Theorem 2.3]). Let 𝐸 be a real normed linear space, 𝐾 nonempty subset of 𝐸, and 𝑇𝐸𝐾 a uniformly continuous generalized Φ-hemicontractive mapping, that is, there exist 𝑥𝐹(𝑇) and a strictly increasing function Φ[0,+)[0,+),Φ(0)=0, such that, for all 𝑥𝐾, there exists 𝑗(𝑥𝑥)𝐽(𝑥𝑥) such that 𝑇𝑥𝑥,𝑗𝑥𝑥𝑥𝑥2Φ𝑥𝑥.(1.12)(a)If 𝑦𝐾 is a fixed point of 𝑇, then 𝑦=𝑥 and so 𝑇 has at most one fixed point in 𝐾.(b)Suppose there exists 𝑥0𝐾 such that the sequence {𝑥𝑛} defined by 𝑥𝑛+1=𝑎𝑛𝑥𝑛+𝑏𝑛𝑇𝑥𝑛+𝑐𝑛𝑢𝑛,𝑛0,(1.13) is contained in 𝐾, where {𝑎𝑛},{𝑏𝑛}, and {𝑐𝑛} are real sequences satisfying the following conditions:(i)𝑎𝑛+𝑏𝑛+𝑐𝑛=1, (ii)𝑛=0(𝑏𝑛+𝑐𝑛)=, (iii)𝑛=0(𝑏𝑛+𝑐𝑛)2<, (iv)𝑛=0𝑐𝑛<. Then, {𝑥𝑛} converges strongly to 𝑥. In particular, if 𝑦 is a fixed point of 𝑇 in 𝐾, then {𝑥𝑛} converges strongly to 𝑦.

Unfortunately, the control conditions (iii) 𝑛=0(𝑏𝑛+𝑐𝑛)2< and (iv) 𝑛=0𝑐𝑛< cannot assure that the result in [11] holds. In the proof course of p552, let 𝜖>0 be any given, we can choose (also in view of conditions (iii) and (iv)) an integer 𝑁1>0 such that for all 𝑛>𝑁1 the following inequality 𝑀1𝑐𝑛<(Φ(𝜖)/4)𝛼𝑛 holds, where 𝛼𝑛=𝑏𝑛+𝑐𝑛. The inequality above 𝑀1𝑐𝑛<(Φ(𝜖)/4)𝛼𝑛 implies that 𝑐𝑛=𝑜(𝛼𝑛). But conditions (iii) and (iv) of [11] can not assure that 𝑐𝑛=𝑜(𝛼𝑛). On the one hand, let 𝑐𝑛=1/𝑛2,𝑛=1,2,3,;𝛼1=0,𝛼2=1/2,𝛼3=0,𝛼4=1/4,𝛼5=0,𝛼6=1/6,; then 𝑛=0𝑐𝑛<, but 𝑐𝑛𝑜(𝛼𝑛). On the other hand, set 𝑐𝑛=1/𝑛,𝛼𝑛=2/𝑛; then 𝑐𝑛=𝑜(𝛼𝑛), but 𝑛=0𝑐𝑛=.

The purpose of this paper is that we obtain the convergence result of the Mann iteration with errors, and we also prove the equivalence of convergence between the Ishikawa iteration with errors defined by (1.10) and the Mann iteration with errors defined by (1.11). We also show that the Ishikawa iteration with errors defined by (1.10) converges to the unique fixed point of 𝑇. Our results extend and improve the corresponding results of [312]. For this, in the sequel, we will need the following lemmas.

Lemma 1.7 (see [13]). Let 𝐸 be a real normed space. Then, for all 𝑥,𝑦𝐸, the following inequality holds: 𝑥+𝑦2𝑥2+2𝑦,𝑗(𝑥+𝑦),𝑗(𝑥+𝑦)𝐽(𝑥+𝑦).(1.14)

Lemma 1.8 (see [14]). Let Φ[0,+)[0,+) be a strictly increasing continuous function with Φ(0)=0 and {𝜃𝑛},{𝜎𝑛}, and {𝜆𝑛} nonnegative three sequences that satisfy the following inequality: 𝜃2𝑛+1𝜃2𝑛2𝜆𝑛Φ𝜃𝑛+1+𝜎𝑛,𝑛𝑁,(1.15) where 𝜆𝑛(0,1),lim𝑛𝜆𝑛=0, and 𝑛=0𝜆𝑛=,𝜎𝑛=𝑜(𝜆𝑛). Then 𝜃𝑛0 as 𝑛.

2. Main Results

Theorem 2.1. Let 𝐾 be a nonempty closed convex subset of a real normed linear space 𝐸. Suppose that 𝑇𝐾𝐾 is a uniformly continuous generalized Φ-hemicontractive mapping with 𝐹(𝑇). Let {𝑢𝑛} be a sequence in 𝐾 defined iteratively from some 𝑢0𝐾 by (1.11), where {𝜀𝑛} is an arbitrary bounded sequence in 𝐾 and {𝛼𝑛},{𝛽𝑛}, and {𝛾𝑛} are three sequences in [0,1] satisfying the following conditions:(i)𝛼𝑛+𝛽𝑛+𝛾𝑛=1, (ii)𝑛=0𝛽𝑛=, (iii)lim𝑛𝛽𝑛=0, (iv)𝛾𝑛=𝑜(𝛽𝑛). Then, the iteration sequence {𝑢𝑛} converges strongly to the unique fixed point of 𝑇.

Proof. Let 𝑞𝐹(𝑇). The uniqueness of the fixed point of 𝑇 comes from Definition 1.4.
First, we prove that there exists 𝑢0𝐾 with 𝑢0𝑇𝑢0 such that 𝑡0=𝑢0𝑇𝑢0𝑢0𝑞𝑅(Φ). In fact, if 𝑢0=𝑇𝑢0, then we are done. Otherwise, there exists the smallest positive integer 𝑛0𝑁 such that 𝑢𝑛0𝑇𝑢𝑛0. We denote 𝑢𝑛0=𝑢0, and then we obtain that 𝑡0=𝑢0𝑇𝑢0𝑢0𝑞𝑅(Φ). Indeed, if 𝑅(Φ)=[0,+), then 𝑡0𝑅(Φ). If 𝑅(Φ)=[0,𝐴) with 0<𝐴<+, then, for 𝑞𝐾, there exists a sequence {𝑤𝑛}𝐾 such that 𝑤𝑛𝑞 as 𝑛 with 𝑤𝑛𝑞, and we also obtain that the sequence {𝑤𝑛𝑇𝑤𝑛} is bounded. So there exists 𝑛0𝑁 such that 𝑤𝑛𝑇𝑤𝑛𝑤𝑛𝑞𝑅(Φ) for 𝑛𝑛0, and then we redefine 𝑢0=𝑤𝑛0. Let 𝜔0=Φ1((𝑢0𝑞)(𝑇𝑢0𝑞)𝑢0𝑞)>0.
Next for 𝑛0 we will prove 𝑢𝑛𝑞𝜔0 by induction. Clearly, 𝑢0𝑞𝜔0 holds. Suppose that 𝑢𝑛𝑞𝜔0, for some 𝑛; then we want to prove 𝑢𝑛+1𝑞𝜔0. If it is not the case, then 𝑢𝑛+1𝑞>𝜔0. Since 𝑇 is a uniformly continuous mapping, setting 𝜖0=Φ(𝜔0)/12𝜔0, there exists 𝛿>0 such that 𝑇𝑥𝑇𝑦<Φ(𝜔0)/12𝜔0 whenever 𝑥𝑦<𝛿 and 𝑇 is a bounded operator. Set 𝑀=sup{𝑇𝑥𝑥𝑞𝜔0}+sup𝑛𝜀𝑛. Since lim𝑛𝛽𝑛=0,𝛾𝑛=𝑜(𝛽𝑛), without loss of generality, let 𝛽𝑛,𝛾𝑛𝛽𝑛𝜔<min0,𝛿4(𝑀+𝑞)2𝑀+2𝜔0,Φ𝜔+2𝑞06𝜔0,Φ𝜔06𝜔0+𝑀+𝑞2,𝑛0.(2.1)
From (1.10), we have 𝑢𝑛+1=𝛼𝑞𝑛𝑢𝑛𝑞+𝛽𝑛𝑇𝑢𝑛+𝛾𝑛𝛽𝜀𝑛+𝛾𝑛𝑞𝑢𝑛+𝛽𝑞𝑛𝑇𝑢𝑛+𝛾𝑛𝜀𝑛+𝛽𝑛+𝛾𝑛𝑞𝜔0+𝛽𝑛𝑇𝑢𝑛+𝜀𝑛+2𝑞𝜔0+𝛽𝑛3(2𝑀+2𝑞)2𝜔0,𝑢𝑛+1𝑢𝑛𝛽=𝑛𝑇𝑢𝑛+𝛾𝑛𝜀𝑛𝛽𝑛+𝛾𝑛𝑢𝑛𝛽𝑛𝑇𝑢𝑛+𝛾𝑛𝜀𝑛+𝛽𝑛+𝛾𝑛𝑢𝑛+𝛽𝑞𝑛+𝛾𝑛𝑞𝛽𝑛𝑇𝑢𝑛+𝜀𝑛𝑢+2𝑛𝑞+2𝑞𝛽𝑛2𝑀+2𝜔0.+2𝑞(2.2)
Applying Lemma 1.7, the recursion (1.11), and the inequalities above, we obtain 𝑢𝑛+1𝑞2=𝑢𝑛𝑞+𝛽𝑛𝑇𝑢𝑛+𝛾𝑛𝜀𝑛𝛽𝑛+𝛾𝑛𝑢𝑛2𝑢𝑛𝑞2𝛽+2𝑛𝑇𝑢𝑛+𝛾𝑛𝜀𝑛𝛽𝑛+𝛾𝑛𝑢𝑛𝑢,𝑗𝑛+1𝑢𝑞𝑛𝑞2+2𝛽𝑛𝑇𝑢𝑛+1𝑢𝑇𝑞,𝑗𝑛+1𝑞+2𝛽𝑛𝑇𝑢𝑛𝑇𝑢𝑛+1+𝑢𝑛+1𝑢𝑛𝑢𝑛+1𝑢𝑞,𝑗𝑛+1𝑞+2𝛾𝑛𝑢𝑛𝜀𝑞+𝑞+𝑛𝑢𝑛+1𝑢𝑞𝑛𝑞22𝛽𝑛Φ𝑢𝑛+1𝑞+2𝛽𝑛𝑇𝑢𝑛𝑇𝑢𝑛+1𝑢𝑛+1𝑞+2𝛽𝑛𝑢𝑛𝑢𝑛+1𝑢𝑛+1𝑞+2𝛾𝑛𝜔0𝑢+𝑞+𝑀𝑛+1𝑞𝜔202𝛽𝑛Φ𝜔0+2𝛽𝑛Φ𝜔012𝜔03𝜔02+2𝛽𝑛Φ𝜔06𝜔03𝜔02+2𝛽𝑛Φ𝜔06𝜔03𝜔02<𝜔20,(2.3) which is a contraction with the assumption 𝑢𝑛+1𝑞>𝜔0. Then, 𝑢𝑛+1𝑞𝜔0, that is, the sequence {𝑢𝑛} is bounded. It leads to lim𝑛𝑢𝑛+1𝑢𝑛=0,lim𝑛𝑇𝑢𝑛+1𝑇𝑢𝑛=0.(2.4)
Again using Lemma 1.7, we have 𝑢𝑛+1𝑞2=𝑢𝑛𝑞+𝛽𝑛𝑇𝑢𝑛+𝛾𝑛𝜀𝑛𝛽𝑛+𝛾𝑛𝑢𝑛2𝑢𝑛𝑞2𝛽+2𝑛𝑇𝑢𝑛+𝛾𝑛𝜀𝑛𝛽𝑛+𝛾𝑛𝑢𝑛𝑢,𝑗𝑛+1𝑢𝑞𝑛𝑞2+2𝛽𝑛𝑇𝑢𝑛+1𝑥𝑇𝑞,𝑗𝑢+1𝑞+2𝛽𝑛𝑇𝑢𝑛𝑇𝑢𝑛+1+𝑢𝑛+1𝑢𝑛𝑢𝑛+1𝑢𝑞,𝑗𝑛+1𝑞+2𝛾𝑛𝑢𝑛𝜀𝑞+𝑞+𝑛𝑢𝑛+1𝑢𝑞𝑛𝑞22𝛽𝑛Φ𝑢𝑛+1𝑞+2𝛽𝑛𝑇𝑢𝑛𝑇𝑢𝑛+1𝑢𝑛+1𝑞+2𝛽𝑛𝑢𝑛𝑢𝑛+1𝑢𝑛+1𝑞+2𝛾𝑛𝜔0𝑢+𝑞+𝑀𝑛+1𝑢𝑞𝑛𝑞22𝛽𝑛Φ𝑢𝑛+1𝑞+𝐴𝑛,(2.5) where 𝐴𝑛=2𝛽𝑛𝑇𝑢𝑛𝑇𝑢𝑛+1𝑢𝑛+1𝑞+2𝛽𝑛𝑢𝑛𝑢𝑛+1𝑢𝑛+1𝑞+2𝛾𝑛𝜔0𝑢+𝑞+𝑀𝑛+1𝛽𝑞=𝑜𝑛.(2.6) Therefore, (2.5) becomes 𝑢𝑛+1𝑞2𝑢𝑛𝑞22𝛽𝑛Φ𝑢𝑛+1𝛽𝑞+𝑜𝑛.(2.7) From Lemma 1.8, we obtain lim𝑛𝑢𝑛𝑞=0.

Theorem 2.2. Let 𝐾 be a nonempty closed convex subset of a real normed linear space 𝐸 and 𝑇𝐾𝐾 a uniformly continuous generalized Φ-pseudocontractive mapping with 𝐹(𝑇). Let {𝑢𝑛},{𝑥𝑛} be two sequences in 𝐾 defined iteratively from some 𝑢0𝑥0𝐾 by (1.11) and (1.10), where {𝛿𝑛},{𝜂𝑛},{𝜀𝑛} are three arbitrary bounded sequences in 𝐾 and {𝛼𝑛},{𝛽𝑛},{𝛾𝑛},{𝛼𝑛},{𝛽𝑛}, and {𝛾𝑛} are six sequences in [0,1] satisfying the following conditions:(i)𝛼𝑛+𝛽𝑛+𝛾𝑛=𝛼𝑛+𝛽𝑛+𝛾𝑛=1, (ii)𝑛=0𝛽𝑛=, (iii)lim𝑛𝛽𝑛=lim𝑛𝛽𝑛=lim𝑛𝛾𝑛=0, (iv)𝛾𝑛=𝑜(𝛽𝑛). Then, the following two assertions are equivalent:(1)Iteration (1.11) converges strongly to the unique fixed point of 𝑇,(2)Iteration (1.10) converges strongly to the unique fixed point of 𝑇.

Proof. Let 𝑞𝐹(𝑇). The uniqueness of 𝑞 comes from Definition 1.4. If the Ishikawa iteration with errors converges to 𝑞, then setting 𝛽𝑛=𝛾𝑛=0, for all 𝑛0 in (1.10), we can get the convergence of the Mann iteration with errors. Conversely, we only prove that (1)(2), that is, if the Mann iteration with errors converges to 𝑞, we want to prove the convergence of the Ishikawa iteration with errors.
First, we will prove that there exists 𝑥0𝐾 with 𝑥0𝑇𝑥0 such that 𝑡0=𝑥0𝑇𝑥0𝑥0𝑞𝑅(Φ).
In fact, if 𝑥0=𝑇𝑥0, then we are done. Otherwise, there exists the smallest positive integer 𝑁0𝑁 such that 𝑥𝑁0𝑇𝑥𝑁0. We denote 𝑥𝑁0=𝑥0, and then we obtain that 𝑡0=𝑥0𝑇𝑥0𝑥0𝑞𝑅(Φ). Indeed if 𝑅(Φ)=[0,+), then 𝑡0𝑅(Φ). If 𝑅(Φ)=[0,𝐴) with 0<𝐴<+, then for 𝑞𝐾 there exists a sequence {𝑤𝑛}𝐾 such that 𝑤𝑛𝑞 as 𝑛 with 𝑤𝑛𝑞, and we also obtain that the sequence {𝑤𝑛𝑇𝑤𝑛} is bounded. So there exists 𝑛0𝑁 such that 𝑤𝑛𝑇𝑤𝑛𝑤𝑛𝑞𝑅(Φ),𝑛𝑛0, and then we redefine 𝑥0=𝑤𝑛0. Let 𝜇0=Φ1((𝑥0𝑞)(𝑇𝑥0𝑞)𝑥0𝑞)>0.
Second, we will prove that the sequence {𝑥𝑛𝑞} is a bounded sequence.
Set 𝐵1=𝑥𝑞𝜇0𝑥𝐾,𝐵2=𝑥𝑞2𝜇0,𝑀𝑥𝐾1=maxsup𝑥𝐵2𝑇𝑥;sup𝑛𝑁𝜀𝑛;sup𝑛𝑁𝛿𝑛;sup𝑛𝑁𝜂𝑛;sup𝑛𝑁𝑢𝑛;sup𝑛𝑁𝑇𝑢𝑛.(2.8) Since 𝑇 is uniformly continuous, for 𝜖=Φ(𝜇0)/5𝜇0, there exists 𝛿>0 such that 𝑇𝑥𝑇𝑦<𝜖 whenever 𝑥𝑦<𝛿. Now let 𝑘=min{1,𝜇0/4(𝑀1+𝜇0+𝑞),𝛿/2(4𝑀1+𝜇0+𝑞),Φ(𝜇0)/20𝜇0(𝑀1+𝜇0+𝑞)}. By the control conditions (iii) and (iv), without loss of generality, set |𝛼𝑛𝛼𝑛|,𝛽𝑛,𝛾𝑛,𝛽𝑛,𝛾𝑛<𝑘, for all𝑛0.
Observe that if 𝑥𝑛𝐵1, we obtain 𝑦𝑛𝐵2. Indeed 𝑦𝑛=𝑞1𝛽𝑛𝛾𝑛𝑥𝑛+𝛽𝑛𝑇𝑥𝑛+𝛾𝑛𝜂𝑛𝑥𝑞𝑛𝑞+𝛽𝑛𝑇𝑥𝑛+𝛽𝑛+𝛾𝑛𝑥𝑛𝑞+𝛾𝑛𝜂𝑛+𝛽𝑛+𝛾𝑛𝑞𝜇0+𝑘2𝑀1+2𝜇0+2𝑞2𝜇0.(2.9)
Next, by induction, we prove 𝑥𝑛𝐵1. Clearly, from (2.1), we obtain 𝑥0𝑞𝜇0, that is, 𝑥0𝐵1. Set 𝑥𝑛𝑞𝜇0, for some 𝑛; then we will prove that 𝑥𝑛+1𝑞𝜇0. If it is not the case, we assume that 𝑥𝑛+1𝑞>𝜇0. From (1.10), we obtain the following inequalities: 𝑥𝑛+1=𝛼𝑞𝑛𝑥𝑛+𝛽𝑛𝑇𝑦𝑛+𝛾𝑛𝛿𝑛𝑞1𝛽𝑛𝛾𝑛𝑥𝑛𝑞+𝛽𝑛𝑇𝑦𝑛+𝛾𝑛𝛿𝑛+𝛽𝑛+𝛾𝑛𝑞𝑥𝑛+𝛽𝑞𝑛+𝛾𝑛𝑥𝑛𝑞+𝛽𝑛𝑇𝑦𝑛+𝛾𝑛𝛿𝑛+𝛽𝑛+𝛾𝑛𝑞𝜇0+𝑘2𝜇0+2𝑀15+2𝑞4𝜇0,𝑥𝑛+1𝑥𝑛=𝛽𝑛𝑇𝑦𝑛+𝛾𝑛𝛿𝑛𝛽𝑛+𝛾𝑛𝑥𝑛𝛽𝑛+𝛾𝑛𝑥𝑛𝑞+𝛽𝑛𝑇𝑦𝑛+𝛾𝑛𝛿𝑛+𝛽𝑛+𝛾𝑛𝑞𝑘2𝜇0+2𝑀1Φ𝜇+2𝑞010𝜇0,𝑥𝑛+1𝑦𝑛=𝛼𝑛𝛼𝑛𝑥𝑛+𝛽𝑛𝑇𝑦𝑛𝛽𝑛𝑇𝑥𝑛+𝛾𝑛𝛿𝑛𝛾𝑛𝜂𝑛||𝛼𝑛𝛼𝑛||𝑥𝑛𝑞+𝛽𝑛𝑇𝑦𝑛+𝛽𝑛𝑇𝑥𝑛+𝛾𝑛𝛿𝑛+𝛾𝑛𝜂𝑛+||𝛼𝑛𝛼𝑛||𝜇𝑞𝑘0+4𝑀1+𝑞𝛿.(2.10)
Since 𝑇 is a uniformly continuous mapping, then 𝑇𝑥𝑛+1𝑇𝑦𝑛Φ𝜇𝜖=05𝜇0.(2.11)
Since 𝛾𝑛=𝑜(𝛽𝑛), we let 𝛾𝑛<𝛽𝑛18𝜇Φ045𝜇02.(2.12)
From (2.9)–(2.12), we have 𝑥𝑛+1𝑞2=1𝛽𝑛𝛾𝑛𝑥𝑛+𝛽𝑛𝑇𝑦𝑛+𝛾𝑛𝛿𝑛𝑞2=𝑥𝑛𝑞+𝛽𝑛𝑇𝑦𝑛+𝛾𝑛𝛿𝑛𝛽𝑛+𝛾𝑛𝑥𝑛2𝑥𝑛𝑞2𝛽+2𝑛𝑇𝑦𝑛+𝛾𝑛𝛿𝑛𝛽𝑛+𝛾𝑛𝑥𝑛𝑥,𝑗𝑛+1𝑥𝑞𝑛𝑞2+2𝛽𝑛𝑇𝑥𝑛+1𝑥𝑇𝑞,𝑗𝑛+1𝑞+2𝛽𝑛𝑇𝑦𝑛𝑇𝑥𝑛+1𝑥,𝑗𝑛+1𝛽𝑞+2𝑛+𝛾𝑛𝑥𝑛+1𝑥𝑛𝑥,𝑗𝑛+1𝑞+2𝛾𝑛𝛿𝑛𝑥𝑛+1𝛽𝑞2𝑛+𝛾𝑛𝑥𝑛+1𝑥𝑞,𝑗𝑛+1𝑥𝑞𝑛𝑞22𝛽𝑛Φ𝑥𝑛+1𝑞+2𝛽𝑛𝑇𝑥𝑛+1𝑇𝑦𝑛𝑥𝑛+1𝛽𝑞+2𝑛+𝛾𝑛𝑥𝑛+1𝑥𝑛𝑥𝑛+1𝑞+2𝛾𝑛𝛿𝑛𝑥𝑛+1𝑞+2𝛾𝑛𝑥𝑛+1𝑞2<𝜇202𝛽𝑛Φ𝜇0+2𝛽𝑛Φ𝜇05𝜇05𝜇04+4𝛽𝑛Φ𝜇010𝜇05𝜇04+2𝛽𝑛Φ𝜇05𝜇05𝜇04+2𝛽𝑛18𝜇Φ045𝜇025𝜇042<𝜇20,(2.13) which is a contradiction with the assumption 𝑥𝑛+1𝑞>𝜇0. Hence, 𝑥𝑛𝐵1, that is, the sequence {𝑥𝑛} is bounded. From (2.9), the sequence {𝑦𝑛} is also bounded. So we obtain lim𝑛𝑥𝑛+1𝑥𝑛=lim𝑛𝑢𝑛+1𝑢𝑛=lim𝑛𝑥𝑛+1𝑦𝑛=0.(2.14) Since 𝑇 is a uniformly continuous mapping, then lim𝑛𝑇𝑢𝑛+1𝑇𝑢𝑛=lim𝑛𝑇𝑥𝑛+1𝑇𝑦𝑛=0.(2.15)
Using the recursion formula (1.10), we compute as follows: 𝑥𝑛+1𝑢𝑛+12𝑥𝑛𝑢𝑛22𝛽𝑛Φ𝑥𝑛+1𝑢𝑛+1+2𝛽𝑛𝑇𝑥𝑛+1𝑇𝑦𝑛𝑥𝑛+1𝑢𝑛+1+2𝛽𝑛𝑇𝑢𝑛𝑇𝑢𝑛+1𝑥𝑛+1𝑢𝑛+1𝛽+2𝑛+𝛾𝑛𝑢𝑛𝑢𝑛+1𝑥𝑛+1𝑢𝑛+1𝛽+2𝑛+𝛾𝑛𝑥𝑛+1𝑥𝑛𝑥𝑛+1𝑢𝑛+1+2𝛾𝑛𝛿𝑛+𝜀𝑛𝑥𝑛+1𝑢𝑛+1=𝑥𝑛𝑢𝑛22𝛽𝑛Φ𝑥𝑛+1𝑢𝑛+1+𝐴𝑛,(2.16) where 𝐴𝑛=2𝛽𝑛𝑇𝑥𝑛+1𝑇𝑦𝑛𝑥𝑛+1𝑢𝑛+1+2𝛽𝑛𝑇𝑢𝑛𝑇𝑢𝑛+1𝑥𝑛+1𝑢𝑛+1𝛽+2𝑛+𝛾𝑛𝑢𝑛𝑢𝑛+1𝑥𝑛+1𝑢𝑛+1𝛽+2𝑛+𝛾𝑛𝑥𝑛+1𝑥𝑛𝑥𝑛+1𝑢𝑛+1+2𝛾𝑛𝛿𝑛+𝜀𝑛𝑥𝑛+1𝑢𝑛+1.(2.17)
Since the sequence {𝑥𝑛𝑢𝑛} is bounded, from (2.15), (2.16), and the control condition (iv), we get 𝐴𝑛=𝑜(𝛽𝑛). Then, in (2.18), set 𝜃𝑛=𝑥𝑛𝑢𝑛,𝜆𝑛=𝛽𝑛, and 𝜎𝑛=𝐴𝑛. From Lemma 1.8, we obtain lim𝑛𝑥𝑛+1𝑢𝑛+1=0. Since lim𝑛𝑢𝑛𝑞=0, using inequality 0𝑥𝑛𝑞𝑥𝑛𝑢𝑛+𝑢𝑛𝑞, then lim𝑛𝑥𝑛𝑞=0, and we complete the proof of Theorem 2.1.

From Theorems 2.1 and 2.2, we obtain the following corollary.

Corollary 2.3. Let 𝐾 be a nonempty convex subset of a real normed linear space 𝐸. 𝑇𝐾𝐾 is a uniformly continuous generalized Φ-hemicontractive mapping with 𝐹(𝑇). Let {𝑥𝑛} be a sequence in 𝐾 defined iteratively from some 𝑥0𝐾 by (1.10), where {𝛿𝑛},{𝜂𝑛} are arbitrary bounded sequences in 𝐾 and {𝛼𝑛},{𝛽𝑛},{𝛾𝑛},{𝛼𝑛},{𝛽𝑛}, and {𝛾𝑛} are sequences in [0,1] satisfying the following conditions:(i)𝛼𝑛+𝛽𝑛+𝛾𝑛=𝛼𝑛+𝛽𝑛+𝛾𝑛=1, (ii)𝑛=0𝛽𝑛=, (iii)lim𝑛𝛽𝑛=lim𝑛𝛽𝑛=lim𝑛𝛾𝑛=0, (iv)𝛾𝑛=𝑜(𝛽𝑛). Then, the iteration sequence {𝑥𝑛} defined by (1.10) converges strongly to the unique fixed point of 𝑇.

Corollary 2.4. Let 𝑇𝐸𝐸 be a uniformly continuous generalized Φ-accretive mapping with 𝑁(𝑇). For some 𝑢0,𝑥0𝐸, the iteration sequences {𝑢𝑛},{𝑥𝑛} in 𝐸 are defined as follows: 𝑢𝑛+1=𝛼𝑛𝑢𝑛+𝛽𝑛𝑆𝑢𝑛+𝛾𝑛𝜀𝑛𝑥,𝑛0,(2.18)𝑛+1=𝛼𝑛𝑥𝑛+𝛽𝑛𝑆𝑦𝑛+𝛾𝑛𝛿𝑛,𝑦𝑛=𝛼𝑛𝑥𝑛+𝛽𝑛𝑆𝑥𝑛+𝛾𝑛𝜂𝑛,𝑛0,(2.19) where 𝑆𝑥=𝑥𝑇𝑥 for all 𝑥𝐸. {𝛿𝑛},{𝜂𝑛}, and {𝜀𝑛} are arbitrary bounded sequences in 𝐾 and {𝛼𝑛},{𝛽𝑛},{𝛾𝑛},{𝛼𝑛},{𝛽𝑛}, and {𝛾𝑛} are sequences in [0,1] satisfying the following conditions:(i)𝛼𝑛+𝛽𝑛+𝛾𝑛=𝛼𝑛+𝛽𝑛+𝛾𝑛=1, (ii)𝑛=0𝛽𝑛=, (iii)lim𝑛𝛽𝑛=lim𝑛𝛽𝑛=lim𝑛𝛾𝑛=0, (iv)𝛾𝑛=𝑜(𝛽𝑛). Then, the following two assertions are equivalent:(1){𝑥𝑛} converges strongly to the unique fixed point of 𝑆,(2){𝑢𝑛} converges strongly to the unique fixed point of 𝑆.

Proof. Let 𝑆=𝐼𝑇, and observe that 𝑇 is a uniformly continuous generalized Φ-accretive mapping if and only if 𝑆 is a uniformly continuous generalized Φ-pseudocontractive mapping. The result follows from Theorem 2.2.

Remark 2.5. Our results improve and extend the corresponding results of [11] in the following sense.(i)We point out the gaps of C. E. Chidume and C. O. Chidume [11] in their proof and provide a counterexample.(ii)In the proof method, Theorem 2.1 differs from Theorem  2.3 in [11].(iii)The control conditions 𝑛=0(𝑏𝑛+𝑐𝑛)2< and 𝑛=0𝑐𝑛< in Theorem  2.3 in [11] are replaced by the condition 𝑏𝑛=𝑜(𝑐𝑛). Under the new condition, we obtain the convergence theorem of the Mann iteration sequence with errors.(iv)We also obtain the equivalence of convergence results between the Ishikawa and Mann iterations with errors.(v)The Mann iteration sequence with errors is extended to the Ishikawa iteration sequence with errors.

Acknowledgments

This paper is supported by Hebei Province Natural Science Foundation (A2011210033) and Shijiazhuang Tiedao University Foundation (Q64).

References

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