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

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,𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑥_𝑛,𝑛≥0,(1.1) where {𝛼_𝑛} 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:𝑥_𝑛+1=𝑥_𝑛+𝜆_𝑛^𝑚_𝑖=1𝜔_𝑖𝑃_𝑖𝑥_𝑛+𝑐_𝑖,𝑛−𝑥_𝑛,(1.2) where {𝜆_𝑛}⊆(0,2), {𝜔_𝑖}^𝑚𝑖=1 are positive weights such that ∑^𝑚𝑖=1𝜔_𝑖=1, 𝑃_𝑖 is the projection of a signal 𝑥∈𝐻 onto a closed convex subset 𝑆_𝑖 of 𝐻, and 𝑐_𝑖,𝑛 stands for the error made in computing the projection onto 𝑆_𝑖 at each iteration 𝑛. He firstly proved that the sequence {𝑥_𝑛} generated by (1.2) converges weakly to a point in 𝐺, where 𝐺∶=⋂^𝑚𝑖=1𝑆_𝑖.

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: 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑥_𝑛+𝑒_𝑛,𝑛≥0,(1.3) where 𝛼_𝑛∶=𝜆_𝑛/2∈(0,1), 𝑒_𝑛∶=2∑^𝑚𝑖=1𝜔_𝑖𝑐_𝑖,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 {𝑥_𝑛} be generated by (1.3) and satisfy the following properties: (i)∑^∞𝑛=0𝛼_𝑛(1−𝛼_𝑛)=∞,(ii)∑^∞𝑛=0𝛼_𝑛‖𝑒_𝑛‖<∞. Then the sequence {𝑥_𝑛} 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 {𝑥_𝑛} be generated by either 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑃_𝐶𝑇𝑥_𝑛+𝑒_𝑛,𝑛≥0,(1.4) or 𝑥_𝑛+1=𝑃_𝐶1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇𝑥_𝑛+𝑒_𝑛,𝑛≥0,(1.5) where the sequences {𝛼_𝑛} and {𝑒_𝑛} are such that (i)∑^∞𝑛=0𝛼_𝑛(1−𝛼_𝑛)=∞,(ii)∑^∞𝑛=0𝛼_𝑛‖𝑒_𝑛‖<∞.Then {𝑥_𝑛} 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: 𝑥_𝑛+1=𝜆_𝑛𝑥_𝑛+1−𝜆_𝑛𝑇𝑥_𝑛+𝑒_𝑛−𝜆_𝑛𝜇_𝑛𝐹𝑥_𝑛,𝑛≥0,(1.6) where 𝜆_𝑛,𝜇_𝑛∈[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 {𝑢_𝑛}⊂[0,+∞), lim_𝑛→∞𝑢_𝑛=0 such that ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤1+𝑢_𝑛‖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)𝜔_𝑤(𝑥_𝑛)={𝑥∶∃𝑥_𝑛𝑗⇀𝑥} denotes the weak 𝜔-limit set of {𝑥_𝑛}.

It is well known that a Hilbert space 𝐻 satisfies Opial's condition [18]; that is, for each sequence {𝑥_𝑛} in 𝐻 which converges weakly to a point 𝑥∈𝐻, one has liminf_𝑛→∞‖‖𝑥_𝑛−𝑥‖‖<liminf_𝑛→∞‖‖𝑥_𝑛−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 𝑃_𝐶 from 𝐻 onto 𝐶 assigns to each 𝑥∈𝐶 its nearest point denoted by 𝑃_𝐶𝑥 in 𝐶 from 𝑥 to 𝐶; that is, 𝑃_𝐶𝑥 is the unique point in 𝑋 with the property ‖‖𝑥−𝑃_𝐶𝑥‖‖≤‖𝑥−𝑦‖,∀𝑦∈𝐶.(2.2)

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

A mapping 𝑇 is said to be demiclosed at zero if whenever {𝑥_𝑛} is a sequence in 𝐷(𝑇) such that {𝑥_𝑛} converges weakly to 𝑥∈𝐷(𝑇) and {𝑇𝑥_𝑛} 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 {𝑢_𝑛}⊂[0,∞) and lim_𝑛→∞𝑢_𝑛=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)𝑦‖²≤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 {𝑥_𝑛} be a bounded sequence in 𝑋 and 𝑞₁,𝑞₂∈𝜔_𝑤({𝑥_𝑛}). Suppose that that lim_𝑛→∞‖‖𝛼𝑥_𝑛+(1−𝛼)𝑞₁−𝑞₂‖‖(2.4) exists for all 𝛼∈[0,1]. Then 𝑞₁=𝑞₂.

Lemma 2.4 (see [23]). Let {𝑎_𝑛}, {𝑏_𝑛}, and {𝑐_𝑛} be sequences of nonnegative real numbers satisfying the inequality 𝑎_𝑛+1≤1+𝑐_𝑛𝑎_𝑛+𝑏_𝑛,𝑛≥1.(2.5) If ∑^∞𝑛=1𝑐_𝑛<+∞,∑^∞𝑛=1𝑏_𝑛<+∞, then (i) lim_𝑛→∞𝑎_𝑛 exists. (ii) In particular, if liminf_𝑛→∞𝑎_𝑛=0, one has lim_𝑛→∞𝑎_𝑛=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=0𝑢_𝑛<∞. For any 𝑥₀∈𝑋, the sequence {𝑥_𝑛} is generated by the following Krasnoselski-Mann's algorithm: 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑥_𝑛+𝑒_𝑛,𝑛≥0,(3.1) where {𝛼_𝑛} and {𝑒_𝑛} satisfy the following conditions: (i)0<𝑎<𝛼_𝑛<𝑏<1, for some 𝑎,𝑏∈(0,1) and for all 𝑛≥0;(ii)∑^∞𝑛=0𝛼_𝑛‖𝑒_𝑛‖<∞.If 𝐹(𝑇)≠∅, then the sequence {𝑥_𝑛} 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=0𝑢_𝑛<∞. Let {𝑥_𝑛} be the sequence as defined in (3.1) and satisfy the conditions in Theorem 3.1. Suppose that 𝐹(𝑇)≠∅; then the limit lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists for 𝑝∈𝐹(𝑇).

Proof. By (3.1), one has ‖‖𝑥_𝑛+1−𝑝‖‖=≤‖‖11−−𝛼𝛼_𝑛𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑇^𝑛𝑥_𝑛−𝑝+𝑒_𝑛‖‖≤1−𝛼_𝑛‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛‖‖𝑇^𝑛𝑥_𝑛−𝑝‖‖+𝛼_𝑛‖‖𝑒_𝑛‖‖≤1+𝑢_𝑛‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛1+𝑢_𝑛‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛‖‖𝑒_𝑛‖‖‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛‖‖𝑒_𝑛‖‖.(3.2) Since ∑^∞𝑛=0𝑢_𝑛<∞ and ∑^∞𝑛=0𝛼_𝑛‖𝑒_𝑛‖<∞, we obtain from Lemma 2.4 that the limit lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists. Furthermore, the sequence {𝑥_𝑛} 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=0𝑢_𝑛<∞. Let {𝑥_𝑛} be the sequence as defined in (3.1) and satisfy the conditions in Theorem 3.1. Suppose that 𝐹(𝑇)≠∅; then lim_𝑛→∞‖𝑡𝑥_𝑛+(1−𝑡)𝑝−𝑞‖ exists for all 𝑡∈[0,1] and 𝑝,𝑞∈𝐹(𝑇).

Proof. Let 𝑑_𝑛(𝑡)=‖𝑡𝑥_𝑛+(1−𝑡)𝑝−𝑞‖; then lim_𝑛→∞𝑑_𝑛(0)=‖𝑝−𝑞‖ exists. It follows from Lemma 3.2 that lim_𝑛→∞𝑑_𝑛(1)=lim_𝑛→∞‖𝑥_𝑛−𝑞‖ exists. Next, we show that lim_𝑛→∞𝑑_𝑛(𝑡) exists for any 𝑡∈(0,1).
Let 𝑇_𝑛𝑥∶=(1−𝛼_𝑛)𝑥+𝛼_𝑛𝑇^𝑛𝑥+𝛼_𝑛𝑒_𝑛, for all 𝑥∈𝑋. For any 𝑥,𝑧∈𝑋, one has ‖‖𝑇_𝑛𝑥−𝑇_𝑛𝑧‖‖≤=‖‖11−−𝛼𝛼_𝑛𝑛(𝑥−𝑧)+𝛼_𝑛(𝑇^𝑛𝑥−𝑇^𝑛𝑧)‖‖≤1−𝛼_𝑛‖𝑥−𝑧‖+𝛼_𝑛‖𝑇^𝑛𝑥−𝑇^𝑛𝑧‖≤1+𝑢_𝑛‖𝑥−𝑧‖+𝛼_𝑛1+𝑢_𝑛‖𝑥−𝑧‖‖𝑥−𝑧‖.(3.3) Set 𝑆_𝑛,𝑚=𝑇_{𝑛+𝑚−1}𝑇_{𝑛+𝑚−2}⋯𝑇_𝑛, 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 ‖‖𝑥_𝑛+1−𝑝‖‖²=‖‖1−𝛼_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑇^𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑒_𝑛‖‖²≤‖‖(1−𝛼_𝑛)(𝑥_𝑛−𝑝)+𝛼_𝑛(𝑇^𝑛𝑥_𝑛−𝑝)‖‖²+2𝛼_𝑛‖‖𝑒_𝑛‖‖⋅‖‖1−𝛼_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑇^𝑛𝑥_𝑛−𝑝‖‖+𝛼^2𝑛‖‖𝑒_𝑛‖‖²≤+12𝛼−_𝑛𝛼_𝑛‖‖𝑥_𝑛−𝑝‖‖²+𝛼_𝑛‖‖𝑇^𝑛𝑥_𝑛−𝑝‖‖²−𝛼_𝑛1−𝛼_𝑛𝜙‖‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖‖‖‖𝑒_𝑛‖‖1−𝛼_𝑛‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛1+𝑢_𝑛‖‖𝑥_𝑛−𝑝‖‖+𝛼^2𝑛‖‖𝑒_𝑛‖‖²≤+12𝛼−_𝑛𝛼_𝑛‖‖𝑥_𝑛−𝑝‖‖²+𝛼_𝑛1+𝑢_𝑛²‖‖𝑥_𝑛−𝑝‖‖²−𝛼_𝑛1−𝛼_𝑛𝜙‖‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖‖‖‖𝑒_𝑛‖‖1+𝑢_𝑛‖‖𝑥_𝑛−𝑝‖‖+𝛼^2𝑛‖‖𝑒_𝑛‖‖²≤+12𝛼+_𝑛𝑢_𝑛²‖‖𝑥_𝑛−𝑝‖‖²−𝛼_𝑛1−𝛼_𝑛𝜙‖‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖‖‖‖𝑒_𝑛‖‖1+𝑢_𝑛‖‖𝑥_𝑛−𝑝‖‖+𝛼^2𝑛‖‖𝑒_𝑛‖‖²,(3.4) which follows that 𝑎(1−𝑏)𝜙‖‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖‖≤≤𝛼1_𝑛+1𝑢−_𝑛𝛼_𝑛𝜙‖‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖‖²‖‖𝑥_𝑛−𝑝‖‖²−‖‖𝑥_𝑛+1−𝑝‖‖²+2𝛼_𝑛‖‖𝑒_𝑛‖‖1+𝑢_𝑛‖‖𝑥_𝑛−𝑝‖‖+𝛼^2𝑛‖‖𝑒_𝑛‖‖².(3.5) This implies that ^∞_𝑛=0𝜙‖‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖‖<∞.(3.6) Therefore lim_𝑛→∞𝜙(‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖)=0. Since 𝜙 is strictly increasing and continuous function with 𝜙(0)=0, then lim_𝑛→∞‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖=0. Also, one has the following inequalities: ‖‖𝑇‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤𝛼_𝑛‖‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖‖+𝛼_𝑛‖‖𝑒_𝑛‖‖⟶0as𝑛⟶∞,(3.7)_𝑛𝑥_𝑛+1−𝑥_𝑛+1‖‖==‖‖‖‖𝑇𝑇^𝑛𝑥_𝑛+1𝑛−1−𝛼_𝑛𝑥_𝑛−𝛼_𝑛𝑇^𝑛𝑥_𝑛+𝑒_𝑛‖‖≤‖‖𝑇_𝑛𝑥_𝑛+1−𝑇^𝑛𝑥_𝑛+1−𝛼_𝑛𝑇^𝑛𝑥_𝑛−𝑥_𝑛−𝛼_𝑛𝑒_𝑛‖‖≤1+𝑥𝑢_𝑛+1𝑛−𝑇^𝑛𝑥_𝑛‖‖+1−𝛼_𝑛‖‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖‖+𝛼_𝑛‖‖𝑒_𝑛‖‖=1+𝑢_𝑛‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+1−𝛼_𝑛‖‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖‖+𝛼_𝑛‖‖𝑒_𝑛‖‖≤1+𝑢_𝑛‖‖𝛼_𝑛𝑇^𝑛𝑥_𝑛−𝑥_𝑛+𝛼_𝑛𝑒_𝑛‖‖+1−𝛼_𝑛‖‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖‖+𝛼_𝑛‖‖𝑒_𝑛‖‖≤1+𝑢_𝑛𝛼_𝑛‖‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖‖+1−𝛼_𝑛‖‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖‖+1+𝑢_𝑛𝛼_𝑛‖‖𝑒_𝑛‖‖+𝛼_𝑛‖‖𝑒_𝑛‖‖‖‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖‖+1+𝑢_𝑛𝛼_𝑛‖‖𝑒_𝑛‖‖+𝛼_𝑛‖‖𝑒_𝑛‖‖⟶0as𝑛⟶∞.(3.8) On the other hand, ‖‖𝑥_𝑛+1−𝑇𝑥_𝑛+1‖‖≤≤‖‖‖‖𝑥𝑥_𝑛+1𝑛+1−𝑇^𝑛+1𝑥_𝑛+1‖‖+‖‖𝑇^𝑛+1𝑥_𝑛+1−𝑇𝑥_𝑛+1‖‖−𝑇^𝑛+1𝑥_𝑛+1‖‖+1+𝑢₁‖‖𝑇^𝑛𝑥_𝑛+1−𝑥_𝑛+1‖‖⟶0as𝑛⟶∞.(3.9) By (3.7)–(3.9), we obtain ‖‖𝑥_𝑛−𝑇𝑥_𝑛‖‖≤≤‖‖2𝑥_𝑛+−𝑢₁𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑇𝑥_𝑛+1‖‖+‖‖𝑇𝑥_𝑛+1−𝑇𝑥_𝑛‖‖‖‖𝑥_𝑛−𝑥_𝑛+1‖‖+‖‖𝑥_𝑛+1−𝑇𝑥_𝑛+1‖‖⟶0as𝑛⟶∞,(3.10) that is, lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0.
From Lemma 3.2, we know that {𝑥_𝑛} is bounded. Since 𝑋 is a uniformly convex Banach space, {𝑥_𝑛} has a convergent subsequence {𝑥_𝑛𝑗}. By the demiclosedness principle of 𝐼−𝑇, we obtain 𝜔_𝑤⊆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=0𝑢_𝑛<∞. For any 𝑥₀∈𝑋, the sequence {𝑥_𝑛} is generated by either 𝑥_𝑛+1=1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑃_𝐶𝑇^𝑛𝑥_𝑛+𝑒_𝑛,𝑛≥0,(3.11) or 𝑥_𝑛+1=𝑃_𝐶1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑇^𝑛𝑥_𝑛+𝑒_𝑛,𝑛≥0,(3.12) where the sequences {𝛼_𝑛} and {𝑒_𝑛} are such that (i)0<𝑎<𝛼_𝑛<𝑏<1, for some 𝑎,𝑏∈(0,1) and for all 𝑛≥0;(ii)∑^∞𝑛=0𝛼_𝑛‖𝑒_𝑛‖<∞. If 𝐹(𝑇)≠∅, then {𝑥_𝑛} converges weakly to a fixed point of  𝑇.

Proof. Let 𝑝∈𝐹(𝑇). By (3.11), one has ‖‖𝑥_𝑛+1−𝑝‖‖=≤‖‖1−1−𝛼_𝑛𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑃_𝐶𝑇^𝑛𝑥_𝑛+𝑒_𝑛−𝑝‖‖≤1−𝛼_𝑛‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛‖‖𝑃_𝐶𝑇^𝑛𝑥_𝑛+𝑒_𝑛−𝑝‖‖≤1−𝛼_𝑛‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛‖‖𝑇^𝑛𝑥_𝑛+𝑒_𝑛−𝑝‖‖≤1+𝑢_𝑛‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛1+𝑢_𝑛‖𝑥_𝑛−𝑝‖+𝛼_𝑛‖‖𝑒_𝑛‖‖‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛‖‖𝑒_𝑛‖‖.(3.13) Notice the condition (ii) and ∑^∞𝑛=0𝑢_𝑛<∞; by Lemma 2.4, lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists. Hence, {𝑥_𝑛} 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 ‖𝑥_𝑛+1−𝑝‖²=‖‖1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑃_𝐶𝑇^𝑛𝑥_𝑛+𝑒_𝑛−𝑝‖‖²=‖‖1−𝛼_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑇^𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑃_𝐶𝑇^𝑛𝑥_𝑛+𝑒_𝑛−𝑇^𝑛𝑥_𝑛‖‖²≤‖‖1−𝛼_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑇^𝑛𝑥_𝑛−𝑝‖‖²+‖‖𝛼_𝑛𝑃_𝐶𝑇^𝑛𝑥_𝑛+𝑒_𝑛−𝑇^𝑛𝑥_𝑛‖‖²≤+12𝛼−_𝑛𝛼‖‖_𝑛1−𝛼_𝑛𝑥_𝑛−𝑝+𝛼_𝑛𝑇^𝑛𝑥_𝑛−𝑝‖‖⋅‖‖𝑃_𝐶𝑇^𝑛𝑥_𝑛+𝑒_𝑛−𝑇^𝑛𝑥_𝑛‖‖‖‖𝑥_𝑛−𝑝‖‖²+𝛼_𝑛‖‖𝑇^𝑛𝑥_𝑛−𝑝‖‖²−𝛼_𝑛1−𝛼_𝑛‖‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖‖²≤+12𝛼+_𝑛𝑢_𝑛1−𝛼_𝑛‖‖𝑥_𝑛−𝑝‖‖+𝛼_𝑛1+𝑢_𝑛‖‖𝑥_𝑛−𝑝‖‖‖‖𝑒_𝑛‖‖²‖‖𝑥_𝑛−𝑝‖‖²−𝛼_𝑛1−𝛼_𝑛‖‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖‖²+21+𝑢_𝑛𝛼_𝑛‖‖𝑒_𝑛‖‖‖‖𝑥_𝑛−𝑝‖‖.(3.14) That is, 𝑎(1−𝑏)‖‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖‖²≤𝛼_𝑛1−𝛼_𝑛‖‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖‖²≤1+𝑢_𝑛²‖‖𝑥_𝑛−𝑝‖‖²−‖‖𝑥_𝑛+1−𝑝‖‖²+21+𝑢_𝑛𝛼_𝑛‖‖𝑒_𝑛‖‖‖‖𝑥_𝑛−𝑝‖‖.(3.15) This implies that ^∞_𝑛=0‖‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖‖²<∞.(3.16) Therefore lim_𝑛→∞‖𝑥_𝑛−𝑇^𝑛𝑥_𝑛‖=0. We also have ‖‖𝑥_𝑛+1−𝑥_𝑛‖‖≤=‖‖𝛼_𝑛1−𝛼_𝑛𝑥_𝑛+𝛼_𝑛𝑃_𝐶𝑇^𝑛𝑥_𝑛+𝑒_𝑛−𝑥_𝑛‖‖‖𝑇_𝑛‖‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖‖+𝛼_𝑛‖‖𝑒_𝑛‖‖⟶0as𝑛⟶∞,𝑥_𝑛+1−𝑥_𝑛+1‖==‖‖‖‖𝑇𝑇^𝑛𝑥_𝑛+1𝑛−1−𝛼_𝑛𝑥_𝑛−𝛼_𝑛𝑃_𝐶𝑇^𝑛𝑥_𝑛+𝑒_𝑛‖‖≤1+𝑥𝑢_𝑛+1𝑛−𝑇^𝑛𝑥_𝑛+𝑇^𝑛𝑥_𝑛−𝑥_𝑛+𝛼_𝑛𝑥_𝑛−𝑃_𝐶𝑇^𝑛𝑥_𝑛+𝑒_𝑛‖‖‖‖𝑥_𝑛+1−𝑥_𝑛‖‖+1+𝛼_𝑛‖‖𝑇^𝑛𝑥_𝑛−𝑥_𝑛‖‖+𝛼_𝑛‖‖𝑒_𝑛‖‖⟶0as𝑛⟶∞.(3.17) It follows from (3.9) and (3.10) that lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=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).