Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2012, Article ID 437430, 13 pages
http://dx.doi.org/10.1155/2012/437430
Research Article

A New Approach to the Approximation of Common Fixed Points of an Infinite Family of Relatively Quasinonexpansive Mappings with Applications

College of Statistics and Mathematics, Yunnan University of Finance and Economics, Yunnan, Kunming 650221, China

Received 23 June 2012; Revised 16 August 2012; Accepted 21 August 2012

Academic Editor: Simeon Reich

Copyright © 2012 Wei-Qi Deng. 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

By using a specific way of choosing the indexes, we propose an iteration algorithm generated by the monotone CQ method for approximating common fixed points of an infinite family of relatively quasinonexpansive mappings. A strong convergence theorem without the stronger assumptions of the AKTT condition and the AKTT condition imposed on the involved mappings is established in the framework of Banach space. As application, an iterative solution to a system of equilibrium problems is studied. The result is more applicable than those of other authors with related interest.

1. Introduction

Let 𝐶 be a nonempty and closed convex subset of a real Banach space 𝐸. A mapping 𝑇𝐶𝐸 is said to be nonexpansive if 𝑇𝑥𝑇𝑦𝑥𝑦,𝑥,𝑦𝐶.(1.1) A mapping 𝑇 is said to be quasi-nonexpansive if 𝐹(𝑇)={𝑥𝐶𝑥=𝑇𝑥} and 𝑇𝑥𝑝𝑥𝑝,𝑥𝐶,𝑝𝐹(𝑇).(1.2) It is easy to see that if 𝑇 is nonexpansive with 𝐹(𝑇), then it is quasi-nonexpansive. There are many methods for approximating fixed points of quasi-nonexpansive mappings. In 1953, Mann [1] introduced the iteration as follows: a sequence {𝑥𝑛} is defined by 𝑥𝑛+1=𝛼𝑛𝑥𝑛+1𝛼𝑛𝑇𝑥𝑛,(1.3) where the initial element 𝑥0𝐶 is arbitrary and {𝛼𝑛} is a sequence of real numbers in [0,1]. Approximation of fixed points of nonexpansive mappings via Mann's algorithm has extensively been investigated. One of the fundamental convergence results was proved by Reich [2]. In infinite-dimensional Hilbert spaces, Mann iteration can yield only weak convergence (see [3, 4]).

Attempts to modify the Mann iteration method (1.3) for strong convergence have recently been made. Nakajo and Takahashi [5] proposed the following modification of Mann iteration method (1.3) for a nonexpansive mapping 𝑇 from 𝐶 into itself in a Hilbert space: from an arbitrary 𝑥0𝐶, 𝑦𝑛=𝛼𝑛𝑥𝑛+1𝛼𝑛𝑇𝑥𝑛,𝐶𝑛=𝑦𝑧𝐶𝑛𝑥𝑧𝑛,𝑄𝑧𝑛=𝑧𝐶𝑥𝑛𝑧,𝑥0𝑥𝑛,𝑥0𝑛+1=𝑃𝐶𝑛𝑄𝑛𝑥0,𝑛0,(1.4) where 𝑃𝐾 denotes the metric projection from a Hilbert space 𝐻 onto a closed convex subset 𝐾 of 𝐻. They proved that the sequence {𝑥𝑛} converges strongly to 𝑃𝐹(𝑇)𝑥0.

Recently, Su and Qin [6] introduced a monotone CQ method for nonexpansive mapping, defined as follows: from an arbitrary 𝑥0𝐶, 𝑦𝑛=𝛼𝑛𝑥𝑛+1𝛼𝑛𝑇𝑥𝑛,𝐶0=𝑦𝑧𝐶0𝑥𝑧0𝑧,𝑄0𝐶=𝐶,𝑛=𝑧𝐶𝑛1𝑄𝑛1𝑦𝑛𝑥𝑧𝑛,𝑄𝑧𝑛=𝑧𝐶𝑛1𝑄𝑛1𝑥𝑛𝑧,𝑥0𝑥𝑛,𝑥0𝑛+1=𝑃𝐶𝑛𝑄𝑛𝑥0,𝑛0,(1.5) and it proved that the sequence {𝑥𝑛} converges strongly to 𝑃𝐹(𝑇)𝑥0.

We now recall some definitions concerning relatively quasi-nonexpansive mappings. Let 𝐸 be a real smooth Banach space with norm and let 𝐸 be the dual of 𝐸. The normalized duality mapping 𝐽 from 𝐸 to 𝐸 is defined by 𝐽𝑥=𝑓𝐸𝑥,𝑓=𝑥2=𝑓2,𝑥𝐸,(1.6) where , denotes the pairing between 𝐸 and 𝐸. Readers are directed to [7] (and its review [8]), where the properties of the duality mapping and several related topics are presented. The function 𝜙𝐸×𝐸R+ is defined by 𝜙(𝑥,𝑦)=𝑥22𝑥,𝐽𝑦+𝑦2,𝑥,𝑦𝐸.(1.7)

Let 𝑇 be a mapping from 𝐶 into 𝐸. A point 𝑝 in 𝐶 is said to be an asymptotic fixed point [9] of 𝑇 if 𝐶 contains a sequence {𝑥𝑛} which converges weakly to 𝑝 and lim𝑛(𝑥𝑛𝑇𝑥𝑛)=0. The set of asymptotic fixed points of 𝑇 is denoted by 𝐹(𝑇).

We say that the mapping 𝑇 is relatively nonexpansive (see [10]) if the following conditions are satisfied:(R1)𝐹(𝑇); (R2)𝜙(𝑝,𝑇𝑥)𝜙(𝑝,𝑥), 𝑝𝐹(𝑇);(R3)𝐹(𝑇)=𝐹(𝑇).If 𝑇 satisfies (R1) and (R2), then 𝑇 is called relatively quasi-nonexpansive.

Several articles have provided methods for approximating fixed points of relatively quasi-nonexpansive mappings [1116]. Employing the ideas of Su and Qin [6], and of Aoyama et al. [17], in 2008, Nilsrakoo and Saejung [18] used the following iterations to obtain strong convergence theorems for common fixed points of a countable family of relatively quasi-nonexpansive mappings in a Banach space 𝑥0𝐶,𝐶1=𝑄1𝑦=𝐶;𝑛=𝐽1𝛼𝑛𝐽𝑥𝑛+1𝛼𝑛𝐽𝑇𝑛𝑥𝑛,𝐶𝑛=𝑧𝐶𝑛1𝑄𝑛1𝜙𝑧,𝑦𝑛𝜙𝑧,𝑥𝑛,𝑄𝑛=𝑧𝐶𝑛1𝑄𝑛1𝑥𝑛𝑧,𝐽𝑥0𝐽𝑥𝑛,𝑥0𝑛+1=Π𝑛𝑥0,𝑛0.(1.8) However, the results were obtained under two stronger assumption conditions, namely, the 𝐴𝐾𝑇𝑇-condition and the 𝐴𝐾𝑇𝑇-condition imposed on the involved mappings.

Inspired and motivated by those studies mentioned above, in this paper, we use a modified type of the iteration scheme (1.8) for approximating common fixed points of an infinite family of relatively quasi-nonexpansive mappings; without stronger assumptions imposed on the involved mappings, a strong convergence theorem in Banach spaces is obtained for solving a system of equilibrium problems. The results improve those of other authors with related interest.

2. Preliminaries

Throughout the paper, let 𝐸 be a real Banach space. We say that 𝐸 is strictly convex if the following implication holds for 𝑥,𝑦𝐸: 𝑥=𝑦=1,𝑥𝑦𝑥+𝑦2<1.(2.1) It is also said to be uniformly convex if for any 𝜖>0, there exists a 𝛿>0 such that 𝑥=𝑦=1,𝑥𝑦𝜖𝑥+𝑦21𝛿.(2.2) It is known that if 𝐸 is uniformly convex Banach space, then 𝐸 is reflexive and strictly convex. A Banach space 𝐸 is said to be smooth if lim𝑡0𝑥+𝑡𝑦𝑥𝑡(2.3) exists for each 𝑥,𝑦𝑆(𝐸)={𝑥𝐸𝑥=1}. In this case, the norm of 𝐸 is said to be Gâteaux differentiable. The space 𝐸 is said to have uniformly Gâteaux differentiable norm if for each 𝑦𝑆(𝐸); the limit (2.3) is attained uniformly for 𝑥𝑆(𝐸). The norm of 𝐸 is said to be Fréchet differentiable if for each 𝑥𝑆(𝐸); the limit (2.3) is attained uniformly for 𝑦𝑆(𝐸). The norm of 𝐸 is said to be uniformly Fréchet differentiable (and 𝐸 is said to be uniformly smooth) if the limit (2.3) is attained uniformly for 𝑥,𝑦𝑆(𝐸).

We also know the following properties (see, e.g., [19] for details). (1)𝐸 (𝐸, resp.) is uniformly convex 𝐸 (𝐸, resp.) is uniformly smooth.(2)𝐽𝑥 for each 𝑥𝐸.(3)If 𝐸 is reflexive, then 𝐽 is a mapping from 𝐸 onto 𝐸.(4)If 𝐸 is strictly convex, then 𝐽𝑥𝐽𝑦= as 𝑥𝑦.(5)If 𝐸 is smooth, then 𝐽 is single-valued.(6)If 𝐸 has a Fréchet differentiable norm, then 𝐽 is norm-to-norm continuous.(7)If 𝐸 is uniformly smooth, then 𝐽 is uniformly norm-to-norm continuous on each bounded subset of 𝐸.(8)If 𝐸 is a Hilbert space, then 𝐽 is the identity operator.

Let 𝐸 be a smooth Banach space. The function 𝜙𝐸×𝐸R+ is defined by 𝜙(𝑥,𝑦)=𝑥22𝑥,𝐽𝑦+𝑦2.(2.4) It is obvious from the definition of the function 𝜙 that ()𝑥𝑦2)𝜙(𝑥,𝑦)(𝑥+𝑦2.(2.5) Moreover, we know the following results.

Lemma 2.1 (see [13]). Let 𝐸 be a strictly convex and smooth Banach space, then 𝜙(𝑥,𝑦)=0 if and only if 𝑥=𝑦.

Lemma 2.2 (see [11]). Let 𝐸 be a uniformly convex and smooth Banach space and let 𝑟>0. Then there exists a continuous, strictly increasing, and convex function 𝑔[0,2𝑟][0,) such that 𝑔(0)=0 and 𝑔(𝑥𝑦)𝜙(𝑥,𝑦)(2.6) for all 𝑥,𝑦𝐵𝑟={𝑧𝐸P𝑧P𝑟}.

Let 𝐶 be a nonempty and closed convex subset of 𝐸. Suppose that 𝐸 is reflexive, strictly convex, and smooth. It is known in [20] that for any 𝑥𝐸, there exists a unique point 𝑥𝐶 such that 𝜙𝑥,𝑥=min𝑦𝐶𝜙(𝑦,𝑥).(2.7) Following Alber [21], we denote such an 𝑥 by Π𝐶𝑥. The mapping Π𝐶 is called the generalized projection from 𝐸 onto 𝐶. It is easy to see that in a Hilbert space, the mapping Π𝐶 coincides with the metric projection 𝑃𝐶. What follows are the well-known facts concerning the generalized projection.

Lemma 2.3 (see [20]). Let 𝐶 be a nonempty closed convex subset of a smooth Banach space 𝐸 and let 𝑥𝐸. Then 𝑥=Π𝐶𝑥𝑥𝑦,𝐽𝑥𝐽𝑥0,𝑦𝐶.(2.8)

Lemma 2.4 (see [20]). Let 𝐸 be a reflexive, strictly convex, and smooth Banach space, let 𝐶 be a nonempty closed convex subset of 𝐸, and let 𝑥𝐸. Then 𝜙𝑦,Π𝐶𝑥Π+𝜙𝐶𝑥,𝑥𝜙(𝑦,𝑥),𝑦𝐶.(2.9)

Dealing with the generalized projection from 𝐸 onto the fixed point set of a relatively quasi-nonexpansive mapping, we have the following result.

Lemma 2.5 (see [18]). Let 𝐸 be a strictly convex and smooth Banach space, let 𝐶 be a nonempty and closed convex subset of 𝐸, and let 𝑇 be a relatively quasi-nonexpansive mapping from 𝐶 into 𝐸. Then 𝐹(𝑇) is closed and convex.

Let 𝐶 be a subset of a Banach space 𝐸 and let {𝑇𝑛} be a family of mappings from 𝐶 into 𝐸. For a subset 𝐵 of 𝐶, we say that (i)({𝑇𝑛},𝐵) satisfies AKTT-condition if 𝑛=1𝑇sup𝑛+1𝑧𝑇𝑛𝑧𝑧𝐵<;(2.10)(ii)({𝑇𝑛},𝐵) satisfies AKTT-condition if 𝑛=1sup𝐽𝑇𝑛+1𝑧𝐽𝑇𝑛𝑧𝑧𝐵<.(2.11)

3. Main Results

Recall that an operator 𝑇 in a Banach space is closed if 𝑥𝑛𝑥 and 𝑇𝑥𝑛𝑦 as 𝑛, then 𝑇𝑥=𝑦.

Theorem 3.1. Let 𝐸 be a uniformly convex and uniformly smooth Banach space, 𝐶 a nonempty and closed convex subset of 𝐸. Let {𝑇𝑖}𝑖=1𝐶𝐸 be a sequence of closed and relatively quasi-nonexpansive mappings with 𝐹=𝑖=1𝐹(𝑇𝑖). Starting from an arbitrary 𝑥1𝐶, the sequence {𝑥𝑛} is define by 𝑥1𝐶,𝐶0=𝑄0𝑦=𝐶;𝑛=𝐽1𝛼𝑛𝐽𝑥𝑛+1𝛼𝑛𝐽𝑇𝑖𝑛𝑥𝑛,𝐶𝑛=𝑧𝐶𝑛1𝑄𝑛1𝜙𝑧,𝑦𝑛𝜙𝑧,𝑥𝑛,𝑄𝑛=𝑧𝐶𝑛1𝑄𝑛1𝑥𝑛𝑧,𝐽𝑥1𝐽𝑥𝑛,𝑥0𝑛+1=Π𝑛𝑥1,𝑛1,(3.1) where Π𝑛=Π𝐶𝑛𝑄𝑛 and {𝛼𝑛} is a sequence in [0,1) with limsup𝑛𝛼𝑛<1; 𝑖𝑛 is the solution to the positive integer equation: 𝑛=𝑖+(𝑚1)𝑚/2(𝑚𝑖,𝑛=1,2,), that is, for each 𝑛1, there exists a unique 𝑖𝑛 such that 𝑖1=1,𝑖2=1,𝑖3=2,𝑖4=1,𝑖5=2,𝑖6=3,𝑖7=1,𝑖8𝑖=2,9=3,𝑖10=4,𝑖11=1,.(3.2) Then {𝑥𝑛} converges strongly to Π𝐹𝑥1.

Proof. We first claim that both 𝐶𝑛 and 𝑄𝑛 are closed and convex. This follows from the fact that 𝜙(𝑧,𝑦𝑛)𝜙(𝑧,𝑥𝑛) is equivalent to the following: 2𝑧,𝐽𝑥𝑛𝐽𝑦𝑛𝑥𝑛2𝑦𝑛2.(3.3)
It is clear that 𝐹𝐶=𝐶0𝑄0. Next, we show that 𝐹𝐶𝑛𝑄𝑛,𝑛1.(3.4) Suppose that 𝐹𝐶𝑘1𝑄𝑘1 for some 𝑘2. Letting 𝑝𝐹, we then have 𝜙𝑝,𝑦𝑘=𝜙𝑝,𝐽1𝛼𝑘𝐽𝑥𝑘+1𝛼𝑘𝐽𝑇𝑖𝑘𝑥𝑘=𝑝22𝑝,𝛼𝑘𝐽𝑥𝑘+1𝛼𝑘𝐽𝑇𝑖𝑘𝑥𝑘𝛼+𝑘𝐽𝑥𝑘+(1𝛼𝑘)𝐽𝑇𝑖𝑘𝑥𝑘2𝑝22𝛼𝑘𝑝,𝐽𝑥𝑘21𝛼𝑘𝑝,𝐽𝑇𝑖𝑘𝑥𝑘+𝛼𝑘𝑥𝑘2+1𝛼𝑘𝑇𝑖𝑘𝑥𝑘2=𝛼𝑘𝑝22𝑝,𝐽𝑥𝑘𝑥+𝑘2+1𝛼𝑘𝑝22𝑝,𝐽𝑇𝑖𝑘𝑥𝑘𝑇+𝑖𝑘𝑥𝑘2=𝛼𝑘𝜙𝑝,𝑥𝑘+1𝛼𝑘𝜙𝑝,𝑇𝑖𝑘𝑥𝑘𝛼𝑘𝜙𝑝,𝑥𝑘+1𝛼𝑘𝜙𝑝,𝑥𝑘=𝜙𝑝,𝑥𝑘.(3.5) This implies that 𝐹𝐶𝑘. It follows from 𝑥𝑘=Π𝑘1𝑥1 and Lemma 2.3 that 𝑥𝑘𝑧,𝐽𝑥1𝐽𝑥𝑘0,𝑧𝐶𝑘1𝑄𝑘1.(3.6) Particularly, 𝑥𝑘𝑧,𝐽𝑥1𝐽𝑥𝑘0,𝑝𝐹(3.7) and hence 𝐹𝑄𝑘, which yields that 𝐹𝐶𝑘𝑄𝑘.(3.8) By induction, (3.4) holds. This implies that {𝑥𝑛} is well defined. It follows from the definition of 𝑄𝑛 and Lemma 2.3 that 𝑥𝑛=Π𝑄𝑛𝑥1. Since 𝑥𝑛+1=Π𝑛𝑥1𝑄𝑛, we have 𝜙𝑥𝑛,𝑥1𝑥𝜙𝑛+1,𝑥1,𝑛1.(3.9) Therefore, {𝜙(𝑥𝑛,𝑥1)} is nondecreasing. Using 𝑥𝑛=Π𝑄𝑛𝑥1 and Lemma 2.4, we have 𝜙𝑥𝑛,𝑥1Π=𝜙𝑄𝑛𝑥1,𝑥1𝜙𝑝,𝑥1𝜙𝑝,𝑥𝑛𝜙𝑝,𝑥1(3.10) for all 𝑝𝐹 and for all 𝑛1, that is, {𝜙(𝑥𝑛,𝑥1)} is bounded. Then lim𝑛𝜙𝑥𝑛,𝑥1exists.(3.11) In particular, by (2.5), the sequence {(𝑥𝑛𝑥1)2} is bounded. This implies that {𝑥𝑛} is bounded. Note again that 𝑥𝑛=Π𝑄𝑛𝑥1 and for any positive integer 𝑘, 𝑥𝑛+𝑘𝑄𝑛+𝑘1𝑄𝑛. By Lemma 2.4, 𝜙𝑥𝑛+𝑘,𝑥𝑛𝑥=𝜙𝑛+𝑘,Π𝑄𝑛𝑥1𝑥𝜙𝑛+𝑘,𝑥1Π𝜙𝑄𝑛𝑥1,𝑥1𝑥=𝜙𝑛+𝑘,𝑥1𝑥𝜙𝑛,𝑥1.(3.12) By Lemma 2.2, we have, for any positive integers 𝑚,𝑛 with 𝑚>𝑛, 𝑔𝑥𝑚𝑥𝑛𝑥𝜙𝑚,𝑥𝑛𝑥𝜙𝑚,𝑥1𝑥𝜙𝑛,𝑥1,(3.13) where 𝑔[0,)[0,) is a continuous, strictly increasing, and convex function with 𝑔(0)=0. Then the properties of the function 𝑔 yield that {𝑥𝑛} is a Cauchy sequence in 𝐶, so there exists an 𝑥𝐶 such that 𝑥𝑛𝑥(𝑛).(3.14) In view of 𝑥𝑛+1=Π𝑛𝑥1𝐶𝑛 and the definition of 𝐶𝑛, we also have 𝜙𝑥𝑛+1,𝑦𝑛𝑥𝜙𝑛+1,𝑥𝑛,𝑛1.(3.15) This implies that lim𝑛𝜙𝑥𝑛+1,𝑦𝑛=lim𝑛𝜙𝑥𝑛+1,𝑥𝑛=0.(3.16) It follows from Lemma 2.2 that lim𝑛𝑥𝑛+1𝑦𝑛=lim𝑛𝑥𝑛+1𝑥𝑛=0.(3.17) Since 𝐽 is uniformly norm-to-norm continuous on bounded sets, we have lim𝑛𝐽𝑥𝑛+1𝐽𝑦𝑛=lim𝑛𝐽𝑥𝑛+1𝐽𝑥𝑛=0.(3.18) On the other hand, we have, for each 𝑛1, 𝐽𝑥𝑛+1𝐽𝑦𝑛=𝐽𝑥𝑛+1𝛼𝑛𝐽𝑥𝑛+1𝛼𝑛𝐽𝑇𝑖𝑛𝑥𝑛=1𝛼𝑛𝐽𝑥𝑛+1𝐽𝑇𝑖𝑛𝑥𝑛𝛼𝑛𝐽𝑥𝑛𝐽𝑥𝑛+11𝛼𝑛𝐽𝑥𝑛+1𝐽𝑇𝑖𝑛𝑥𝑛𝛼𝑛𝐽𝑥𝑛𝐽𝑥𝑛+1,(3.19) and hence 𝐽𝑥𝑛+1𝐽𝑇𝑖𝑛𝑥𝑛11𝛼𝑛𝐽𝑥𝑛+1𝐽𝑦𝑛+𝛼𝑛1𝛼𝑛𝐽𝑥𝑛𝐽𝑥𝑛+1.(3.20) From (3.18) and limsup𝑛𝛼𝑛<1, we obtain that lim𝑛𝐽𝑥𝑛+1𝐽𝑇𝑖𝑛𝑥𝑛=0.(3.21) Since 𝐽1 is uniformly norm-to-norm continuous on bounded sets, we have lim𝑛𝑥𝑛+1𝑇𝑖𝑛𝑥𝑛=lim𝑛𝐽1𝐽𝑥𝑛+1𝐽1𝐽𝑇𝑖𝑛𝑥𝑛=0.(3.22) It follows from (3.17) that, as 𝑛, 𝑥𝑛𝑇𝑖𝑛𝑥𝑛𝑥𝑛𝑥𝑛+1+𝑥𝑛+1𝑇𝑖𝑛𝑥𝑛0.(3.23) Now, set 𝒦𝑖={𝑘1𝑘=𝑖+(𝑚1)𝑚/2,𝑚𝑖,𝑚+} for each 𝑖1. Note that 𝑇𝑖𝑘=𝑇𝑖 whenever 𝑘𝒦𝑖. For example, by the definition of 𝒦1, we have 𝒦1={1,2,4,7,11,16,} and 𝑖1=𝑖2=𝑖4=𝑖7=𝑖11=𝑖16==1. Then it follows from (3.23) that lim𝒦𝑖𝑘𝑇𝑖𝑥𝑘𝑥𝑘=0,𝑖1.(3.24) Since {𝑥𝑘}𝑘𝒦𝑖 is a subsequence of {𝑥𝑛}, (3.14) implies that 𝑥𝑘𝑥 as 𝒦𝑖𝑘. It immediately follows from (3.24) and the closedness of 𝑇𝑖 that 𝑥𝐹(𝑇𝑖) for each 𝑖1, and hence 𝑥𝐹. Furthermore, by (3.10), 𝜙𝑥,𝑥1=lim𝑛𝜙𝑥𝑛,𝑥1𝜙𝑝,𝑥1,𝑝𝐹.(3.25) This implies that 𝑥=Π𝐹𝑥1. The proof is completed.

Remark 3.2. Note that the algorithm (3.1) is based on the projection onto an intersection of two closed and convex sets. An example [22] of how to compute such a projection is given as follows.

Dykstra's Algorithm
Let Ω1,Ω2,,Ω𝑝 be closed and convex subsets of 𝑛. For any 𝑖=1,2,,𝑝 and 𝑥0𝑛, the sequences {𝑥𝑘𝑖} are defined by the following recursive formulae: 𝑥𝑘0=𝑥𝑝𝑘1,𝑥𝑘𝑖=𝑃Ω𝑖𝑥𝑘𝑖1𝑦𝑖𝑘1𝑦,𝑖=1,2,,𝑝,𝑘𝑖=𝑥𝑘𝑖𝑥𝑘𝑖1𝑦𝑖𝑘1,𝑖=1,2,,𝑝,(3.26) for 𝑘=1,2, with initial values 𝑥0𝑝=𝑥0 and 𝑦0𝑖=0 for 𝑖=1,2,,𝑝. If Ω=𝑝𝑖=1Ω𝑖, then {𝑥𝑘𝑖} converges to 𝑥=𝑃Ω(𝑥0), where 𝑃Ω(𝑥)=arginf𝑦Ω𝑦𝑥2,forall𝑥𝑛.

4. Applications

The so-called convex feasibility problem for a family of mappings {𝑇𝑖}𝑖=1 is to find a point in the nonempty intersection 𝑖=1𝐹(𝑇𝑖), which exactly illustrates the importance of finding fixed points of infinite families. The following example also clarifies the same thing.

Example 4.1. Let 𝐸 be a smooth, strictly convex, and reflexive Banach space, 𝐶 a nonempty and closed convex subset of 𝐸, and {𝑓𝑖}𝑖=1𝐶𝐶 a countable family of bifunctions satisfying the conditions: for each 𝑖1, (𝐴1)𝑓𝑖(𝑥,𝑥)=0;(𝐴2)𝑓𝑖 is monotone, that is, 𝑓𝑖(𝑥,𝑦)+𝑓𝑖(𝑦,𝑥)0;(𝐴3)limsup𝑡0𝑓𝑖(𝑥+𝑡(𝑧𝑥),𝑦)𝑓𝑖(𝑥,𝑦);(𝐴4) the mapping 𝑦𝑓𝑖(𝑥,𝑦) is convex and lower semicontinuous. A system of equilibrium problems for {𝑓𝑖}𝑖=1 is to find an 𝑥𝐶 such that 𝑓𝑖𝑥,𝑦0,𝑦𝐶,𝑖1,(4.1) whose set of common solutions is denoted by 𝐸𝑃=𝑖=1𝐸𝑃(𝑓𝑖), where 𝐸𝑃(𝑓𝑖) denotes the set of solutions to the equilibrium problem for 𝑓𝑖(𝑖=1,2,). It will be shown in Theorem 4.3 that such a system of problems can be reduced to approximation of some fixed points of a countable family of nonexpansive mappings.

Example 4.2 (see [23]). Let 𝑟>0. Define a countable family of mappings {𝑇𝑟,𝑖}𝑖=1𝐸𝐶 as follows: 𝑇𝑟,𝑖(𝑥)=𝑧𝐶𝑓𝑖1(𝑧,𝑦)+𝑟𝑦𝑧,𝐽𝑧𝐽𝑥0,𝑦𝐶,𝑖1.(4.2) Then we have that(1){𝑇𝑟,𝑖}𝑖=1 is a sequence of single-valued mappings;(2){𝑇𝑟,𝑖}𝑖=1 is a sequence of closed relatively quasi-nonexpansive mappings;(3)𝐹=𝑖=1𝐹(𝑇𝑟,𝑖)=𝐸𝑃.

Now, we have the following result.

Theorem 4.3. Let 𝐶,𝐸, and {𝛼𝑛} be the same as those in Theorem 3.1. Let {𝑓𝑖}𝑖=1𝐶𝐶 be a countable family of bifunctions satisfying the conditions (𝐴1)–(𝐴4). Let {𝑇𝑟,𝑖}𝑖=1𝐸𝐶 be a countable family of mappings defined by (4.2). Let {𝑥𝑛} be the sequence generated by 𝑥1𝐶,𝐶0=𝑄0𝑓=𝐶;𝑖𝑛𝑢𝑛+1,𝑦𝑟𝑦𝑢𝑛,𝐽𝑢𝑛𝐽𝑥𝑛𝑦0,𝑦𝐶,𝑛=𝐽1𝛼𝑛𝐽𝑥𝑛+1𝛼𝑛𝐽𝑢𝑛,𝐶𝑛=𝑧𝐶𝑛1𝑄𝑛1𝜙𝑧,𝑦𝑛𝜙𝑧,𝑥𝑛,𝑄𝑛=𝑧𝐶𝑛1𝑄𝑛1𝑥𝑛𝑧,𝐽𝑥1𝐽𝑥𝑛,𝑥0𝑛+1=Π𝑛𝑥1,𝑛1,(4.3) where 𝑖𝑛 satisfies the positive integer equation: 𝑛=𝑖+(𝑚1)𝑚/2(𝑚𝑖,𝑛=1,2,). If 𝐹=𝑖=1𝐹(𝑇𝑟,𝑖), then {𝑥𝑛} strongly converges to Π𝐹𝑥1 which is a common solution of the system of equilibrium problems for {𝑓𝑖}𝑖=1.

Proof. Since each 𝑇𝑟,𝑖 is single-valued, 𝑢𝑛=𝑇𝑟,𝑖𝑛𝑥𝑛 for all 𝑛1. In addition, we have pointed out in Example 4.2 that 𝐹=𝐸𝑃 and {𝑇𝑟,𝑖}𝑖=1 is a sequence of closed relatively quasi-nonexpansive mappings. Hence, (4.3) can be rewritten as follows: 𝑥1𝐶,𝐶0=𝑄0𝑦=𝐶;𝑛=𝐽1𝛼𝑛𝐽𝑥𝑛+1𝛼𝑛𝐽𝑇𝑟,𝑖𝑛𝑥𝑛,𝐶𝑛=𝑧𝐶𝑛1𝑄𝑛1𝜙𝑧,𝑦𝑛𝜙𝑧,𝑥𝑛,𝑄𝑛=𝑧𝐶𝑛1𝑄𝑛1𝑥𝑛𝑧,𝐽𝑥1𝐽𝑥𝑛,𝑥0𝑛+1=Π𝑛𝑥1,𝑛1.(4.4) Therefore, this conclusion can be obtained immediately from Theorem 3.1.

Acknowledgments

The author is greatly grateful to the referees for their useful suggestions by which the contents of this article are improved. This study was supported by the General Project of Scientific Research Foundation of Yunnan University of Finance and Economics (YC2012A02).

References

  1. W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. S. Reich, “Weak convergence theorems for nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 274–276, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. H. H. Bauschke, E. Matoušková, and S. Reich, “Projection and proximal point methods: convergence results and counterexamples,” Nonlinear Analysis. Theory, Methods & Applications, vol. 56, no. 5, pp. 715–738, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. A. Genel and J. Lindenstrauss, “An example concerning fixed points,” Israel Journal of Mathematics, vol. 22, no. 1, pp. 81–86, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372–379, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. Y. Su and X. Qin, “Monotone CQ iteration processes for nonexpansive semigroups and maximal monotone operators,” Nonlinear Analysis. Theory, Methods & Applications, vol. 68, no. 12, pp. 3657–3664, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, The Netherlands,, 1990. View at Publisher · View at Google Scholar
  8. S. Reich, “Book Review: geometry of Banach spaces, duality mappings and nonlinear problems,” American Mathematical Society, vol. 26, no. 2, pp. 367–370, 1992. View at Publisher · View at Google Scholar
  9. S. Reich, “A weak convergence theorem for the alternating method with Bregman distances,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 313–318, Dekker, New York, NY, USA, 1996. View at Google Scholar · View at Zentralblatt MATH
  10. D. Butnariu, S. Reich, and A. J. Zaslavski, “Asymptotic behavior of relatively nonexpansive operators in Banach spaces,” Journal of Applied Analysis, vol. 7, no. 2, pp. 151–174, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. F. Kohsaka and W. Takahashi, “Block iterative methods for a finite family of relatively nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2007, Article ID 21972, 18 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. S.-Y. Matsushita and W. Takahashi, “Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, no. 1, pp. 37–47, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. S.-Y. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. S.-Y. Matsushita and W. Takahashi, “An iterative algorithm for relatively nonexpansive mappings by a hybrid method and applications,” in Nonlinear Analysis and Convex Analysis, pp. 305–313, Yokohama Publishers, Yokohama, Japan, 2004. View at Google Scholar · View at Zentralblatt MATH
  15. S. Plubtieng and K. Ungchittrakool, “Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space,” Journal of Approximation Theory, vol. 149, no. 2, pp. 103–115, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. X. Qin and Y. Su, “Strong convergence theorems for relatively nonexpansive mappings in a Banach space,” Nonlinear Analysis. Theory, Methods & Applications, vol. 67, no. 6, pp. 1958–1965, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis. Theory, Methods & Applications, vol. 67, no. 8, pp. 2350–2360, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. W. Nilsrakoo and S. Saejung, “Strong convergence to common fixed points of countable relatively quasi-nonexpansive mappings,” Fixed Point Theory and Applications, vol. 2008, Article ID 312454, 19 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, Japan, 2000.
  20. S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach space,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 938–945, 2002. View at Publisher · View at Google Scholar
  21. Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15–50, Dekker, New York, NY, USA, 1996. View at Google Scholar · View at Zentralblatt MATH
  22. J. P. Boyle and R. L. Dykstra, “A method for finding projections onto the intersection of convex sets in Hilbert spaces,” in Advances in Order Restricted Statistical Inference, vol. 37 of Lecture Notes in Statistics, pp. 28–47, Springer, Berlin, Germany, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,” The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994. View at Google Scholar · View at Zentralblatt MATH