Hybrid Iteration Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings in Banach Spaces

Abstract

Let 𝐸 be a real uniformly convex Banach space, and let{𝑇_𝑖∶𝑖∈𝐼} be 𝑁 nonexpansive mappings from 𝐸 into itself with 𝐹={𝑥∈𝐸∶𝑇_𝑖𝑥=𝑥,𝑖∈𝐼}≠𝜙, where 𝐼={1,2,…,𝑁}. From an arbitrary initial point 𝑥₁∈𝐸, hybrid iteration scheme {𝑥_𝑛} is defined as follows: 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)(𝑇_𝑛𝑥_𝑛−𝜆_𝑛+1𝜇𝐴(𝑇_𝑛𝑥_𝑛)), 𝑛≥1, where 𝐴∶𝐸→𝐸 is an 𝐿-Lipschitzian mapping, 𝑇_𝑛=𝑇_𝑖, 𝑖=𝑛(mod𝑁), 1≤𝑖≤𝑁, 𝜇>0, {𝜆_𝑛}⊂[0,1), and {𝛼_𝑛}⊂[𝑎,𝑏] for some 𝑎,𝑏∈(0,1). Under some suitable conditions, the strong and weak convergence theorems of {𝑥_𝑛} to a common fixed point of the mappings {𝑇_𝑖∶𝑖∈𝐼} are obtained. The results presented in this paper extend and improve the results of Wang (2007) and partially improve the results of Osilike, Isiogugu, and Nwokoro (2007).

1. Introduction

Let 𝐸 be a Banach space endowed with the norm ‖⋅‖. A mapping 𝑇∶𝐸→𝐸 is said to be nonexpansive if ‖𝑇𝑥−𝑇𝑦‖≤‖𝑥−𝑦‖ for any 𝑥,𝑦∈𝐸. 𝐹∶𝐸→𝐸 is said to be 𝐿-Lipschitzian if there exists constant 𝐿>0 such that ‖F𝑥−𝐹𝑦‖≤𝐿‖𝑥−𝑦‖ for any 𝑥,𝑦∈𝐸.

Let 𝐻 be a Hilbert space with inner product ⟨⋅,⋅⟩ and associated with norm ‖⋅‖, 𝐴∶𝐻→𝐻 is said to be 𝜂-strong monotone if there exists 𝜂>0 such that

⟨𝐴𝑥−𝐴𝑦,𝑥−𝑦⟩≥𝜂‖𝑥−𝑦‖²,∀𝑥,𝑦∈𝐻. (1.1)

The interest and importance of construction of fixed points of nonexpansive mappings stem mainly from the fact that it may be applied in many areas, such as imagine recovery and signal processing (see, e.g., [1–3]). Especially, numerous problems in physics, optimization, economics, traffic analysis, and mechanics reduce to find a solution of equilibrium problem. The equilibrium problem is to find

𝑥^∗∈𝐶suchthat𝑓(𝑥^∗,𝑦)≥0,∀𝑦∈𝐶, (1.2) where 𝐶 is a nonempty closed convex subset of a Hilbert space 𝐻, 𝑓 is a bifunction from 𝐶×𝐶 to 𝑅, and 𝑅 is the set of real numbers.

It has been shown by Blum and Oettli [4] and Noor and Oettli [5] that variational inequalities and mathematical programming problems can be viewed as a special realization of the abstract equilibrium problems. Given a mapping 𝑇∶𝐶→𝐻, let 𝑓(𝑥,𝑦)=⟨𝑇𝑥,𝑦−𝑥⟩ for all 𝑥,𝑦∈𝐶. It is well-known that 𝑥^∗∈𝐶 is a solution of (1.2) if and only if ⟨𝑇𝑥^∗,𝑦−𝑥^∗⟩≥0 for all 𝑦∈𝐶. Very recently, Yao et al. [6] find a common element of the set of solutions of equilibrium problem (1.2) and the set of common fixed points of a finite family of nonexpansive mappings by using an iterative scheme of a finite family of nonexpansive mappings. See the references therein for more details. Therefore, the topic on construction of fixed points of nonexpansive mappings is useful for equilibrium problems in physics, optimization, traffic analysis, and so forth.

Motivated by earlier results of Xu and Kim [7] and Yamada [8], some authors [9–14] further extended hybrid iteration method used this method to approximate fixed points of nonexpansive mappings, and obtained some strong and weak convergence theorems for nonexpansive mappings.

Recently, Wang [12] introduced an explicit hybrid iteration method for nonexpansive mappings and obtained the following convergence theorem.

Theorem 1.1 ([12]). Let 𝐻 be a Hilbert space, let 𝑇∶𝐻→𝐻 be a nonexpansive mapping with 𝐹(𝑇)={𝑥∈𝐻∶𝑇𝑥=𝑥}≠𝜙, and let 𝐴∶𝐻→𝐻 be a 𝜂-strong monotone and 𝐿-Lipschizian mapping. For any given 𝑥₁∈𝐻, {𝑥_𝑛} is defined by 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝑇^𝜆𝑛+1𝑥_𝑛,𝑛≥1, (1.3) where 𝑇^𝜆𝑛+1𝑥_𝑛=𝑇𝑥_𝑛−𝜆_𝑛+1𝜇𝐴(𝑇𝑥_𝑛). If {𝛼_𝑛}⊂[0,1) and {𝜆_𝑛}⊂[0,1) satisfy the following conditions: (1)  𝛼≤𝛼_𝑛≤𝛽 for some 𝛼,𝛽∈(0,1); (2)   ∑^∞_𝑛=2𝜆_𝑛<∞; (3)   0<𝜇<2𝜂/𝐿², then,(1){𝑥_𝑛} converges weakly to a fixed point of 𝑇.(2){𝑥_𝑛} converges strongly to a fixed point of 𝑇 if only if liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹(𝑇))=0.

Very recently, Osilike et al. [11] extended Wang's results to arbitrary Banach spaces without the strong monotonicity assumption imposed on the hybrid operator and obtained the following result.

Theorem 1.2 ([11]). Let 𝐸 be an arbitrary Banach space endowed with the norm ‖⋅‖, let 𝑇∶𝐸→𝐸 be a nonexpansive mapping with 𝐹(𝑇)≠𝜙, and let 𝐴∶𝐸→𝐸 be an 𝐿-Lipschitzian mapping. Let {𝑥_𝑛} be the sequence generated from an arbitrary 𝑥₁∈𝐸 by 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝑇^𝜆𝑛+1𝑥_𝑛,𝑛≥1, (1.4) where 𝑇^𝜆𝑛+1𝑥_𝑛=𝑇𝑥_𝑛−𝜆_𝑛+1𝜇𝐴(𝑇𝑥_𝑛), 𝜇>0, and {𝛼_𝑛}⊂[0,1) and {𝜆_𝑛}⊂[0,1) satisfy the following conditions: (1)   0<𝛼≤𝛼_𝑛≤1 for all 𝑛≥1 and some 𝛼∈(0,1); (2)   ∑^∞_𝑛=1(1−𝛼_𝑛)=∞; (3)   ∑^∞_𝑛=2𝜆_𝑛<∞, then,(1)lim_𝑛→∞‖𝑥_𝑛−𝑝‖ exists for each 𝑝∈𝐹(𝑇),(2)lim_𝑛→∞‖𝑥_𝑛−𝑇𝑥_𝑛‖=0,(3){𝑥_𝑛} converges strongly to a fixed point of 𝑇 if and only if liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹(𝑇))=0.

Motivated by above work, we obtain the strong and weak convergence theorems for a finite family of nonexpansive mappings in uniformly convex Banach space by using hybrid iteration method. The results presented in this paper extend and improve the results of Wang [12] and partially improve the results of Osilike et al. [11].

2. Preliminaries

Throughout this paper, we denote 𝐼={1,2,…,𝑁}.

A mapping 𝑇∶𝐾→𝐸 is said to be demicompact if, for any sequence {𝑥_𝑛} in 𝐾 such that ‖𝑥_𝑛−𝑇𝑥_𝑛‖→0(𝑛→∞), there exists subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} such that {𝑥_𝑛𝑗} converges strongly to 𝑥^∗∈𝐾.

For studying the strong convergence of fixed points of a nonexpansive mapping, Senter and Dotson [15] introduced condition (𝐴). Later on, Maiti and Ghosh [16], Tan and Xu [17] studied condition (𝐴) and pointed out that Condition (𝐴) is weaker than the requirement of demicompactness for nonexpansive mappings. A mapping 𝑇∶𝐾→𝐾 with 𝐹(𝑇)={𝑥∈𝐾∶𝑇𝑥=𝑥}≠𝜙 is said to satisfy condition (𝐴) if there exists a nondecreasing function 𝑓∶[0,∞)→[0,∞) with 𝑓(0)=0 and 𝑓(𝑡)>0 for all 𝑡∈(0,∞) such that ‖𝑥−𝑇𝑥‖≥𝑓(𝑑(𝑥,𝐹(𝑇))) for all 𝑥∈𝐾, where 𝑑(𝑥,𝐹(𝑇))=inf{‖𝑥−𝑞‖∶𝑞∈𝐹(𝑇)}.

A family of mappings {𝑇_𝑖∶𝑖∈𝐼} from 𝐸 into itself with 𝐹={𝑥∈𝐸∶𝑇_𝑖𝑥=𝑥,𝑖∈𝐼} is said to satisfy condition (B) if there exists a nondecreasing function 𝑓∶[0,∞)→[0,∞) with 𝑓(0)=0 and 𝑓(𝑡)>0 for all 𝑡∈(0,∞) such that ∑^𝑁_𝑖=1‖𝑥−𝑇_𝑖𝑥‖/𝑁≥𝑓(𝑑(𝑥,𝐹)) for all 𝑥∈𝐸.

A Banach space 𝐸 is said to satisfy Opial's condition if, for any sequence {𝑥_𝑛} in 𝐸, 𝑥_𝑛0𝑥00086⇀𝑥 implies that limsup_𝑛→∞‖𝑥_𝑛−𝑥‖<limsup_𝑛→∞‖𝑥_𝑛−𝑦‖ for all 𝑦∈𝐸 with 𝑦≠𝑥, where 𝑥_𝑛⇀0𝑥00086𝑥 denotes that {𝑥_𝑛} converges weakly to 𝑥.

A mapping 𝑇 with domain 𝐷(𝑇) and range 𝑅(𝑇) in 𝐸 is said to be demi-closed at 𝑝 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 𝑝, then 𝑇𝑥^∗=𝑝.

In the coming Lemma we will use the following well-known results.

Lemma 2.1 ([18]). Let {𝛼_𝑛} and {𝑡_𝑛} be two nonnegative sequences satisfying 𝛼_𝑛+1≤(1+𝑎_𝑛)𝛼_𝑛+𝑏_𝑛,∀𝑛≥1. (2.1) If ∑^∞_𝑛=1𝑎_𝑛<∞ and ∑^∞_𝑛=1𝑏_𝑛<∞, then lim_𝑛→∞𝛼_𝑛 exists.

Lemma 2.2 (see [19]). Let 𝐸 be a real uniformly convex Banach space and let 𝑎, 𝑏 be two constant with 0<𝑎<𝑏<1. Suppose that {𝑡_𝑛}⊂[𝑎,𝑏] is a real sequence and {𝑥_𝑛}, {𝑦_𝑛} are two sequences in 𝐸. Then the conditions lim_𝑛→∞‖𝑡_𝑛𝑥_𝑛+(1−𝑡_𝑛)𝑦_𝑛‖=𝑑,limsup_𝑛→∞‖𝑥_𝑛‖≤𝑑,limsup_𝑛→∞‖𝑦_𝑛‖≤𝑑, (2.2) imply that lim_𝑛→∞‖𝑥_n−𝑦_𝑛‖=0, where 𝑑≥0 is a constant.

Lemma 2.3 (see [20]). Let 𝐸 be a real uniformly convex Banach space, let 𝐾 be a nonempty closed convex subset of 𝐸, and let 𝑇∶𝐾→𝐾 be a nonexpansive mapping. Then 𝐼−𝑇 is demiclosed at zero.

3. Main Results

Theorem 3.1. Let 𝐸 be a real uniformly convex Banach space endowed with the norm ‖⋅‖, let 𝐼={1,2,…,𝑁}, {𝑇∶𝑖∈𝐼} be 𝑁 nonexpansive mappings from 𝐸 into itself with 𝐹={𝑥∈𝐸∶𝑇_𝑖𝑥=𝑥,𝑖∈𝐼}≠𝜙, and let 𝐴∶𝐸→𝐸 be an 𝐿-Lipschitzian mapping. For any given 𝑥₁∈𝐸, {𝑥_𝑛} is defined by 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)𝑇^𝜆𝑛+1𝑥_𝑛,𝑛≥1, (3.1) where 𝑇^𝜆𝑛+1𝑥_𝑛=𝑇_𝑛𝑥_𝑛−𝜆_𝑛+1𝜇𝐴(𝑇_𝑛𝑥_𝑛), 𝜇>0, 𝑇_𝑛=𝑇_𝑖, 𝑖=𝑛(mod𝑁),1≤𝑖≤𝑁. If {𝛼_𝑛} and {𝜆_𝑛}⊂[0,1) satisfy the following conditions:(1)𝑎≤𝛼_𝑛≤𝑏 for all 𝑛≥1 and some 𝑎,𝑏∈(0,1),(2)∑^∞_𝑛=2𝜆_𝑛<∞, then (1)lim_𝑛→∞‖𝑥_𝑛−𝑞‖ exists for each 𝑞∈𝐹,(2)lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑖𝑥_𝑛‖=0 for each 𝑖∈𝐼, (3){𝑥_𝑛} converges strongly to a common fixed point of {𝑇_𝑖∶𝑖∈𝐼} if and only if liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0.

Proof. (1) For any 𝑞∈𝐹, we have ‖𝑥_𝑛+1−𝑞‖=‖𝛼_𝑛(𝑥_𝑛−𝑞)+(1−𝛼_𝑛)(𝑇^𝜆𝑛+1𝑥_𝑛−𝑞)‖≤𝛼_𝑛‖𝑥_𝑛−𝑞‖+(1−𝛼_𝑛)‖𝑥_𝑛−𝑞‖+𝜆_𝑛+1(1−𝛼_𝑛)𝜇‖𝐴(𝑇_𝑛𝑥_𝑛)‖≤‖𝑥_𝑛−𝑞‖+(1−𝛼_𝑛)𝜆_𝑛+1𝜇‖𝐴(𝑇_𝑛𝑥_𝑛)−𝐴(𝑞)‖+(1−𝛼_𝑛)𝜆_𝑛+1𝜇‖𝐴(𝑞)‖≤[1+(1−𝑎)𝜇𝐿𝜆_𝑛+1]‖𝑥_𝑛−𝑞‖+(1−𝑎)𝜆_𝑛+1𝜇‖𝐴(𝑞)‖. (3.2) Since ∑^∞_𝑛=2𝜆_𝑛<∞, it follows from Lemma 2.1 that lim_𝑛→∞‖𝑥_𝑛−𝑞‖ exists.
(2) Since lim_𝑛→∞‖𝑥_𝑛−𝑞‖ exists for any 𝑞∈𝐹, {𝑥_𝑛} is bounded. So are {𝐴(𝑇_𝑛𝑥_𝑛)} and {𝑇_𝑛𝑥_𝑛}. Thus we may assume that lim_𝑛→∞‖𝑥_𝑛−𝑞‖=𝑑, that is, lim_𝑛→∞‖𝛼_𝑛(𝑥_𝑛−𝑞)+(1−𝛼_𝑛)(𝑇^𝜆𝑛+1𝑥_𝑛−𝑞)‖=𝑑. (3.3) Since lim_𝑛→∞‖𝑥_𝑛−𝑞‖=𝑑, lim_𝑛→∞𝜆_𝑛=0 and ‖𝑇^𝜆𝑛+1𝑥_𝑛−𝑞‖=‖𝑇_𝑛𝑥_𝑛−𝜆_𝑛+1𝜇𝐴(𝑇_𝑛𝑥_𝑛)−𝑞‖≤‖𝑥_𝑛−𝑞‖+𝜆_𝑛+1𝜇‖𝐴(𝑇_𝑛𝑥_𝑛)‖, (3.4) we have limsup_𝑛→∞‖𝑇^𝜆𝑛+1𝑥_𝑛−𝑞‖≤𝑑. (3.5) Thus, it follows from Lemma 2.2 that lim_𝑛→∞‖𝑥_𝑛−𝑇^𝜆𝑛+1𝑥_𝑛‖=0. (3.6)
In addition, ‖𝑥_𝑛−𝑇^𝜆𝑛+1𝑥_𝑛‖=‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛+𝜆_𝑛+1𝜇𝐴(𝑇_𝑛𝑥_𝑛)‖≥||𝑥_𝑛−𝑇_𝑛𝑥_𝑛||−𝜆_𝑛+1𝜇||𝐴(𝑇_𝑛𝑥_𝑛)||, (3.7) so ‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖≤‖𝑥_𝑛−𝑇^𝜆𝑛+1𝑥_𝑛‖+𝜆_𝑛+1𝜇‖𝐴(𝑇_𝑛𝑥_𝑛)‖. (3.8) Therefore, it follows from (3.6) that lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑛𝑥_𝑛‖=0. (3.9)
On the other hand, since 𝑥_𝑛+1=𝛼_𝑛𝑥_𝑛+(1−𝛼_𝑛)[𝑇_𝑛𝑥_𝑛−𝜆_𝑛+1𝜇𝐴(𝑇_𝑛𝑥_𝑛)], (3.10) we have 𝑥_𝑛+1−𝑇_𝑛𝑥_𝑛=𝛼_𝑛(𝑥_𝑛−𝑇_𝑛𝑥_𝑛)−(1−𝛼_𝑛)𝜆_𝑛+1𝜇𝐴(𝑇_𝑛𝑥_𝑛). (3.11) Thus, it follows from (3.9) that lim_𝑛→∞‖𝑥_𝑛+1−𝑇_𝑛𝑥_𝑛‖=0. (3.12) From (3.9) and (3.12), we can obtain lim_𝑛→∞‖𝑥_𝑛+1−𝑥_𝑛‖=0. (3.13) Further, for any positive integer 𝑘, we also have lim_𝑛→∞‖𝑥_𝑛+𝑘−𝑥_𝑛‖=0. (3.14)
For each 𝑖∈𝐼, ‖𝑥_𝑛−T_𝑛+𝑖𝑥_𝑛‖=‖𝑥_𝑛−𝑥_𝑛+𝑖+𝑥_𝑛+𝑖−𝑇_𝑛+𝑖𝑥_𝑛+𝑖+𝑇_𝑛+𝑖𝑥_𝑛+𝑖−𝑇_𝑛+𝑖𝑥_𝑛‖≤2||𝑥_𝑛−𝑥_𝑛+𝑖||+||𝑥_𝑛+𝑖−𝑇_𝑛+𝑖𝑥_𝑛+𝑖||. (3.15) It follows from (3.9) and (3.14) that lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑛+𝑖𝑥_𝑛‖=0. This implies that lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑖𝑥_𝑛‖=0 for each 𝑖∈𝐼.
(3) Suppose that {𝑥_𝑛} converges strongly to a common fixed point 𝑞 of the mappings {𝑇_𝑖∶𝑖∈𝐼}, then lim_𝑛→∞‖𝑥_𝑛−𝑞‖=0. Since 0≤𝑑(𝑥_𝑛,𝐹)≤‖𝑥_𝑛−𝑞‖, we have liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0.
Conversely, suppose that liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0. Since {𝑥_𝑛} is bounded, there exists constant 𝑀>0 such that ‖𝑥_𝑛−𝑞‖≤𝑀. From (3.2), for any 𝑞∈𝐹, we obtain ‖𝑥_𝑛+1−𝑞‖≤[1+(1−𝑎)𝜇𝐿𝜆_𝑛+1]‖𝑥_𝑛−𝑞‖+(1−𝑎)𝜆_𝑛+1𝜇‖𝐴(𝑞)‖≤‖𝑥_𝑛−𝑞‖+𝜆_𝑛+1𝛿, (3.16) where 𝛿=(1−𝑎)𝜇[𝐿𝑀+‖𝐴(𝑞)‖]. Furthermore, we have 𝑑(𝑥_𝑛+1,𝐹)≤𝑑(𝑥_𝑛,𝐹)+𝜆_𝑛+1𝛿. (3.17) It follows from Lemma 2.1 that lim_𝑛→∞𝑑(𝑥_𝑛,𝐹) exists. Since liminf_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0, we obtain that lim_𝑛→∞𝑑(𝑥_𝑛,𝐹)=0. We now show that {x_𝑛} is a Cauchy sequence.
For arbitrary 𝜀>0, there exists positive integer 𝑁₁ such that 𝑑(𝑥_𝑛,𝐹)<𝜀/4 for all 𝑛≥𝑁₁. In addition, since ∑^∞_𝑛=2𝜆_𝑛<∞, there exists positive integer 𝑁₂ such that ∑^∞_𝑗=𝑛𝜆_𝑗<𝜀/4𝛿 for all 𝑛≥𝑁₂. Taking 𝑁=max{𝑁₁,𝑁₂}, for any 𝑛,𝑚≥𝑁, from (3.16), we have ‖𝑥_𝑛−𝑥_𝑚‖≤‖𝑥_𝑛−𝑞‖+‖𝑥_𝑚−𝑞‖≤2‖𝑥_𝑁−𝑞‖+2𝛿^∞_𝑗=𝑁𝜆_𝑗+1. (3.18) Taking the infimum in above inequalities for all 𝑞∈𝐹, we obtain ‖𝑥_𝑛−𝑥_𝑚‖≤2𝑑(𝑥_𝑁,𝐹)+2𝛿^∞_𝑗=𝑁𝜆_𝑗+1<𝜀. (3.19) This implies that {𝑥_𝑛} is a Cauchy sequence. Therefore there exists 𝑝∈𝐸 such that {𝑥_𝑛} converges strongly to 𝑝. Since lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑖𝑥_𝑛‖=0 for each 𝑖∈𝐼, it follows from Lemma 2.3 that 𝑝∈𝐹. This completes the proof.

From Lemma 2.3 and lim_𝑛→∞‖𝑥_𝑛−𝑇_𝑖𝑥_𝑛‖=0 for each 𝑖∈𝐼, using routine method, we can easily show the following weak convergence theorem, whose proof is omitted.

Theorem 3.2. Let 𝐸 be a real uniformly convex Banach space satisfying Opial's condition, let {𝑇_𝑖∶𝑖∈𝐼} be 𝑁 nonexpansive mappings from 𝐸 into itself with 𝐹={𝑥∈𝐸∶𝑇𝑥=𝑥}≠𝜙, and let 𝐴∶𝐻→𝐻 be an 𝐿-Lipschitzian mapping. For any given 𝑥₁∈𝐸, {𝑥_𝑛} is defined as in Theorem 3.1, and {𝛼_𝑛}⊂[0,1) and {𝜆_𝑛}⊂[0,1) satisfy the conditions appeared in Theorem 3.1. Then {𝑥_𝑛} converges weakly to a common fixed point of the mappings of {𝑇_𝑖∶𝑖∈𝐼}.

Example 3.3. Let 𝐸=𝑅 be endowed with standard norm ‖⋅‖=|⋅|, where 𝑅 is real number set. Define 𝑇₁∶[0,∞)→[0,∞) and 𝑇₂∶[0,∞)→[0,∞) by 𝑇₁𝑥=1/2+𝑥/(1+𝑥) and 𝑇₂𝑥=1/2+𝑥²/(1+𝑥) for all 𝑥∈[0,∞), respectively. Obviously, 𝑇₁ and 𝑇₂ are nonexpansive mappings, and 1 is a common fixed point of 𝑇₁ and 𝑇₂. Let 𝐴∶[0,∞)→[0,∞) be defined by 𝐴𝑥=2𝑥+1 for all 𝑥∈[0,∞). We now chose parameters {𝛼_𝑛}, {𝜆_𝑛} and 𝜇 as follows: 𝛼_𝑛1=0.8−2𝑛,𝜆_𝑛=1(𝑛+1)²,𝑛≥1;𝜇=1. (3.20) It is easy to see that 𝐴 is a 2-Lipschitzian mapping and {𝛼_𝑛}, {𝜆_𝑛}, and 𝜇 satisfy the conditions of Theorem 3.2. Then {𝑥_𝑛} is generated by 𝑥_𝑛+11=(0.8−2𝑛)𝑥_𝑛1+(0.2+2𝑛)𝑇_𝑛𝑥_𝑛−(1−𝛼_𝑛)1(𝑛+1)²𝐴(𝑇_𝑛𝑥_𝑛),𝑛≥1, (3.21) where 𝑇_2𝑛−1=𝑇₁ and 𝑇_2𝑛=𝑇₂. It follows from Theorem 3.2 that {𝑥_𝑛} converges strongly to the common fixed point 1 of 𝑇₁ and 𝑇₂. As 𝑥₁=2, by using Mathematical 5.0 to compute {𝑥_𝑛}, we know that 𝑥₁₀=0.82893, 𝑥₂₀=0.935807, 𝑥₅₀=0.994619, and 𝑥₁₀₀=0.999272. This example shows that the algorithm is efficient for approximating common fixed points of nonexpansive mappings.

Remark 3.4. By using Theorem 3.1 and Lemma 2.3, we can easily prove that {𝑥_𝑛} converges strongly to a common fixed point of the mappings of {𝑇_𝑖∶𝑖∈𝐼} if one of the mappings {𝑇_𝑖∶𝑖∈𝐼} is demicompact or {𝑇_𝑖∶𝑖∈𝐼} satisfies condition (B). Therefore the results presented in this paper improve and extend the results of Wang [12] and partially improve the results of Osilike et al. [11].

Remark 3.5. We do not know how to overcome the constraint condition ∑^∞_𝑛=1‖𝑇_𝑛𝑥_𝑛−𝑇_𝑛+1𝑥_𝑛‖<∞ when we try to extend Theorem 3.1 to arbitrary Banach spaces.

Acknowledgments

The authors are extremely grateful to the referees for their useful suggestions that improved the content of the paper. This work was supported by Scientific Research Foundation of Yunnan University of Finance and Economics.