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

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 [13] and the split feasibility problem [46]. 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. 𝑇𝑋𝑋 is called a nonexpansive mapping if for any 𝑥,𝑦𝑋, 𝑇𝑥𝑇𝑦𝑥𝑦. Krasnoselski-Mann's iteration method for finding fixed points of 𝑇 is defined by foranyinitial𝑥0𝑋,𝑥𝑛+1=1𝛼𝑛𝑥𝑛+𝛼𝑛𝑇𝑥𝑛,𝑛0,(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,𝑛𝑥𝑛,(1.2) where {𝜆𝑛}(0,2), {𝜔𝑖}𝑚𝑖=1 are positive weights such that 𝑚𝑖=1𝜔𝑖=1, 𝑃𝑖 is the projection of a signal 𝑥𝐻 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=1𝛼𝑛𝑥𝑛+𝛼𝑛𝑇𝑥𝑛+𝑒𝑛,𝑛0,(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 𝑇𝑋𝑋 be a nonexpansive mapping such that 𝐹(𝑇)(𝐹(𝑇) denotes the set of fixed points of  𝑇, that is, 𝐹(𝑇)={𝑥𝑋𝑇𝑥=𝑥}). Given an initial guess 𝑥0𝑋. Let {𝑥𝑛} be generated by (1.3) and satisfy the following properties: (i)𝑛=0𝛼𝑛(1𝛼𝑛)=,(ii)𝑛=0𝛼𝑛𝑒𝑛<. 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 𝑇𝐶𝐶 a nonexpansive mapping with 𝐹(𝑇). Given an initial guess 𝑥0𝑋. Let {𝑥𝑛} be generated by either 𝑥𝑛+1=1𝛼𝑛𝑥𝑛+𝛼𝑛𝑃𝐶𝑇𝑥𝑛+𝑒𝑛,𝑛0,(1.4) or 𝑥𝑛+1=𝑃𝐶1𝛼𝑛𝑥𝑛+𝛼𝑛𝑇𝑥𝑛+𝑒𝑛,𝑛0,(1.5) where the sequences {𝛼𝑛} and {𝑒𝑛} are such that (i)𝑛=0𝛼𝑛(1𝛼𝑛)=,(ii)𝑛=0𝛼𝑛𝑒𝑛<.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+1𝜆𝑛𝑇𝑥𝑛+𝑒𝑛𝜆𝑛𝜇𝑛𝐹𝑥𝑛,𝑛0,(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 𝑇𝐶𝐶, if there exists a sequence {𝑢𝑛}[0,+), lim𝑛𝑢𝑛=0 such that 𝑇𝑛𝑥𝑇𝑛𝑦1+𝑢𝑛𝑥𝑦,(1.7) for all 𝑥,𝑦𝐶 and 𝑛0), 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 [1217] 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)={𝑥𝑥𝑛𝑗𝑥} 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 𝑥𝐻, one has liminf𝑛𝑥𝑛𝑥<liminf𝑛𝑥𝑛𝑦,(2.1) for all 𝑦𝐻, 𝑦𝑥.

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 𝑥𝐶 its nearest point denoted by 𝑃𝐶𝑥 in 𝐶 from 𝑥 to 𝐶; that is, 𝑃𝐶𝑥 is the unique point in 𝑋 with the property 𝑥𝑃𝐶𝑥𝑥𝑦,𝑦𝐶.(2.2)

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

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 𝑥𝐷(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 𝑇𝐶𝑋 be an asymptotically nonexpansive mapping with a sequence {𝑢𝑛}[0,) and lim𝑛𝑢𝑛=0; then (𝐼𝑇) 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+(1𝜆)𝑦2𝜆𝑥2+(1𝜆)𝑦2𝜆(1𝜆)𝜙(𝑥𝑦),(2.3) for all 𝜆[0,1] and 𝑥,𝑦𝑋 such that 𝑥𝑟 and 𝑦𝑟.

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,𝑞2𝜔𝑤({𝑥𝑛}). Suppose that that lim𝑛𝛼𝑥𝑛+(1𝛼)𝑞1𝑞2(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+11+𝑐𝑛𝑎𝑛+𝑏𝑛,𝑛1.(2.5) If 𝑛=1𝑐𝑛<+,𝑛=1𝑏𝑛<+, then (i) lim𝑛𝑎𝑛 exists. (ii) In particular, if liminf𝑛𝑎𝑛=0, one has lim𝑛𝑎𝑛=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 𝑇𝑋𝑋 be an asymptotically nonexpansive mapping with 𝑛=0𝑢𝑛<. For any 𝑥0𝑋, the sequence {𝑥𝑛} is generated by the following Krasnoselski-Mann's algorithm: 𝑥𝑛+1=1𝛼𝑛𝑥𝑛+𝛼𝑛𝑇𝑛𝑥𝑛+𝑒𝑛,𝑛0,(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 𝑛0;(ii)𝑛=0𝛼𝑛𝑒𝑛<.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 𝑇𝑋𝑋 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 lim𝑛𝑥𝑛𝑝 exists for 𝑝𝐹(𝑇).

Proof. By (3.1), one has 𝑥𝑛+1=𝑝1𝛼𝑛𝑥𝑛𝑝+𝛼𝑛𝑇𝑛𝑥𝑛𝑝+𝑒𝑛1𝛼𝑛𝑥𝑛𝑝+𝛼𝑛𝑇𝑛𝑥𝑛𝑝+𝛼𝑛𝑒𝑛1𝛼𝑛𝑥𝑛𝑝+𝛼𝑛1+𝑢𝑛𝑥𝑛𝑝+𝛼𝑛𝑒𝑛1+𝑢𝑛𝑥𝑛𝑝+𝛼𝑛𝑒𝑛.(3.2) Since 𝑛=0𝑢𝑛< and 𝑛=0𝛼𝑛𝑒𝑛<, we obtain from Lemma 2.4 that the limit lim𝑛𝑥𝑛𝑝 exists. Furthermore, the sequence {𝑥𝑛} is bounded.

Lemma 3.3. Let 𝑋 be a real uniformly convex Banach space and let 𝑇𝑋𝑋 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 lim𝑛𝑡𝑥𝑛+(1𝑡)𝑝𝑞 exists for all 𝑡[0,1] and 𝑝,𝑞𝐹(𝑇).

Proof. Let 𝑑𝑛(𝑡)=𝑡𝑥𝑛+(1𝑡)𝑝𝑞; then lim𝑛𝑑𝑛(0)=𝑝𝑞 exists. It follows from Lemma 3.2 that lim𝑛𝑑𝑛(1)=lim𝑛𝑥𝑛𝑞 exists. Next, we show that lim𝑛𝑑𝑛(𝑡) exists for any 𝑡(0,1).
Let 𝑇𝑛𝑥=(1𝛼𝑛)𝑥+𝛼𝑛𝑇𝑛𝑥+𝛼𝑛𝑒𝑛, for all 𝑥𝑋. For any 𝑥,𝑧𝑋, one has 𝑇𝑛𝑥𝑇𝑛𝑧=1𝛼𝑛(𝑥𝑧)+𝛼𝑛(𝑇𝑛𝑥𝑇𝑛𝑧)1𝛼𝑛𝑥𝑧+𝛼𝑛𝑇𝑛𝑥𝑇𝑛𝑧1𝛼𝑛𝑥𝑧+𝛼𝑛1+𝑢𝑛𝑥𝑧1+𝑢𝑛𝑥𝑧.(3.3) Set 𝑆𝑛,𝑚=𝑇𝑛+𝑚1𝑇𝑛+𝑚2𝑇𝑛, 𝑚1. 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 𝑝𝐹(𝑇). With the help of Lemma 2.2 and the inequality 𝑎+𝑏2𝑎2+2𝑎𝑏+𝑏2, one has 𝑥𝑛+1𝑝2=1𝛼𝑛𝑥𝑛𝑝+𝛼𝑛𝑇𝑛𝑥𝑛𝑝+𝛼𝑛𝑒𝑛2(1𝛼𝑛)(𝑥𝑛𝑝)+𝛼𝑛(𝑇𝑛𝑥𝑛𝑝)2+2𝛼𝑛𝑒𝑛1𝛼𝑛𝑥𝑛𝑝+𝛼𝑛𝑇𝑛𝑥𝑛𝑝+𝛼2𝑛𝑒𝑛21𝛼𝑛𝑥𝑛𝑝2+𝛼𝑛𝑇𝑛𝑥𝑛𝑝2𝛼𝑛1𝛼𝑛𝜙𝑥𝑛𝑇𝑛𝑥𝑛+2𝛼𝑛𝑒𝑛1𝛼𝑛𝑥𝑛𝑝+𝛼𝑛1+𝑢𝑛𝑥𝑛𝑝+𝛼2𝑛𝑒𝑛21𝛼𝑛𝑥𝑛𝑝2+𝛼𝑛1+𝑢𝑛2𝑥𝑛𝑝2𝛼𝑛1𝛼𝑛𝜙𝑥𝑛𝑇𝑛𝑥𝑛+2𝛼𝑛𝑒𝑛1+𝑢𝑛𝑥𝑛𝑝+𝛼2𝑛𝑒𝑛21+𝑢𝑛2𝑥𝑛𝑝2𝛼𝑛1𝛼𝑛𝜙𝑥𝑛𝑇𝑛𝑥𝑛+2𝛼𝑛𝑒𝑛1+𝑢𝑛𝑥𝑛𝑝+𝛼2𝑛𝑒𝑛2,(3.4) which follows that 𝑎𝑥(1𝑏)𝜙𝑛𝑇𝑛𝑥𝑛𝛼𝑛1𝛼𝑛𝜙𝑥𝑛𝑇𝑛𝑥𝑛1+𝑢𝑛2𝑥𝑛𝑝2𝑥𝑛+1𝑝2+2𝛼𝑛𝑒𝑛1+𝑢𝑛𝑥𝑛𝑝+𝛼2𝑛𝑒𝑛2.(3.5) This implies that 𝑛=0𝜙𝑥𝑛𝑇𝑛𝑥𝑛<.(3.6) Therefore lim𝑛𝜙(𝑥𝑛𝑇𝑛𝑥𝑛)=0. Since 𝜙 is strictly increasing and continuous function with 𝜙(0)=0, then lim𝑛𝑥𝑛𝑇𝑛𝑥𝑛=0. Also, one has the following inequalities: 𝑥𝑛+1𝑥𝑛𝛼𝑛𝑇𝑛𝑥𝑛𝑥𝑛+𝛼𝑛𝑒𝑛𝑇0as𝑛,(3.7)𝑛𝑥𝑛+1𝑥𝑛+1=𝑇𝑛𝑥𝑛+11𝛼𝑛𝑥𝑛𝛼𝑛𝑇𝑛𝑥𝑛+𝑒𝑛=𝑇𝑛𝑥𝑛+1𝑇𝑛𝑥𝑛+1𝛼𝑛𝑇𝑛𝑥𝑛𝑥𝑛𝛼𝑛𝑒𝑛𝑇𝑛𝑥𝑛+1𝑇𝑛𝑥𝑛+1𝛼𝑛𝑇𝑛𝑥𝑛𝑥𝑛+𝛼𝑛𝑒𝑛1+𝑢𝑛𝑥𝑛+1𝑥𝑛+1𝛼𝑛𝑇𝑛𝑥𝑛𝑥𝑛+𝛼𝑛𝑒𝑛=1+𝑢𝑛𝛼𝑛𝑇𝑛𝑥𝑛𝑥𝑛+𝛼𝑛𝑒𝑛+1𝛼𝑛𝑇𝑛𝑥𝑛𝑥𝑛+𝛼𝑛𝑒𝑛1+𝑢𝑛𝛼𝑛𝑇𝑛𝑥𝑛𝑥𝑛+1𝛼𝑛𝑇𝑛𝑥𝑛𝑥𝑛+1+𝑢𝑛𝛼𝑛𝑒𝑛+𝛼𝑛𝑒𝑛1+𝑢𝑛𝑇𝑛𝑥𝑛𝑥𝑛+1+𝑢𝑛𝛼𝑛𝑒𝑛+𝛼𝑛𝑒𝑛0as𝑛.(3.8) On the other hand, 𝑥𝑛+1𝑇𝑥𝑛+1𝑥𝑛+1𝑇𝑛+1𝑥𝑛+1+𝑇𝑛+1𝑥𝑛+1𝑇𝑥𝑛+1𝑥𝑛+1𝑇𝑛+1𝑥𝑛+1+1+𝑢1𝑇𝑛𝑥𝑛+1𝑥𝑛+10as𝑛.(3.9) By (3.7)–(3.9), we obtain 𝑥𝑛𝑇𝑥𝑛𝑥𝑛𝑥𝑛+1+𝑥𝑛+1𝑇𝑥𝑛+1+𝑇𝑥𝑛+1𝑇𝑥𝑛2+𝑢1𝑥𝑛𝑥𝑛+1+𝑥𝑛+1𝑇𝑥𝑛+10as𝑛,(3.10) that is, lim𝑛𝑥𝑛𝑇𝑥𝑛=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 𝐼𝑇, we obtain 𝜔𝑤𝐹(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 𝑇𝐶𝐶 be an asymptotically nonexpansive mapping with 𝑛=0𝑢𝑛<. For any 𝑥0𝑋, the sequence {𝑥𝑛} is generated by either 𝑥𝑛+1=1𝛼𝑛𝑥𝑛+𝛼𝑛𝑃𝐶𝑇𝑛𝑥𝑛+𝑒𝑛,𝑛0,(3.11) or 𝑥𝑛+1=𝑃𝐶1𝛼𝑛𝑥𝑛+𝛼𝑛𝑇𝑛𝑥𝑛+𝑒𝑛,𝑛0,(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 𝑛0;(ii)𝑛=0𝛼𝑛𝑒𝑛<. If 𝐹(𝑇), then {𝑥𝑛} converges weakly to a fixed point of  𝑇.

Proof. Let 𝑝𝐹(𝑇). By (3.11), one has 𝑥𝑛+1=𝑝1𝛼𝑛𝑥𝑛+𝛼𝑛𝑃𝐶𝑇𝑛𝑥𝑛+𝑒𝑛𝑝1𝛼𝑛𝑥𝑛𝑝+𝛼𝑛𝑃𝐶𝑇𝑛𝑥𝑛+𝑒𝑛𝑝1𝛼𝑛𝑥𝑛𝑝+𝛼𝑛𝑇𝑛𝑥𝑛+𝑒𝑛𝑝1𝛼𝑛𝑥𝑛𝑝+𝛼𝑛1+𝑢𝑛𝑥𝑛𝑝+𝛼𝑛𝑒𝑛1+𝑢𝑛𝑥𝑛𝑝+𝛼𝑛𝑒𝑛.(3.13) Notice the condition (ii) and 𝑛=0𝑢𝑛<; by Lemma 2.4, lim𝑛𝑥𝑛𝑝 exists. Hence, {𝑥𝑛} is bounded.
By the well-known inequality 𝑡𝑥+(1𝑡)𝑦2=𝑡𝑥2+(1𝑡)𝑦2𝑡(1𝑡)𝑥𝑦2, for all 𝑥,𝑦𝐻 and 𝑡[0,1], we obtain 𝑥𝑛+1𝑝2=1𝛼𝑛𝑥𝑛+𝛼𝑛𝑃𝐶𝑇𝑛𝑥𝑛+𝑒𝑛𝑝2=1𝛼𝑛𝑥𝑛𝑝+𝛼𝑛𝑇𝑛𝑥𝑛𝑝+𝛼𝑛𝑃𝐶𝑇𝑛𝑥𝑛+𝑒𝑛𝑇𝑛𝑥𝑛21𝛼𝑛𝑥𝑛𝑝+𝛼𝑛𝑇𝑛𝑥𝑛𝑝2+𝛼𝑛𝑃𝐶𝑇𝑛𝑥𝑛+𝑒𝑛𝑇𝑛𝑥𝑛2+2𝛼𝑛1𝛼𝑛𝑥𝑛𝑝+𝛼𝑛𝑇𝑛𝑥𝑛𝑃𝑝𝐶𝑇𝑛𝑥𝑛+𝑒𝑛𝑇𝑛𝑥𝑛1𝛼𝑛𝑥𝑛𝑝2+𝛼𝑛𝑇𝑛𝑥𝑛𝑝2𝛼𝑛1𝛼𝑛𝑥𝑛𝑇𝑛𝑥𝑛2+2𝛼𝑛1𝛼𝑛𝑥𝑛𝑝+𝛼𝑛1+𝑢𝑛𝑥𝑛𝑒𝑝𝑛1+𝑢𝑛2𝑥𝑛𝑝2𝛼𝑛1𝛼𝑛𝑥𝑛𝑇𝑛𝑥𝑛2+21+𝑢𝑛𝛼𝑛𝑒𝑛𝑥𝑛.𝑝(3.14) That is, 𝑥𝑎(1𝑏)𝑛𝑇𝑛𝑥𝑛2𝛼𝑛1𝛼𝑛𝑥𝑛𝑇𝑛𝑥𝑛21+𝑢𝑛2𝑥𝑛𝑝2𝑥𝑛+1𝑝2+21+𝑢𝑛𝛼𝑛𝑒𝑛𝑥𝑛.𝑝(3.15) This implies that 𝑛=0𝑥𝑛𝑇𝑛𝑥𝑛2<.(3.16) Therefore lim𝑛𝑥𝑛𝑇𝑛𝑥𝑛=0. We also have 𝑥𝑛+1𝑥𝑛=1𝛼𝑛𝑥𝑛+𝛼𝑛𝑃𝐶𝑇𝑛𝑥𝑛+𝑒𝑛𝑥𝑛𝛼𝑛𝑇𝑛𝑥𝑛𝑥𝑛+𝛼𝑛𝑒𝑛0as𝑛,𝑇𝑛𝑥𝑛+1𝑥𝑛+1𝑇=𝑛𝑥𝑛+11𝛼𝑛𝑥𝑛𝛼𝑛𝑃𝐶𝑇𝑛𝑥𝑛+𝑒𝑛=𝑇𝑛𝑥𝑛+1𝑇𝑛𝑥𝑛+𝑇𝑛𝑥𝑛𝑥𝑛+𝛼𝑛𝑥𝑛𝑃𝐶𝑇𝑛𝑥𝑛+𝑒𝑛1+𝑢𝑛𝑥𝑛+1𝑥𝑛+1+𝛼𝑛𝑇𝑛𝑥𝑛𝑥𝑛+𝛼𝑛𝑒𝑛0as𝑛.(3.17) It follows from (3.9) and (3.10) that lim𝑛𝑥𝑛𝑇𝑥𝑛=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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