Strong Convergence of Hybrid Algorithm for Asymptotically Nonexpansive Mappings in Hilbert Spaces

Abstract

The hybrid algorithms for constructing fixed points of nonlinear mappings have been studied extensively in recent years. The advantage of this methods is that one can prove strong convergence theorems while the traditional iteration methods just have weak convergence. In this paper, we propose two types of hybrid algorithm to find a common fixed point of a finite family of asymptotically nonexpansive mappings in Hilbert spaces. One is cyclic Mann's iteration scheme, and the other is cyclic Halpern's iteration scheme. We prove the strong convergence theorems for both iteration schemes.

1. Introduction

Let 𝐻 be a real Hilbert space and 𝐶 be a nonempty closed convex subset of 𝐻, ⟨⋅,⋅⟩ and ‖⋅‖ denote the inner product and norm in 𝐻, respectively. Let 𝑇 be a self-mapping of 𝐶. Then, 𝑇 is said to be a Lipschitzian mapping if for each 𝑛≥1 there exists an nonnegative real number 𝑘_𝑛 such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦≤𝑘_𝑛𝑥−𝑦, (1.1) for all 𝑥,𝑦∈𝐶. A Lipschitzian mapping is said to be nonexpansive mapping if 𝑘_𝑛=1 for all 𝑛≥1 and asymptotically nonexpansive mapping [1] if lim_𝑛→∞𝑘_𝑛=1, respectively. We use 𝐹(𝑇) to denote the set of fixed points of 𝑇 (i.e., 𝐹(𝑇)={𝑥∈𝐶∶𝑇𝑥=𝑥}). It is well known that if 𝑇 is asymptotically nonexpansive mapping with 𝐹(𝑇)≠∅, then 𝐹(𝑇) is closed and convex.

Iterative methods for finding fixed points of nonexpansive mappings are an important topic in the theory of nonexpansive mappings and have wide applications in a number of applied areas, such as the convex feasibility problem [2–4], the split feasibility problem [5–7] and image recovery and signal processing [8–10]. The Mann's iteration is defined by the following: 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝑛≥0, (1.2) where 𝑥₀∈𝐶 is chosen arbitrarily and {𝛼_𝑛}⊆[0,1]. Reich [11] proved that if 𝑋 is a uniformly convex Banach space with a Fréchet differentiable norm and if {𝛼_𝑛} is chosen such that ∑^∞_𝑛=0𝛼_𝑛(1−𝛼_𝑛)=∞, then the sequence {𝑥_𝑛} defined by (1.2) converges weakly to a fixed point of nonexpansive mapping 𝑇. However, we highlight that the Mann's iterations have only weak convergence even in a Hilbert space (see e.g., [12]).

In order to obtain the strong convergence theorem for the Mann iteration method (1.2) to nonexpansive mappings, in 2003, Nakajo and Takahashi [13] proved the following theorem in a Hilbert space by using an idea of the hybrid method in mathematical programming.

Theorem 1.1 (see [13]). Let 𝐶 be a closed convex subset of a Hilbert space 𝐻 and let 𝑇 be a nonexpansive mapping of 𝐶 into itself such that 𝐹(𝑇) is nonempty. Let 𝑃 be the metric projection of 𝐻 onto 𝐹(𝑇). Let 𝑥₀∈𝐶 and 𝑦_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀, (1.3) where {𝛼_𝑛}⊆[0,1] satisfies sup_𝑛≥0𝛼_𝑛<1 and 𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀ is the metric projection of 𝐻 onto 𝐶_𝑛∩𝑄_𝑛. Then {𝑥_𝑛} converges strongly to 𝑃𝑥₀∈𝐹(𝑇).

The iterative algorithm (1.3) is often referred to as hybrid algorithm or CQ algorithm in the literature. We call it hybrid algorithm. Since then, the hybrid algorithm has been studied extensively by many authors (see, e.g., [14–18]). Specifically, Kim and Xu [19] extended the results of Nakajo and Takahashi [13] from nonexpansive mapping to asymptotically nonexpansive mapping; they proposed the following hybrid algorithm: 𝑥₀𝑦∈𝐶ischosenarbitrarily,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇^𝑛𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑧²+𝜃_𝑛,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥₀,𝑥−𝑧≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀, (1.4) where 𝜃_𝑛=(1−𝛼_𝑛)(𝑘²_𝑛−1)(diam𝐶)². Zhang and Chen in [20], studied the following hybrid algorithm of Halpern's type for asymptotically nonexpansive mappings: 𝑥₀𝑦∈𝐶bechosenarbitrarily,_𝑛=𝛼_𝑛𝑥₀+1−𝛼_𝑛𝑇^𝑛𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑧²+𝜃_𝑛,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀, (1.5) where 𝜃_𝑛=(1−𝛼_𝑛)(𝑘²_𝑛−1)(diam𝐶)². Some other related works can be found in [21–25].

The hybrid algorithm of (1.3)–(1.5) just considered a single nonexpansive and asymptotically nonexpansive mapping. In order to extend them to a finite family of mappings. Recall that in 1996, Bauschke [26] investigated the following cyclic Halpern's type algorithm for a finite family of nonexpansive mappings {𝑇_𝑗}^𝑁−1_𝑗=0: 𝐶∋𝑥₀⟼𝑥₁∶=𝛼₀𝑢+1−𝛼₀𝑇₀𝑥₀⟼⋯⟼𝑥_𝑁∶=𝛼_𝑁−1𝑢+1−𝛼_𝑁−1𝑇_𝑁−1𝑥_𝑁−1⟼𝑥_𝑁+1∶=𝛼_𝑁𝑢+1−𝛼_𝑁𝑇₀𝑥_𝑁⟼⋯, (1.6) or, more compactly, 𝑢,𝑥₀𝑥∈𝐶,_𝑛+1=𝛼_𝑛𝑢+1−𝛼_𝑛𝑇_[𝑛]𝑥_𝑛,𝑛≥0, (1.7) where 𝑇_[𝑛]∶=𝑇_𝑛mod𝑁, and the mod𝑁 function takes values in {0,1,…,𝑁−1}.

If 𝛼_𝑛=0 and each nonexpansive mapping {𝑇_𝑗}^𝑁−1_𝑗=0 is a projection onto a closed convex set, then (1.7) reduces to the famous Algebraic Reconstruction Technique (ART), which has numerous applications from computer tomograph to image reconstruction.

For the cyclic Mann's type algorithm, a finite family of asymptotically nonexpansive mappings was introduced by Qin et al. [17] and Osilike and Shehu [14], independently. Let {𝑇_𝑗}^𝑁−1_𝑗=0 be a finite family of asymptotically nonexpansive self-mappings of 𝐶. For a given 𝑥₀∈𝐶, and a real sequence {𝛼_𝑛}^∞_𝑛=0⊆(0,1), the sequence {𝑥_𝑛}^∞_𝑛=0 is generated as follows: 𝑥₁=𝛼₀𝑥₀+1−𝛼₀𝑇₀𝑥₀,𝑥₂=𝛼₁𝑥₁+1−𝛼₁𝑇₁𝑥₁,⋮𝑥_𝑁=𝛼_𝑁−1𝑥_𝑁−1+1−𝛼_𝑁−1𝑇_𝑁−1𝑥_𝑁−1,𝑥_𝑁+1=𝛼_𝑁𝑥_𝑁+1−𝛼_𝑁𝑇²₀𝑥_𝑁,𝑥_𝑁+2=𝛼_𝑁+1𝑥_𝑁+1+1−𝛼_𝑁+1𝑇²₁𝑥_𝑁+1,⋮𝑥_2𝑁=𝛼_2𝑁−1𝑥_2𝑁−1+1−𝛼_2𝑁−1𝑇²_𝑁−1𝑥_2𝑁−1,𝑥_2𝑁+1=𝛼_2𝑁𝑥_2𝑁+1−𝛼_2𝑁𝑇³₀𝑥_2𝑁,𝑥_2𝑁+2=𝛼_2𝑁+1𝑥_2𝑁+1+1−𝛼_2𝑁+1𝑇³₁𝑥_2𝑁+1,⋮ (1.8) The algorithm can be expressed in a compact form as 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇_𝑖(𝑛)^𝑘(𝑛)𝑥_𝑛,𝑛≥0, (1.9) where 𝑛=(𝑘−1)𝑁+𝑖, 𝑖=𝑖(𝑛)∈𝐽={0,1,2,…,𝑁−1}, 𝑘=𝑘(𝑛)≥1 positive integer and lim_𝑛→∞𝑘(𝑛)=∞. Similarly, we can define the cyclic Halpern's type algorithm for asymptotically nonexpansive mappings as follows: 𝑥_𝑛+1=𝛼_𝑛𝑥₀+1−𝛼_𝑛𝑇_𝑖(𝑛)^𝑘(𝑛)𝑥_𝑛,𝑛≥0. (1.10)

The purpose of this paper is to extend the hybrid algorithms (1.4) and (1.5) to the cyclic Mann's type (1.9) and the cyclic Halpern's type (1.10). Our results generalize the corresponding results of Kim and Xu [19] and Zhang and Chen [20] from a single asymptotically nonexpansive mapping to a finite family of asymptotically nonexpansive mappings, respectively.

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 the Opial's condition [27]; that is, for each sequence {𝑥_𝑛} in 𝐻 which converges weakly to a point 𝑥∈𝐻, we have limsup_𝑛→∞𝑥_𝑛−𝑥<limsup_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 form 𝐻 onto 𝐶 assigns to each 𝑥∈𝐶 its nearest point denoted 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) The following Lemmas 2.1 and 2.2 are well known.

Lemma 2.1. Let 𝐶 be a closed convex subset of a real Hilbert space 𝐻. Given 𝑥∈𝐻 and 𝑧∈𝐶, then 𝑧=𝑃_𝐶𝑥 if and only if there holds the relation 𝑥−𝑧,𝑦−𝑧≤0,∀𝑦∈𝐶. (2.3)

Lemma 2.2. Let 𝐻 be a real Hilbert space, then for all 𝑥,𝑦∈𝐻𝑥−𝑦²=‖𝑥‖²−𝑦²−2⟨𝑥−𝑦,𝑦⟩. (2.4)

Lemma 2.3 (see [28]). Let 𝑋 be a uniformly convex Banach space, 𝐶 a nonempty closed convex subset of 𝑋, and 𝑇∶𝐶→𝐶 an asymptotically nonexpansive mapping. Then (𝐼−𝑇) is demiclosed at zero, that is, if 𝑥_𝑛⇀𝑥 and 𝑥_𝑛−𝑇𝑥_𝑛→0, then 𝑥∈𝐹(𝑇).

Lemma 2.4 (see [22]). Let 𝐶 be a closed convex subset of a real Hilbert space 𝐻. Let {𝑥_𝑛} be sequences in 𝐻 and 𝑢∈𝐻. Let 𝑞=𝑃_𝐶𝑢. If {𝑥_𝑛} is such that 𝜔_𝑤(𝑥_𝑛)⊂𝐶 and satisfies the condition 𝑥_𝑛−𝑢≤𝑢−𝑞,∀𝑛≥1, (2.5) then {𝑥_𝑛} converges strongly to 𝑞.

Lemma 2.5 (see [22]). Let 𝐶 be a closed convex subset of a real Hilbert space 𝐻. For any 𝑥,𝑦,𝑧∈𝐻 and real number 𝑎∈ℝ, the set 𝑣∈𝐶∶𝑦−𝑣²≤‖𝑥−𝑣‖²+⟨𝑧,𝑣⟩+𝑎 (2.6) is convex and closed.

3. Main Results

In this section, we consider a finite family of asymptotically nonexpansive mappings {𝑇_𝑗}^𝑁−1_𝑗=0; that is, there exists {𝑢_𝑗𝑛}⊆[0,∞), 𝑗∈𝐽∶={0,1,2,…,𝑁−1} with lim_𝑛→∞𝑢_𝑗𝑛=0, for all 𝑗∈𝐽 such that 𝑇^𝑛_𝑗𝑥−𝑇^𝑛_𝑗𝑦≤1+𝑢_𝑗𝑛𝑥−𝑦, (3.1) for all 𝑛≥1 and 𝑥,𝑦∈𝐶. Let 𝑢_𝑛∶=max_𝑗∈𝐽{𝑢_𝑗𝑛}, then lim_𝑛→∞𝑢_𝑛=0, and 𝑇^𝑛_𝑗𝑥−𝑇^𝑛_𝑗𝑦≤1+𝑢_𝑛𝑥−𝑦, (3.2) for all 𝑛≥1, and for all 𝑥,𝑦∈𝐶 and 𝑗∈𝐽.

We prove the following theorems.

Theorem 3.1. Let 𝐶 be a bounded closed convex subset of a Hilbert space 𝐻, and let {𝑇_𝑗}^𝑁−1_𝑗=0∶𝐶→𝐶 be a finite family of asymptotically nonexpansive mappings with 𝐹∶=⋂^𝑁−1_𝑗=0𝐹(𝑇_𝑗)≠∅. Assume that {𝛼_𝑛}⊆(0,1) such that lim_𝑛→∞𝛼_𝑛=0. Suppose the sequence {𝑥_𝑛} generated by 𝑥₀𝑦∈𝐶ischosenarbitrary,_𝑛=𝛼_𝑛𝑥₀+1−𝛼_𝑛𝑇^𝑘(𝑛)_𝑖(𝑛)𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑧²+𝛼_𝑛𝑥₀²+2𝑥_𝑛−𝑥₀,𝑧+𝜃_𝑛,𝑄_𝑛=𝑧∈𝐶∶⟨𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛,𝑥⟩≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀, (3.3) where 𝜃_𝑛=(2+𝑢_𝑛)𝑢_𝑛(1−𝛼_𝑛)(diam𝐶)²→0, as 𝑛→∞. Then {𝑥_𝑛} converges strongly to 𝑃_𝐹𝑥₀.

Proof. By Lemma 2.5, we conclude that 𝐶_𝑛 is closed and convex. It is obvious that 𝑄_𝑛 and 𝐹 are closed and convex. Then, the projection mappings 𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀ and 𝑃_𝐹𝑥₀ are well defined. We divide the proof into several steps.
Step 1. We show that 𝐹⊆𝐶_𝑛∩𝑄_𝑛, for all 𝑛. Let 𝑝∈𝐹. By the hybrid algorithm (3.3) and that ‖⋅‖² is convex, we have 𝑦_𝑛−𝑝²≤𝛼_𝑛𝑥₀−𝑝²+1−𝛼_𝑛𝑇_𝑖(𝑛)^𝑘(𝑛)−𝑝²≤𝛼_𝑛𝑥₀−𝑝²+1−𝛼_𝑛1+𝑢_𝑛²𝑥_𝑛−𝑝²=𝑥_𝑛−𝑝²+𝛼_𝑛𝑥₀−𝑝²−𝑥_𝑛−𝑝²+1−𝛼_𝑛2+𝑢_𝑛𝑢_𝑛𝑥_𝑛−𝑝²≤𝑥_𝑛−𝑝²+𝛼_𝑛𝑥₀²+2𝑥_𝑛−𝑥₀,𝑝+𝜃_𝑛, (3.4) where 𝜃_𝑛=(2+𝑢_𝑛)𝑢_𝑛(1−𝛼_𝑛)(diam𝐶)². Hence, 𝑝∈𝐶_𝑛, that is, F⊆𝐶_𝑛, for all 𝑛.
Next, we prove that 𝐹⊆𝑄_𝑛, for all 𝑛≥0. Indeed, for 𝑛=0, 𝑄₀=𝐶, then 𝐹⊆𝑄₀. Assuming that 𝐹⊆𝑄_𝑚, we show that 𝐹⊆𝑄_𝑚+1. Since 𝑥_𝑚+1 is the projection of 𝑥₀ onto 𝐶_𝑚∩𝑄_𝑚, it follows from Lemma 2.1 that 𝑥_𝑚+1−𝑧,𝑥₀−𝑥_𝑚+1≥0,∀𝑧∈𝐶_𝑚∩𝑄_𝑚. (3.5) As 𝐹⊆𝐶_𝑚∩𝑄_𝑚, in particular, we have 𝑥_𝑛+1−𝑝,𝑥₀−𝑥_𝑚+1≥0,∀𝑝∈𝐹. (3.6) Thus, 𝐹⊆𝑄_𝑚+1. Therefore, 𝐹⊆𝐶_𝑛∩𝑄_𝑛, for all 𝑛≥0.
Step 2. We prove that 𝑥_𝑛+𝑗−𝑥_𝑛⟶0as𝑛⟶∞,∀𝑗=0,1,2,…,𝑁−1. (3.7)
Since the definition of 𝑄_𝑛 implies that 𝑥_𝑛=𝑃_𝑄𝑛𝑥₀, we have 𝑥_𝑛−𝑥₀≤𝑥₀−𝑦,∀𝑦∈𝑄_𝑛. (3.8) By Step 1, 𝐹⊆𝑄_𝑛, we have 𝑥_𝑛−𝑥₀≤𝑥₀−𝑝,∀𝑝∈𝐹. (3.9) In particular, 𝑥_𝑛−𝑥₀≤𝑥₀−𝑞,𝑞=𝑃_𝐹𝑥₀. (3.10) Since 𝑥_𝑛+1∈𝑄_𝑛, we have ⟨𝑥_𝑛+1−𝑥_𝑛,𝑥_𝑛−𝑥₀⟩≥0 and ‖𝑥_𝑛−𝑥₀‖≤‖𝑥_𝑛+1−𝑥₀‖. The second inequality shows that the sequence {‖𝑥_𝑛−𝑥₀‖} is nondecreasing. Since 𝐶 is bounded, we obtain that the lim_𝑛→∞‖𝑥_𝑛−𝑥₀‖ exists.
With the help of Lemma 2.2, we obtain 𝑥_𝑛+1−𝑥_𝑛²=𝑥_𝑛+1−𝑥₀−𝑥_𝑛−𝑥₀²=𝑥_𝑛+1−𝑥₀²−𝑥_𝑛−𝑥₀²−2𝑥_𝑛+1−𝑥_𝑛,𝑥_𝑛−𝑥₀≤𝑥_𝑛+1−𝑥₀²−𝑥_𝑛−𝑥₀²⟶0as𝑛⟶∞. (3.11) Consequently, 𝑥_𝑛+𝑗−𝑥_𝑛⟶0as𝑛⟶∞,∀𝑗∈𝐽. (3.12)
Step 3. We now claim that ‖𝑥_𝑛−𝑇_𝑗𝑥_𝑛‖→0, as 𝑛→∞, for all 𝑗∈𝐽. Notice that for all 𝑛>𝑁, 𝑛=(𝑛−𝑁)(mod𝑁), since 𝑛=(𝑘(𝑛)−1)𝑁+𝑖(𝑛), we obtain 𝑛−𝑁=(𝑘(𝑛)−1)𝑁+𝑖(𝑛)−𝑁=(𝑘(𝑛−𝑁)−1)𝑁+𝑖(𝑛−𝑁). So that 𝑛−𝑁=[(𝑘(𝑛)−1)−1]𝑁+𝑖(𝑛)=(𝑘(𝑛−𝑁)−1)𝑁+𝑖(𝑛−𝑁). Hence 𝑘(𝑛)−1=𝑘(𝑛−𝑁) and 𝑖(𝑛)=𝑖(𝑛−𝑁).
By the hybrid algorithm (3.3) and the condition lim_𝑛→∞𝛼_𝑛=0, we get 𝑦_𝑛−𝑇_𝑖(𝑛)^𝑘(𝑛)𝑥_𝑛=𝛼_𝑛𝑥₀−𝑇_𝑖(𝑛)^𝑘(𝑛)𝑥_𝑛⟶0as𝑛⟶∞. (3.13) It follows from the fact 𝑥_𝑛+1∈𝐶_𝑛 that we have 𝑦_𝑛−𝑥_𝑛+1²≤𝑥_𝑛−𝑥_𝑛+1²+𝛼_𝑛𝑥₀²+2𝑥_𝑛−𝑥₀,𝑥_𝑛+1+𝜃_𝑛𝑇⟶0as𝑛⟶∞,_{𝑖(𝑛)𝑘(𝑛)}𝑥_𝑛−𝑥_𝑛≤𝑇_𝑖(𝑛)^𝑘(𝑛)𝑥_𝑛−𝑦_𝑛+𝑦_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑥_𝑛⟶0as𝑛⟶∞. (3.14) Putting 𝐿=sup_𝑛≥0{1+𝑢_𝑛}, we deduce that 𝑥_𝑛+1−𝑇_𝑖(𝑛)𝑥_𝑛≤𝑥_𝑛+1−𝑇_𝑖(𝑛)^𝑘(𝑛)𝑥_𝑛+𝑇_𝑖(𝑛)^𝑘(𝑛)𝑥_𝑛−𝑇_𝑖(𝑛)𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛−𝑇_𝑖(𝑛)^𝑘(𝑛)𝑥_𝑛+𝐿𝑇_𝑖(𝑛)^{𝑘(𝑛)−1}𝑥_𝑛−𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛−𝑇_𝑖(𝑛)^𝑘(𝑛)𝑥_𝑛+𝐿𝑇_𝑖(𝑛)^{𝑘(𝑛)−1}𝑥_𝑛−𝑇_𝑖()^{𝑘(𝑛)−1}_𝑛−𝑁𝑥_𝑛−𝑁+𝑇_𝑖()^{𝑘(𝑛)−1}_𝑛−𝑁𝑥_𝑛−𝑁−𝑥_{𝑛−𝑁−1}+𝑥_{𝑛−𝑁−1}−𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝑥_𝑛−𝑇_𝑖(𝑛)^𝑘(𝑛)𝑥_𝑛+𝐿²𝑥_𝑛−𝑥_𝑛−𝑁+𝐿𝑇_𝑖()^{𝑘(𝑛)−1}_𝑛−𝑁𝑥_𝑛−𝑁−𝑥_{𝑛−𝑁−1}+𝐿𝑥_{𝑛−𝑁−1}−𝑥_𝑛⟶0as𝑛⟶∞. (3.15) Hence, 𝑥_𝑛−𝑇_𝑖(𝑛)𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−𝑇_𝑖(𝑛)𝑥_𝑛⟶0as𝑛⟶∞. (3.16) Consequently, for all 𝑗=0,1,…,𝑁−1, we have 𝑥_𝑛−𝑇_𝑛+𝑗𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+𝑗+𝑥_𝑛+𝑗−𝑇_𝑛+𝑗𝑥_𝑛+𝑗+𝐿𝑥_𝑛+𝑗−𝑥_𝑛⟶0as𝑛⟶∞. (3.17) Thus, ‖𝑥_𝑛−𝑇_𝑗𝑥_𝑛‖→0, as 𝑛→∞, for all 𝑗∈𝐽.
Step 4. Since {𝑥_𝑛} is bounded, then {𝑥_𝑛} has a weakly convergent subsequence {𝑥_𝑛𝑗}. Suppose {𝑥_𝑛𝑗} converges weakly to 𝑝. Since C is weakly closed and {𝑥_𝑛𝑗}⊂𝐶, we have 𝑝∈𝐶. By Lemma 2.3, 𝐼−𝑇_𝑗 is demiclosed at 0 for all 𝑗∈𝐽, and we get 𝑝−𝑇_𝑗𝑝=0(𝑗∈𝐽), that is 𝑝∈𝐹. Suppose {𝑥_𝑛} does not converge weakly to 𝑝, then there exists another subsequence {𝑥_𝑛𝑘}of {𝑥_𝑛} which converges weakly to some 𝑝₁. Similarly we can prove that 𝑝₁∈𝐹. It follows from the proof of above that we know that lim_𝑛→∞‖𝑥_𝑛−𝑝‖ and lim_𝑛→∞‖𝑥_𝑛−𝑝₁‖ exist. Since every Hilbert space satisfies Opial’s condition, we have lim_𝑛→∞𝑥_𝑛−𝑝=lim_𝑗→∞𝑥_𝑛𝑗−𝑝<lim_𝑗→∞𝑥_𝑛𝑗−𝑝₁=lim_𝑛→∞𝑥_𝑛−𝑝₁=lim_𝑘→∞𝑥_𝑛𝑘−𝑝₁<lim_𝑘→∞𝑥_𝑛𝑘−𝑝<lim_𝑛→∞𝑥_𝑛−𝑝. (3.18) This is a contradiction. Hence, 𝜔_𝑤(𝑥_𝑛)⊆𝐹. Then by virtue of (3.10) and Lemma 2.4, we conclude that 𝑥_𝑛→𝑞 as 𝑛→∞, where 𝑞=𝑃_𝐹𝑥₀.

Recall that a mapping 𝑇 is said to be asymptotically strictly pseudocontractive [29], if there exist 𝜆∈[0,1) and a sequence {𝑢_𝑛}⊆[0,∞) with lim_𝑛→∞𝑢_𝑛=0 such that 𝑇^𝑛𝑥−𝑇^𝑛𝑦²≤1+𝑢_𝑛²𝑥−𝑦²+𝜆𝐼−𝑇^𝑛𝑥−𝐼−𝑇^𝑛𝑦², (3.19) for all 𝑛 and 𝑥,𝑦∈𝐶.

Theorem 3.2. Let 𝐶 be a bounded closed convex subset of a Hilbert space 𝐻, and let {𝑇_𝑗}^𝑁−1_𝑗=0∶𝐶→𝐶 be a finite family of asymptotically nonexpansive mappings with 𝐹∶=⋂^𝑁−1_𝑗=0𝐹(𝑇_𝑗)≠∅. Assume that {𝛼_𝑛}⊆(0,𝑎), for some 0<𝑎<1. Suppose the sequence {𝑥_𝑛} generated by 𝑥₀𝑦∈𝐶ischosenarbitrary,_𝑛=𝛼_𝑛𝑥_𝑛+1−𝛼_𝑛𝑇_𝑖(𝑛)^𝑘(𝑛)𝑥_𝑛,𝐶_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧²≤𝑥_𝑛−𝑧²+𝜃_𝑛,𝑄_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛,𝑥≥0_𝑛+1=𝑃_{𝐶𝑛∩𝑄𝑛}𝑥₀, (3.20) where 𝜃_𝑛=(2+𝑢_𝑛)𝑢_𝑛(diam𝐶)². Then {𝑥_𝑛} converges strongly to 𝑃_𝐹𝑥₀.

Proof. Since 𝑇_𝑗 is asymptotically nonexpansive if and only if 𝑇_𝑗 is asymptotically strictly pseudocontractive mapping with 𝜆=0. Then, the rest of proof follows from Theorem 3.2 of Osilike and Shehu [14] and Theorem 2.2 of Qin et al. [17] directly by letting 𝜆=0.

Acknowledgments

The authors are deeply grateful to Professor Rudong Chen (Editor) for managing the review process. This work was supported by the Natural Science Foundations of Jiangxi Province (2009GZS0021, CA201107114) and the Youth Science Funds of the Education Department of Jiangxi Province (GJJ12141).