Table of Contents 
ISRN Mathematical Analysis
Volume 2011, Article ID 687184, 9 pages
http://dx.doi.org/10.5402/2011/687184
Research Article

Robustness of Krasnoselski-Mann's Algorithm for Asymptotically Nonexpansive Mappings

Yu-Chao Tang1,2 and Li-Wei Liu1

1Department of Mathematics, Nanchang University, Nanchang 330031, China
2Department of Mathematics, Xi'an Jiaotong University, Xi'an 710049, China

Received 22 February 2011; Accepted 11 April 2011

Academic Editors: V. Kravchenko and A. Peris

Copyright © 2011 Yu-Chao Tang and Li-Wei Liu. 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

Iterative approximation of fixed points of nonexpansive mapping is a very active theme in many aspects of mathematical and engineering areas, in particular, in image recovery and signal processing. Because the errors usually occur in few places, it is necessary to show that whether the iterative algorithm is robust or not. In the present work, we prove that Krasnoselski-Mann's algorithm is robust for asymptotically nonexpansive mapping in a Banach space setting. Our results generalize the corresponding results existing in the literature.

1. Introduction

Many practical problems can be formulated as the fixed point problem of 𝑥=𝑇𝑥, where 𝑇 is a nonexpansive mapping. Iterative methods as a powerful tool are often used to approximate the fixed points of such mapping. It has been show that the methods used to find fixed points of nonexpansive mapping covered a widely applied mathematics problems, such as the convex feasibility problem [1–3] and the split feasibility problem [4–6]. It is recommended for interested reader to [7] for an extensive study on the theory about iterative fixed point theory.

Let 𝑋 be a real Banach space. 𝑇∶𝑋→𝑋 is called a nonexpansive mapping if for any 𝑥,𝑦∈𝑋, ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖. Krasnoselski-Mann's iteration method for finding fixed points of 𝑇 is defined by foranyinitial𝑥0∈𝑋,𝑥𝑛+1=1−𝛼𝑛𝑥𝑛+𝛼𝑛𝑇𝑥𝑛,𝑛≥0,(1.1) where {𝛼𝑛} is a sequence in [0,1].

In 2001, Combettes [8] considered a parallel projection method algorithm in signal synthesis problems in a real Hilbert space 𝐻 as follows:𝑥𝑛+1=𝑥𝑛+𝜆𝑛𝑚𝑖=1𝜔𝑖𝑃𝑖𝑥𝑛+𝑐𝑖,𝑛−𝑥𝑛,(1.2) where {𝜆𝑛}⊆(0,2), {𝜔𝑖}𝑚𝑖=1 are positive weights such that ∑𝑚𝑖=1𝜔𝑖=1, 𝑃𝑖 is the projection of a signal 𝑥∈𝐻 onto a closed convex subset 𝑆𝑖 of 𝐻, and 𝑐𝑖,𝑛 stands for the error made in computing the projection onto 𝑆𝑖 at each iteration 𝑛. He firstly proved that the sequence {𝑥𝑛} generated by (1.2) converges weakly to a point in 𝐺, where ⋂𝐺∶=𝑚𝑖=1𝑆𝑖.

Kim and Xu [9] generalized the results of Combettes [8] from Hilbert spaces to uniformly convex Banach spaces and obtained its equivalent form as follows: 𝑥𝑛+1=1−𝛼𝑛𝑥𝑛+𝛼𝑛𝑇𝑥𝑛+𝑒𝑛,𝑛≥0,(1.3) where 𝛼𝑛∶=𝜆𝑛/2∈(0,1), 𝑒𝑛∑∶=2𝑚𝑖=1𝜔𝑖𝑐𝑖,𝑛, and 𝑇 is nonexpansive. They proved that the weak convergence of the (1.3) in a uniformly convex Banach space. More precisely, they proved that the following main theorems.

Theorem 1.1 (see [9]). Assume that 𝑋 is a uniformly convex Banach space. Assume, in addition, that either 𝑋∗ has the Kadec-Klee property or 𝑋 satisfies Opial's property. Let 𝑇∶𝑋→𝑋 be a nonexpansive mapping such that 𝐹(𝑇)≠∅(𝐹(𝑇) denotes the set of fixed points of  𝑇, that is, 𝐹(𝑇)={𝑥∈𝑋∶𝑇𝑥=𝑥}). Given an initial guess 𝑥0∈𝑋. Let {𝑥𝑛} be generated by (1.3) and satisfy the following properties: (i)∑∞𝑛=0𝛼𝑛(1−𝛼𝑛)=∞,(ii)∑∞𝑛=0𝛼𝑛‖𝑒𝑛‖<∞. Then the sequence {𝑥𝑛} converges weakly to a fixed point of  𝑇.

Theorem 1.2 (see [9]). Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻 and 𝑇∶𝐶→𝐶 a nonexpansive mapping with 𝐹(𝑇)≠∅. Given an initial guess 𝑥0∈𝑋. Let {𝑥𝑛} be generated by either 𝑥𝑛+1=1−𝛼𝑛𝑥𝑛+𝛼𝑛𝑃𝐶𝑇𝑥𝑛+𝑒𝑛,𝑛≥0,(1.4) or 𝑥𝑛+1=𝑃𝐶1−𝛼𝑛𝑥𝑛+𝛼𝑛𝑇𝑥𝑛+𝑒𝑛,𝑛≥0,(1.5) where the sequences {𝛼𝑛} and {𝑒𝑛} are such that (i)∑∞𝑛=0𝛼𝑛(1−𝛼𝑛)=∞,(ii)∑∞𝑛=0𝛼𝑛‖𝑒𝑛‖<∞.Then {𝑥𝑛} converges weakly to a fixed point of  𝑇.

Very recently, Ceng et al. [10] extended the algorithm (1.3) of Kim and Xu [9] to Krasnoselski-Mann's algorithm with perturbed mapping defined by the following: 𝑥𝑛+1=𝜆𝑛𝑥𝑛+1−𝜆𝑛𝑇𝑥𝑛+𝑒𝑛−𝜆𝑛𝜇𝑛𝐹𝑥𝑛,𝑛≥0,(1.6) where 𝜆𝑛,𝜇𝑛∈[0,1], and 𝐹 is a strongly accretive and strictly pseudocontractive mapping.

An important generalization of the class of nonexpansive mapping is asymptotically nonexpansive mapping (i.e., for 𝑇∶𝐶→𝐶, if there exists a sequence {𝑢𝑛}⊂[0,+∞), lim𝑛→∞𝑢𝑛=0 such that ‖𝑇𝑛𝑥−𝑇𝑛𝑦‖≤1+𝑢𝑛‖𝑥−𝑦‖,(1.7) for all 𝑥,𝑦∈𝐶 and 𝑛≥0), which was introduced by Goebel and Kirk [11]; they proved that if 𝐶 is a nonempty closed, convex, and bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive self-mapping has a fixed point. The class of asymptotically nonexpansive mapping has been studied by many authors and some recent results can be found in [12–17] and references cited therein.

Inspired and motivated by the above works, the purpose of this paper is to extend the results of Kim and Xu [9] from nonexpansive mapping to asymptotically nonexpansive mapping. We prove that the Krasnoselski-Mann iterative sequence converges weakly to the fixed point of asymptotically nonexpansive mapping.

2. Preliminaries

In this section, we collect some useful results which will be used in the following section.

We use the following notations:(i)⇀ for weak convergence and → for strong convergence,(ii)𝜔𝑤(𝑥𝑛)={𝑥∶∃𝑥𝑛𝑗⇀𝑥} denotes the weak 𝜔-limit set of {𝑥𝑛}.

It is well known that a Hilbert space 𝐻 satisfies Opial's condition [18]; that is, for each sequence {𝑥𝑛} in 𝐻 which converges weakly to a point 𝑥∈𝐻, one has liminf𝑛→∞‖‖𝑥𝑛‖‖−𝑥<liminf𝑛→∞‖‖𝑥𝑛‖‖−𝑦,(2.1) for all 𝑦∈𝐻, 𝑦≠𝑥.

Recall that given a closed convex subset of 𝐶 of a real Hilbert space 𝐻, the nearest point projection 𝑃𝐶 from 𝐻 onto 𝐶 assigns to each 𝑥∈𝐶 its nearest point denoted by 𝑃𝐶𝑥 in 𝐶 from 𝑥 to 𝐶; that is, 𝑃𝐶𝑥 is the unique point in 𝑋 with the property ‖‖𝑥−𝑃𝐶𝑥‖‖≤‖𝑥−𝑦‖,∀𝑦∈𝐶.(2.2)

A Banach space 𝑋 is said to have the Kadec-Klee property [19] if for any sequence {𝑥𝑛} in 𝑋, 𝑥𝑛⇀𝑥 and ‖𝑥𝑛‖→‖𝑥‖ imply that 𝑥𝑛→𝑥.

A mapping 𝑇 is said to be demiclosed at zero if whenever {𝑥𝑛} is a sequence in 𝐷(𝑇) such that {𝑥𝑛} converges weakly to 𝑥∈𝐷(𝑇) and {𝑇𝑥𝑛} converges strongly to zero, then 𝑇𝑥=0.

Lemma 2.1 (see [20]). Let 𝑋 be a real uniformly convex Banach space, let 𝐶 be a nonempty closed convex subset of 𝑋, and let 𝑇∶𝐶→𝑋 be an asymptotically nonexpansive mapping with a sequence {𝑢𝑛}⊂[0,∞) and lim𝑛→∞𝑢𝑛=0; then (𝐼−𝑇) is demiclosed at zero.

Lemma 2.2 (see [21]). Given a number 𝑟>0, a real Banach space is uniformly convex if and only if there exists a continuous strictly increasing function 𝜙∶[0,∞)→[0,∞), 𝜙(0)=0, such that ‖‖𝜆𝑥+(1−𝜆)𝑦2≤𝜆‖𝑥‖2+(1−𝜆)‖𝑦‖2−𝜆(1−𝜆)𝜙(‖𝑥−𝑦‖),(2.3) for all 𝜆∈[0,1] and 𝑥,𝑦∈𝑋 such that ‖𝑥‖≤𝑟 and ‖𝑦‖≤𝑟.

Lemma 2.3 (see [22]). Let 𝑋 be a real uniformly convex Banach space such that its dual 𝑋∗ has Kadec-Klee property. Let {𝑥𝑛} be a bounded sequence in 𝑋 and 𝑞1,𝑞2∈𝜔𝑤({𝑥𝑛}). Suppose that that lim𝑛→∞‖‖𝛼𝑥𝑛+(1−𝛼)𝑞1−𝑞2‖‖(2.4) exists for all 𝛼∈[0,1]. Then 𝑞1=𝑞2.

Lemma 2.4 (see [23]). Let {𝑎𝑛}, {𝑏𝑛}, and {𝑐𝑛} be sequences of nonnegative real numbers satisfying the inequality 𝑎𝑛+1≤1+𝑐𝑛𝑎𝑛+𝑏𝑛,𝑛≥1.(2.5) If ∑∞𝑛=1𝑐𝑛∑<+∞,∞𝑛=1𝑏𝑛<+∞, then (i) lim𝑛→∞𝑎𝑛 exists. (ii) In particular, if liminf𝑛→∞𝑎𝑛=0, one has lim𝑛→∞𝑎𝑛=0.

3. Main Results

We state our first theorem as follows.

Theorem 3.1. Suppose that 𝑋 is a uniformly convex Banach space, and 𝑋∗ has the Kadec-Klee property or 𝑋 satisfies Opial's property. Let 𝑇∶𝑋→𝑋 be an asymptotically nonexpansive mapping with ∑∞𝑛=0𝑢𝑛<∞. For any 𝑥0∈𝑋, the sequence {𝑥𝑛} is generated by the following Krasnoselski-Mann's algorithm: 𝑥𝑛+1=1−𝛼𝑛𝑥𝑛+𝛼𝑛𝑇𝑛𝑥𝑛+𝑒𝑛,𝑛≥0,(3.1) where {𝛼𝑛} and {𝑒𝑛} satisfy the following conditions: (i)0<𝑎<𝛼𝑛<𝑏<1, for some 𝑎,𝑏∈(0,1) and for all 𝑛≥0;(ii)∑∞𝑛=0𝛼𝑛‖𝑒𝑛‖<∞.If 𝐹(𝑇)≠∅, then the sequence {𝑥𝑛} converges weakly to a fixed point of  𝑇.

For the sake of convenience, we need the following lemmas.

Lemma 3.2. Let 𝑋 be a real normed linear space and let 𝑇∶𝑋→𝑋 be an asymptotically nonexpansive mapping with ∑∞𝑛=0𝑢𝑛<∞. Let {𝑥𝑛} be the sequence as defined in (3.1) and satisfy the conditions in Theorem 3.1. Suppose that 𝐹(𝑇)≠∅; then the limit lim𝑛→∞‖𝑥𝑛−𝑝‖ exists for 𝑝∈𝐹(𝑇).

Proof. By (3.1), one has ‖‖𝑥𝑛+1‖‖=‖‖−𝑝1−𝛼𝑛𝑥𝑛−𝑝+𝛼𝑛𝑇𝑛𝑥𝑛−𝑝+𝑒𝑛‖‖≤1−𝛼𝑛‖‖𝑥𝑛‖‖−𝑝+𝛼𝑛‖‖𝑇𝑛𝑥𝑛‖‖−𝑝+𝛼𝑛‖‖𝑒𝑛‖‖≤1−𝛼𝑛‖‖𝑥𝑛‖‖−𝑝+𝛼𝑛1+𝑢𝑛‖‖𝑥𝑛‖‖−𝑝+𝛼𝑛‖‖𝑒𝑛‖‖≤1+𝑢𝑛‖‖𝑥𝑛‖‖−𝑝+𝛼𝑛‖‖𝑒𝑛‖‖.(3.2) Since ∑∞𝑛=0𝑢𝑛<∞ and ∑∞𝑛=0𝛼𝑛‖𝑒𝑛‖<∞, we obtain from Lemma 2.4 that the limit lim𝑛→∞‖𝑥𝑛−𝑝‖ exists. Furthermore, the sequence {𝑥𝑛} is bounded.

Lemma 3.3. Let 𝑋 be a real uniformly convex Banach space and let 𝑇∶𝑋→𝑋 be an asymptotically nonexpansive mapping with ∑∞𝑛=0𝑢𝑛<∞. Let {𝑥𝑛} be the sequence as defined in (3.1) and satisfy the conditions in Theorem 3.1. Suppose that 𝐹(𝑇)≠∅; then lim𝑛→∞‖𝑡𝑥𝑛+(1−𝑡)𝑝−𝑞‖ exists for all 𝑡∈[0,1] and 𝑝,𝑞∈𝐹(𝑇).

Proof. Let 𝑑𝑛(𝑡)=‖𝑡𝑥𝑛+(1−𝑡)𝑝−𝑞‖; then lim𝑛→∞𝑑𝑛(0)=‖𝑝−𝑞‖ exists. It follows from Lemma 3.2 that lim𝑛→∞𝑑𝑛(1)=lim𝑛→∞‖𝑥𝑛−𝑞‖ exists. Next, we show that lim𝑛→∞𝑑𝑛(𝑡) exists for any 𝑡∈(0,1).
Let 𝑇𝑛𝑥∶=(1−𝛼𝑛)𝑥+𝛼𝑛𝑇𝑛𝑥+𝛼𝑛𝑒𝑛, for all 𝑥∈𝑋. For any 𝑥,𝑧∈𝑋, one has ‖‖𝑇𝑛𝑥−𝑇𝑛𝑧‖‖=‖‖1−𝛼𝑛(𝑥−𝑧)+𝛼𝑛(𝑇𝑛𝑥−𝑇𝑛𝑧)‖‖≤1−𝛼𝑛‖𝑥−𝑧‖+𝛼𝑛‖𝑇𝑛𝑥−𝑇𝑛≤𝑧‖1−𝛼𝑛‖𝑥−𝑧‖+𝛼𝑛1+𝑢𝑛≤‖𝑥−𝑧‖1+𝑢𝑛‖𝑥−𝑧‖.(3.3) Set 𝑆𝑛,𝑚=𝑇𝑛+𝑚−1𝑇𝑛+𝑚−2⋯𝑇𝑛, 𝑚≥1. The rest of the proof is the same as Lemma  3.3 of [14, 16]. This completes the proof of Lemma 3.3.

Now, we give the proof of Theorem 3.1.

Proof. Let 𝑝∈𝐹(𝑇). With the help of Lemma 2.2 and the inequality ‖𝑎+𝑏‖2≤‖𝑎‖2+2‖𝑎‖⋅‖𝑏‖+‖𝑏‖2, one has ‖‖𝑥𝑛+1‖‖−𝑝2=‖‖1−𝛼𝑛𝑥𝑛−𝑝+𝛼𝑛𝑇𝑛𝑥𝑛−𝑝+𝛼𝑛𝑒𝑛‖‖2≤‖‖(1−𝛼𝑛)(𝑥𝑛−𝑝)+𝛼𝑛(𝑇𝑛𝑥𝑛‖‖−𝑝)2+2𝛼𝑛‖‖𝑒𝑛‖‖⋅‖‖1−𝛼𝑛𝑥𝑛−𝑝+𝛼𝑛𝑇𝑛𝑥𝑛‖‖−𝑝+𝛼2𝑛‖‖𝑒𝑛‖‖2≤1−𝛼𝑛‖‖𝑥𝑛‖‖−𝑝2+𝛼𝑛‖‖𝑇𝑛𝑥𝑛‖‖−𝑝2−𝛼𝑛1−𝛼𝑛𝜙‖‖𝑥𝑛−𝑇𝑛𝑥𝑛‖‖+2𝛼𝑛‖‖𝑒𝑛‖‖1−𝛼𝑛‖‖𝑥𝑛‖‖−𝑝+𝛼𝑛1+𝑢𝑛‖‖𝑥𝑛‖‖−𝑝+𝛼2𝑛‖‖𝑒𝑛‖‖2≤1−𝛼𝑛‖‖𝑥𝑛‖‖−𝑝2+𝛼𝑛1+𝑢𝑛2‖‖𝑥𝑛‖‖−𝑝2−𝛼𝑛1−𝛼𝑛𝜙‖‖𝑥𝑛−𝑇𝑛𝑥𝑛‖‖+2𝛼𝑛‖‖𝑒𝑛‖‖1+𝑢𝑛‖‖𝑥𝑛‖‖−𝑝+𝛼2𝑛‖‖𝑒𝑛‖‖2≤1+𝑢𝑛2‖‖𝑥𝑛‖‖−𝑝2−𝛼𝑛1−𝛼𝑛𝜙‖‖𝑥𝑛−𝑇𝑛𝑥𝑛‖‖+2𝛼𝑛‖‖𝑒𝑛‖‖1+𝑢𝑛‖‖𝑥𝑛‖‖−𝑝+𝛼2𝑛‖‖𝑒𝑛‖‖2,(3.4) which follows that 𝑎‖‖𝑥(1−𝑏)𝜙𝑛−𝑇𝑛𝑥𝑛‖‖≤𝛼𝑛1−𝛼𝑛𝜙‖‖𝑥𝑛−𝑇𝑛𝑥𝑛‖‖≤1+𝑢𝑛2‖‖𝑥𝑛‖‖−𝑝2−‖‖𝑥𝑛+1‖‖−𝑝2+2𝛼𝑛‖‖𝑒𝑛‖‖1+𝑢𝑛‖‖𝑥𝑛‖‖−𝑝+𝛼2𝑛‖‖𝑒𝑛‖‖2.(3.5) This implies that ∞𝑛=0𝜙‖‖𝑥𝑛−𝑇𝑛𝑥𝑛‖‖<∞.(3.6) Therefore lim𝑛→∞𝜙(‖𝑥𝑛−𝑇𝑛𝑥𝑛‖)=0. Since 𝜙 is strictly increasing and continuous function with 𝜙(0)=0, then lim𝑛→∞‖𝑥𝑛−𝑇𝑛𝑥𝑛‖=0. Also, one has the following inequalities: ‖‖𝑥𝑛+1−𝑥𝑛‖‖≤𝛼𝑛‖‖𝑇𝑛𝑥𝑛−𝑥𝑛‖‖+𝛼𝑛‖‖𝑒𝑛‖‖‖‖𝑇⟶0as𝑛⟶∞,(3.7)𝑛𝑥𝑛+1−𝑥𝑛+1‖‖=‖‖𝑇𝑛𝑥𝑛+1−1−𝛼𝑛𝑥𝑛−𝛼𝑛𝑇𝑛𝑥𝑛+𝑒𝑛‖‖=‖‖𝑇𝑛𝑥𝑛+1−𝑇𝑛𝑥𝑛+1−𝛼𝑛𝑇𝑛𝑥𝑛−𝑥𝑛−𝛼𝑛𝑒𝑛‖‖≤‖‖𝑇𝑛𝑥𝑛+1−𝑇𝑛𝑥𝑛‖‖+1−𝛼𝑛‖‖𝑇𝑛𝑥𝑛−𝑥𝑛‖‖+𝛼𝑛‖‖𝑒𝑛‖‖≤1+𝑢𝑛‖‖𝑥𝑛+1−𝑥𝑛‖‖+1−𝛼𝑛‖‖𝑇𝑛𝑥𝑛−𝑥𝑛‖‖+𝛼𝑛‖‖𝑒𝑛‖‖=1+𝑢𝑛‖‖𝛼𝑛𝑇𝑛𝑥𝑛−𝑥𝑛+𝛼𝑛𝑒𝑛‖‖+1−𝛼𝑛‖‖𝑇𝑛𝑥𝑛−𝑥𝑛‖‖+𝛼𝑛‖‖𝑒𝑛‖‖≤1+𝑢𝑛𝛼𝑛‖‖𝑇𝑛𝑥𝑛−𝑥𝑛‖‖+1−𝛼𝑛‖‖𝑇𝑛𝑥𝑛−𝑥𝑛‖‖+1+𝑢𝑛𝛼𝑛‖‖𝑒𝑛‖‖+𝛼𝑛‖‖𝑒𝑛‖‖≤1+𝑢𝑛‖‖𝑇𝑛𝑥𝑛−𝑥𝑛‖‖+1+𝑢𝑛𝛼𝑛‖‖𝑒𝑛‖‖+𝛼𝑛‖‖𝑒𝑛‖‖⟶0as𝑛⟶∞.(3.8) On the other hand, ‖‖𝑥𝑛+1−𝑇𝑥𝑛+1‖‖≤‖‖𝑥𝑛+1−𝑇𝑛+1𝑥𝑛+1‖‖+‖‖𝑇𝑛+1𝑥𝑛+1−𝑇𝑥𝑛+1‖‖≤‖‖𝑥𝑛+1−𝑇𝑛+1𝑥𝑛+1‖‖+1+𝑢1‖‖𝑇𝑛𝑥𝑛+1−𝑥𝑛+1‖‖⟶0as𝑛⟶∞.(3.9) By (3.7)–(3.9), we obtain ‖‖𝑥𝑛−𝑇𝑥𝑛‖‖≤‖‖𝑥𝑛−𝑥𝑛+1‖‖+‖‖𝑥𝑛+1−𝑇𝑥𝑛+1‖‖+‖‖𝑇𝑥𝑛+1−𝑇𝑥𝑛‖‖≤2+𝑢1‖‖𝑥𝑛−𝑥𝑛+1‖‖+‖‖𝑥𝑛+1−𝑇𝑥𝑛+1‖‖⟶0as𝑛⟶∞,(3.10) that is, lim𝑛→∞‖𝑥𝑛−𝑇𝑥𝑛‖=0.
From Lemma 3.2, we know that {𝑥𝑛} is bounded. Since 𝑋 is a uniformly convex Banach space, {𝑥𝑛} has a convergent subsequence {𝑥𝑛𝑗}. By the demiclosedness principle of 𝐼−𝑇, we obtain 𝜔𝑤⊆𝐹(𝑇). The rest of proof is followed by the standard argument in Theorem  3.3 of Kim and Xu [9]. This completes the proof.

Theorem 3.4. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻 and let 𝑇∶𝐶→𝐶 be an asymptotically nonexpansive mapping with ∑∞𝑛=0𝑢𝑛<∞. For any 𝑥0∈𝑋, the sequence {𝑥𝑛} is generated by either 𝑥𝑛+1=1−𝛼𝑛𝑥𝑛+𝛼𝑛𝑃𝐶𝑇𝑛𝑥𝑛+𝑒𝑛,𝑛≥0,(3.11) or 𝑥𝑛+1=𝑃𝐶1−𝛼𝑛𝑥𝑛+𝛼𝑛𝑇𝑛𝑥𝑛+𝑒𝑛,𝑛≥0,(3.12) where the sequences {𝛼𝑛} and {𝑒𝑛} are such that (i)0<𝑎<𝛼𝑛<𝑏<1, for some 𝑎,𝑏∈(0,1) and for all 𝑛≥0;(ii)∑∞𝑛=0𝛼𝑛‖𝑒𝑛‖<∞. If 𝐹(𝑇)≠∅, then {𝑥𝑛} converges weakly to a fixed point of  𝑇.

Proof. Let 𝑝∈𝐹(𝑇). By (3.11), one has ‖‖𝑥𝑛+1‖‖=‖‖−𝑝1−𝛼𝑛𝑥𝑛+𝛼𝑛𝑃𝐶𝑇𝑛𝑥𝑛+𝑒𝑛‖‖≤−𝑝1−𝛼𝑛‖‖𝑥𝑛‖‖−𝑝+𝛼𝑛‖‖𝑃𝐶𝑇𝑛𝑥𝑛+𝑒𝑛‖‖≤−𝑝1−𝛼𝑛‖‖𝑥𝑛‖‖−𝑝+𝛼𝑛‖‖𝑇𝑛𝑥𝑛+𝑒𝑛‖‖≤−𝑝1−𝛼𝑛‖‖𝑥𝑛‖‖−𝑝+𝛼𝑛1+𝑢𝑛‖𝑥𝑛−𝑝‖+𝛼𝑛‖‖𝑒𝑛‖‖≤1+𝑢𝑛‖‖𝑥𝑛‖‖−𝑝+𝛼𝑛‖‖𝑒𝑛‖‖.(3.13) Notice the condition (ii) and ∑∞𝑛=0𝑢𝑛<∞; by Lemma 2.4, lim𝑛→∞‖𝑥𝑛−𝑝‖ exists. Hence, {𝑥𝑛} is bounded.
By the well-known inequality ‖𝑡𝑥+(1−𝑡)𝑦‖2=𝑡‖𝑥‖2+(1−𝑡)‖𝑦‖2−𝑡(1−𝑡)‖𝑥−𝑦‖2, for all 𝑥,𝑦∈𝐻 and 𝑡∈[0,1], we obtain ‖𝑥𝑛+1−𝑝‖2=‖‖1−𝛼𝑛𝑥𝑛+𝛼𝑛𝑃𝐶𝑇𝑛𝑥𝑛+𝑒𝑛‖‖−𝑝2=‖‖1−𝛼𝑛𝑥𝑛−𝑝+𝛼𝑛𝑇𝑛𝑥𝑛−𝑝+𝛼𝑛𝑃𝐶𝑇𝑛𝑥𝑛+𝑒𝑛−𝑇𝑛𝑥𝑛‖‖2≤‖‖1−𝛼𝑛𝑥𝑛−𝑝+𝛼𝑛𝑇𝑛𝑥𝑛‖‖−𝑝2+‖‖𝛼𝑛𝑃𝐶𝑇𝑛𝑥𝑛+𝑒𝑛−𝑇𝑛𝑥𝑛‖‖2+2𝛼𝑛‖‖1−𝛼𝑛𝑥𝑛−𝑝+𝛼𝑛𝑇𝑛𝑥𝑛‖‖⋅‖‖𝑃−𝑝𝐶𝑇𝑛𝑥𝑛+𝑒𝑛−𝑇𝑛𝑥𝑛‖‖≤1−𝛼𝑛‖‖𝑥𝑛‖‖−𝑝2+𝛼𝑛‖‖𝑇𝑛𝑥𝑛‖‖−𝑝2−𝛼𝑛1−𝛼𝑛‖‖𝑥𝑛−𝑇𝑛𝑥𝑛‖‖2+2𝛼𝑛1−𝛼𝑛‖‖𝑥𝑛‖‖−𝑝+𝛼𝑛1+𝑢𝑛‖‖𝑥𝑛‖‖‖‖𝑒−𝑝𝑛‖‖≤1+𝑢𝑛2‖‖𝑥𝑛‖‖−𝑝2−𝛼𝑛1−𝛼𝑛‖‖𝑥𝑛−𝑇𝑛𝑥𝑛‖‖2+21+𝑢𝑛𝛼𝑛‖‖𝑒𝑛‖‖‖‖𝑥𝑛‖‖.−𝑝(3.14) That is, ‖‖𝑥𝑎(1−𝑏)𝑛−𝑇𝑛𝑥𝑛‖‖2≤𝛼𝑛1−𝛼𝑛‖‖𝑥𝑛−𝑇𝑛𝑥𝑛‖‖2≤1+𝑢𝑛2‖‖𝑥𝑛‖‖−𝑝2−‖‖𝑥𝑛+1‖‖−𝑝2+21+𝑢𝑛𝛼𝑛‖‖𝑒𝑛‖‖‖‖𝑥𝑛‖‖.−𝑝(3.15) This implies that ∞𝑛=0‖‖𝑥𝑛−𝑇𝑛𝑥𝑛‖‖2<∞.(3.16) Therefore lim𝑛→∞‖𝑥𝑛−𝑇𝑛𝑥𝑛‖=0. We also have ‖‖𝑥𝑛+1−𝑥𝑛‖‖=‖‖1−𝛼𝑛𝑥𝑛+𝛼𝑛𝑃𝐶𝑇𝑛𝑥𝑛+𝑒𝑛−𝑥𝑛‖‖≤𝛼𝑛‖‖𝑇𝑛𝑥𝑛−𝑥𝑛‖‖+𝛼𝑛‖‖𝑒𝑛‖‖⟶0as𝑛⟶∞,‖𝑇𝑛𝑥𝑛+1−𝑥𝑛+1‖‖𝑇‖=𝑛𝑥𝑛+1−1−𝛼𝑛𝑥𝑛−𝛼𝑛𝑃𝐶𝑇𝑛𝑥𝑛+𝑒𝑛‖‖=‖‖𝑇𝑛𝑥𝑛+1−𝑇𝑛𝑥𝑛+𝑇𝑛𝑥𝑛−𝑥𝑛+𝛼𝑛𝑥𝑛−𝑃𝐶𝑇𝑛𝑥𝑛+𝑒𝑛‖‖≤1+𝑢𝑛‖‖𝑥𝑛+1−𝑥𝑛‖‖+1+𝛼𝑛‖‖𝑇𝑛𝑥𝑛−𝑥𝑛‖‖+𝛼𝑛‖‖𝑒𝑛‖‖⟶0as𝑛⟶∞.(3.17) It follows from (3.9) and (3.10) that lim𝑛→∞‖𝑥𝑛−𝑇𝑥𝑛‖=0. Since a Hilbert space 𝐻 must be a uniformly convex Banach space and satisfy Opial's property, then the rest of proof is the same as Theorem 3.1. So it is omitted.

Acknowledgment

This work was supported by The National Natural Science Foundations of China (60970149) and The Natural Science Foundations of Jiangxi Province (2009GZS0021, 2007GQS2063).

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