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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 498487, 12 pages
http://dx.doi.org/10.1155/2012/498487
Research Article

A Strong Convergence Theorem for Relatively Nonexpansive Mappings and Equilibrium Problems in Banach Spaces

1Leshan College of Profession and Technology, Sichuan, Leshan 614000, China
2Department of Mathematics, Sichuan University, Sichuan, Chengdu 610064, China
3College of Business, City University of Hong Kong, Hong Kong

Received 3 July 2012; Revised 24 August 2012; Accepted 24 August 2012

Academic Editor: Yeong-Cheng Liou

Copyright © 2012 Mei Yuan et al. 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

Relatively nonexpansive mappings and equilibrium problems are considered based on a shrinking projection method. Using properties of the generalized f-projection operator, a strong convergence theorem for relatively nonexpansive mappings and equilibrium problems is proved in Banach spaces under some suitable conditions.

1. Introduction

It is well known that metric projection operator in Hilbert and Banach spaces is widely used in different areas of mathematics such as functional analysis and numerical analysis, theory of optimization and approximation, and also for the problems of optimal control and operations research, nonlinear and stochastic programming and game theory.

Let 𝑋 be a real Banach space with its dual 𝑋, and let 𝐾 be a nonempty, closed, and convex subset of 𝑋. In 1994, Alber [1] introduced the generalized projections 𝜋𝐾𝑋𝐾 and Π𝐾𝑋𝐾 in uniformly convex and uniformly smooth Banach spaces based on the function 𝜙(𝑦,𝑥) defined on p.3 and studied their properties in detail. In 2005, Li [2] extended the definition of the generalized projection operator from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces and studied some properties of the generalized projection operator. Recently, Wu and Huang [3] introduced a new generalized 𝑓-projection operator in a Banach space. By making use of (2.5), they extended the definition of the generalized projection operators introduced by Abler [1] and proved some properties of the generalized 𝑓-projection operator. Wu and Huang [4] studied a relation between the generalized projection operator and the resolvent operator for the subdifferential of a proper convex and lower semicontinuous functional in reflexive and smooth Banach spaces (see [59]). Very recently, Li et al. [10] studied some properties of the generalized 𝑓-projection operator, and proved the strong convergence theorems for relatively nonexpansive mappings in Banach spaces.

On the other hand, equilibrium problem was introduced by Blum and Oettli [11], in 1994. It is a hot topic of intensive research efforts, because it has a great impact and influence in the development of several branches of pure and applied sciences. It has been shown that equilibrium problem theory provides a novel and unified treatment of a wide class of problems arisen in economics, finance, physics, image reconstruction, ecology, transportation, network, elasticity, and optimization problems. Numerous issues in physics, optimization, and economics reduce to finding a solution of equilibrium problem. Some methods have been proposed to solve the equilibrium problems (see, e.g, [1214] and the references therein).

In this paper, motivated and inspired by the work mentioned above, we introduce a new hybrid projection algorithm based on the shrinking projection method for relatively nonexpansive mapping and equilibrium problem. Using the new algorithm, we prove a strong convergence theorem for relatively nonexpansive mappings and equilibrium problems in Banach spaces. The result presented in this paper extends and improves the main result of Li et al. [10].

2. Preliminaries

Let 𝑋 be a real Banach space with its dual 𝑋 and 𝑅=(,+). We denote the duality between 𝑋 and 𝑋 by ,, and the norms of Banach space 𝑋 and 𝑋 by 𝑋 and 𝑋, respectively. A Banach space 𝑋 is said to be strictly convex if (𝑥+𝑦)/2<1 for all 𝑥,𝑦𝑋 with 𝑥=𝑦=1 and 𝑥𝑦. It is also said to be uniformly convex if lim𝑛𝑥𝑛𝑦𝑛=0 for any two sequences {𝑥𝑛}, {𝑦𝑛} in 𝑋, such that 𝑥𝑛=𝑦𝑛=1 and lim𝑛(𝑥𝑛+𝑦𝑛)/2=1. The function 𝛿𝑋(𝜀)=inf1𝑥+𝑦2𝑥=1,𝑦=1,𝑥𝑦𝜀(2.1) is called the modulus of convexity of 𝑋.

A Banach space 𝑋 is said to be smooth provided that lim𝑡0(𝑥+𝑡𝑦𝑥)/𝑡 exists for all 𝑥,𝑦𝑋 with 𝑥=𝑦=1. It is also said to be uniformly smooth if the limit is attained uniformly for 𝑥=𝑦=1. The function 𝜌𝑋(𝑡)=sup𝑥+𝑦+𝑥𝑦21𝑥=1,𝑦𝑡(2.2) is called the modulus of smoothness of 𝑋.

When {𝑥𝑛} is a sequence in 𝑋, we denote the strong convergence of {𝑥𝑛} with a cluster 𝑥𝑋 by 𝑥𝑛𝑥 and the weak convergence of {𝑥𝑛} with a weak cluster 𝑥𝑋 by 𝑥𝑛𝑥. A Banach space 𝑋 is said to have the Kadec-Klee property if a sequence {𝑥𝑛} of 𝑋 satisfies that 𝑥𝑛𝑥𝑋 and 𝑥𝑛𝑥, then 𝑥𝑛𝑥. It is known that if 𝑋 is uniformly convex, then 𝑋 has the Kadec-Klee property.

The normalized duality mapping 𝐽 from 𝑋 to 𝑋 is defined by 𝑥𝐽𝑥=𝑋𝑥,𝑥=𝑥2=𝑥2(2.3) for any 𝑥𝑋. We list some properties of mapping 𝐽 as follows. (i) If 𝑋 is a smooth Banach space (with Gâteaux differential norm), then 𝐽 is single-valued and demicontinuous. If 𝑋 is a smooth reflexive Banach space, then 𝐽 is single-valued and hemicontinuous. If 𝑋 is a strongly smooth Banach space (with Fréchet differential norm), then 𝐽 is single-valued and continuous. (ii)𝐽 is uniformly continuous on every bounded set of a uniformly smooth Banach space. (iii) If 𝑋 is a reflexive, smooth and strictly convex Banach space, 𝐽𝑋𝑋 is the duality mapping of 𝑋, then 𝐽1=𝐽,𝐽𝐽=𝐼𝑋,𝐽𝐽=𝐼𝑋.

Let 𝑋 be a smooth Banach space and 𝐾 be a nonempty, closed and convex subset of 𝑋. The function 𝜙𝑋×𝑋𝑅 is defined by 𝜙(𝑦,𝑥)=𝑦22𝑦,𝐽𝑥+𝑥2(2.4) for all 𝑥,𝑦𝑋.

Next, we recall the concept of the generalized 𝑓-projector operator, together with its properties. Let 𝐺𝐾×𝑋𝑅{+} be a functional defined as follows: 𝐺(𝜉,𝜑)=𝜉22𝜉,𝜑+𝜑2+2𝜌𝑓(𝜉),(2.5) where 𝜉𝐾,𝜑𝑋, 𝜌 is a positive number and 𝑓𝐾𝑅{+} is proper, convex, and lower semicontinuous.

From the definitions of 𝐺 and 𝑓, it is easy to have the following properties: (i)𝐺(𝜉,𝜑) is convex and continuous with respect to 𝜑 when 𝜉 is fixed; (ii)𝐺(𝜉,𝜑) is convex and lower semicontinuous with respect to 𝜉 when 𝜑 is fixed.

Definition 2.1. Let 𝑋 be a real smooth Banach space and 𝐾 be a nonempty, closed and convex subset of 𝑋. We say that Π𝑓𝐾𝑋2𝐾 is a generalized 𝑓-projection operator if Π𝑓𝐾𝑥=𝑢𝐾𝐺(𝑢,𝐽𝑥)=inf𝜉𝐾𝐺(𝜉,𝐽𝑥),𝑥𝑋.(2.6)

In order to obtain our results, the following lemmas are crucial to us.

Lemma 2.2 (see [15]). Let 𝑋 be a real Banach space and 𝑓𝑋𝑅{+} be a lower semicontinuous convex functional. Then there exist 𝑥𝑋 and 𝛼𝑅 such that 𝑓(𝑥)𝑥,𝑥+𝛼,𝑥𝑋.(2.7)

Lemma 2.3 (see [16]). Let 𝑋 be a uniformly convex and smooth Banach space and let {𝑦𝑛}, {𝑧𝑛} be two sequences of 𝑋. If 𝜙(𝑦𝑛,𝑧𝑛)0 and either {𝑦𝑛} or {𝑧𝑛} is bounded, then 𝑦𝑛𝑧𝑛0.

Let 𝐾 be a closed subset of a real Banach space 𝑋, and let 𝑇 be a mapping from 𝐾 to 𝐾. We denote by 𝐹(𝑇) the set of all fixed points of 𝑇. A point 𝑝 in 𝐾 is said to be an asymptotic fixed point of 𝑇, if 𝐾 contains a sequence {𝑥𝑛} which converges weakly to 𝑝 such that lim𝑛(𝑥𝑛𝑇𝑥𝑛)=0. The set of all asymptotic fixed points of 𝑇 will be denoted by 𝐹(𝑇). 𝑇 is called nonexpansive if 𝑇𝑥𝑇𝑦𝑥𝑦 for all 𝑥,𝑦𝐾, and relatively nonexpansive if 𝐹(𝑇)=𝐹(𝑇) and 𝜙(𝑝,𝑇𝑥)𝜙(𝑝,𝑥) for all 𝑥𝐾 and 𝑝𝐹(𝑇). Obviously, the definition of relatively nonexpansive mapping 𝑇 is equivalent to 𝐹(𝑇)=𝐹(𝑇) and 𝐺(𝑝,𝐽𝑇𝑥)𝐺(𝑝,𝐽𝑥) for all 𝑥𝐾 and 𝑝𝐹(𝑇).

Lemma 2.4 (see [17]). Let 𝑋 be a strictly convex and smooth Banach space, let 𝐾 be a closed, and convex subset of 𝑋, and let 𝑇 be a relatively nonexpansive mapping from 𝐾 into itself. Then 𝐹(𝑇) is closed, and convex.

Lemma 2.5 (see [10]). Let 𝑋 be a real reflexive and smooth Banach space and let 𝐾 be a nonempty, closed, and convex subset of 𝑋. The following statements hold: (i)Π𝑓𝐾𝑥 is a nonempty, closed, and convex subset of 𝐾 for all 𝑥𝑋;(ii) for all 𝑥𝑋,̂𝑥Π𝑓𝐾𝑥 if and only if ̂𝑥𝑦,𝐽𝑥𝐽̂𝑥+𝜌𝑓(𝑦)𝜌𝑓(̂𝑥)0,𝑦𝐾;(2.8)(iii) if 𝑋 is strictly convex, then Π𝑓𝐾 is a single-valued mapping.

Lemma 2.6 (see [10]). Let 𝑋 be a real reflexive and smooth Banach space, let 𝐾 be a nonempty, closed, and convex subset of 𝑋, and let 𝑥𝑋, ̂𝑥Π𝑓𝐾𝑥. Then 𝜙(𝑦,̂𝑥)+𝐺(̂𝑥,𝐽𝑥)𝐺(𝑦,𝐽𝑥),𝑦𝐾.(2.9)

Lemma 2.7 (see [10]). Let 𝑋 be a Banach space and 𝑦𝑋. Let 𝑓𝑋𝑅{+} be a proper, convex and lower semicontinuous functional with convex domain 𝐷(𝑓). If {𝑥𝑛} is a sequence in 𝐷(𝑓) such that 𝑥𝑛̂𝑥int(𝐷(𝑓)) and limn𝐺(𝑥𝑛,𝐽𝑦)=𝐺(̂𝑥,𝐽𝑦), then lim𝑛𝑥𝑛=̂𝑥.

Let 𝑀 be a closed and convex subset of a real Banach space 𝑋 and 𝑔𝑀×𝑀𝑅 be a bifunction. The equilibrium problem for 𝑔 is as follows. Find ̂𝑥𝑀 such that 𝑔(̂𝑥,𝑦)0,𝑦𝑀.(2.10) The set of all solutions for the above equilibrium problem is denoted by EP(𝑔). For solving the equilibrium problem, one always assumes that the bifunction 𝑔 satisfies the following conditions:(A1)𝑔(𝑥,𝑥)=0, for all 𝑥𝑀; (A2)𝑔 is monotone, that is, 𝑔(𝑥,𝑦)+𝑔(𝑦,𝑥)0, for all 𝑥,𝑦𝑀; (A3) for all 𝑥,𝑦,𝑧𝑀,limsup𝑡0𝑔(𝑡𝑧+(1𝑡)𝑥,𝑦)𝑔(𝑥,𝑦); (A4) for all 𝑥𝑀, 𝑔(𝑥,) is convex and lower semicontinuous.

In order to prove our results, we present several necessary lemmas.

Lemma 2.8 (see [14]). Let 𝑀 be a closed and convex subset of a uniformly smooth, strictly convex and reflexive Banach space 𝑋, and 𝑔(,) be a bifunction from 𝑀×𝑀𝑅 satisfying the conditions (A1)–(A4). For all 𝑟>0 and 𝑥𝑋, define the mappingas follows. 𝑇𝑟1𝑥=𝑧𝑀𝑔(𝑧,𝑦)+𝑟.𝐽𝑧𝐽𝑥,𝑦𝑧0,𝑦𝑀(2.11) Then, the following statements hold: (B1)𝑇𝑟 is single-valued; (B2)𝑇𝑟 is a firmly nonexpansive-type mapping, that is, for all 𝑥,𝑦𝑋, 𝐽𝑇𝑟𝑥𝐽𝑇𝑟𝑦,𝑇𝑟𝑥𝑇𝑟𝑦𝐽𝑥𝐽𝑦,𝑇𝑟𝑥𝑇𝑟𝑦;(2.12)(B3)𝐹(𝑇𝑟)=𝐹(𝑇𝑟)=EP(𝑔); (B4)EP(𝑔) is closed and convex.

Lemma 2.9 (see [14]). Let 𝑀 be a closed and convex subset of a smooth, strictly convex, and reflexive Banach space 𝑋, 𝑔 be a bifunction from 𝑀×𝑀 to 𝑅 satisfying the conditions (A1)–(A4), and 𝑟>0. Then, for any 𝑥𝑋 and 𝑞𝐹(𝑇𝑟), 𝜙𝑞,𝑇𝑟𝑥𝑇+𝜙𝑟𝑥,𝑥𝜙(𝑞,𝑥).(2.13)

3. The Main Result

In this section, we prove a strong convergence theorem for relatively nonexpansive mappings and equilibrium problems in Banach spaces.

Theorem 3.1. Let 𝑋 be a uniformly convex and uniformly smooth Banach space, 𝐾 and 𝑀 be two nonempty, closed and convex subsets of 𝑋 such that 𝐾𝑀. Let 𝑇𝐾𝐾 be a relatively nonexpansive mapping and 𝑓𝑋𝑅 a convex and lower semicontinuous mapping with 𝐾int(𝐷(𝑓)). Let 𝑔(,) be a bifunction from 𝑀×𝑀𝑅, which satisfies the conditions (A1)–(A4). Assume that {𝛼𝑛}𝑛=0 is a sequence in [0,1) such that limsup𝑛𝛼𝑛<1, and {𝑟𝑛}[𝑎,) for some 𝑎>0. Define a sequence {𝑥𝑛} in 𝐾𝑀 by the following algorithm: 𝑥0=𝑥𝐾𝑀,𝐻0𝑦=𝐾𝑀,𝑛=𝐽1𝛼𝑛𝐽𝑥𝑛+1𝛼𝑛𝐽𝑇𝑥𝑛,𝑢𝑛𝑢𝑀𝑠𝑢𝑐𝑡𝑎𝑡𝑔𝑛+1,𝑦𝑟𝑛𝐽𝑢𝑛𝐽𝑦𝑛,𝑦𝑢𝑛𝐻0,𝑦𝑀,𝑛+1=𝑧𝐻𝑛𝐺𝑧,𝐽𝑢𝑛𝐺𝑧,𝐽𝑦𝑛𝐺𝑧,𝐽𝑥𝑛,𝑥n+1=Π𝑓𝐻𝑛+1𝑥,𝑛=0,1,2,.(3.1) If 𝐹=𝐹(𝑇)EP(𝑔) is nonempty, then {𝑥𝑛} converges strongly to Π𝑓𝐹𝑥.

Proof . The proof is divided into the following four steps.
(I) First, we prove the following conclusion: 𝐻𝑛 is a closed convex set and 𝐹𝐻𝑛 for all 𝑛0.
It is obvious that 𝐻0 is a closed convex set and 𝐹𝐻0. Thus, we only need to show that 𝐻𝑛 is a closed convex set and 𝐹𝐻𝑛 for all 𝑛1.
Since 𝐺(𝑧,𝐽𝑢𝑛)𝐺(𝑧,𝐽𝑦𝑛) and 𝐺(𝑧,𝐽𝑦𝑛)𝐺(𝑧,𝐽𝑥𝑛) are respectively equivalent to 2𝑧,𝐽𝑦𝑛𝐽𝑢𝑛𝑢+𝑛2𝑦𝑛20,2𝑧,𝐽𝑥𝑛𝐽𝑦𝑛𝑦+𝑛2𝑥𝑛20,(3.2) it follows that 𝐻𝑛+1 is closed and convex for all 𝑛0. Thus, we know that {𝑥𝑛} is well defined. Further, for any 𝑢𝐹 and 𝑛0, we have 𝐺𝑢,𝐽𝑦𝑛=𝑢22𝑢,𝛼𝑛𝐽𝑥𝑛+1𝛼𝑛𝐽𝑇𝑥𝑛+𝛼𝑛𝐽𝑥𝑛+1𝛼𝑛𝐽𝑇𝑥𝑛2+2𝜌𝑓(𝑢)𝑢22𝛼𝑛𝑢,𝐽𝑥𝑛21𝛼𝑛𝑢,𝐽𝑇𝑥𝑛+𝛼𝑛𝑥𝑛2+1𝛼𝑛𝑇𝑥𝑛2+2𝜌𝑓(𝑢)=𝛼𝑛𝑢22𝑢,𝐽𝑥𝑛𝑥+𝑛2++2𝜌𝑓(𝑢)1𝛼𝑛𝑢22𝑢,𝐽𝑇𝑥𝑛+𝑇𝑥𝑛2+2𝜌𝑓(𝑢)=𝛼𝑛𝐺𝑢,𝐽𝑥𝑛+1𝛼𝑛𝐺𝑢,𝐽𝑇𝑥𝑛𝐺𝑢,𝐽𝑥𝑛.(3.3) On the other hand, it follows from the definition of {𝑢𝑛} and Lemma 2.8 that 𝑢𝑛=𝑇𝑟𝑛𝑦𝑛. From Lemma 2.9, we obtain 𝜙𝑢,𝑢𝑛=𝜙𝑢,𝑇𝑟𝑛𝑦𝑛𝜙𝑢,𝑦𝑛,(3.4) which implies that 𝐺𝑢,𝐽𝑢𝑛𝐺𝑢,𝐽𝑦𝑛.(3.5) Therefore, 𝑢𝐻𝑛+1 for all 𝑛0.
(II) Second, we show that {𝑥𝑛} is bounded and lim𝑛𝐺(𝑥𝑛,𝐽𝑥) exists.
Since 𝑓𝑋𝑅 is a convex and lower semicontinuous mapping, a direct application of Lemma 2.2 yields that there exist 𝑥𝑋 and 𝛼𝑅 such that 𝑓(𝑦)𝑦,𝑥+𝛼,𝑦𝑋.(3.6) It follows that 𝐺𝑥𝑛=𝑥,𝐽𝑥𝑛22𝑥𝑛,𝐽𝑥+𝑥2𝑥+2𝜌𝑓𝑛𝑥𝑛22𝑥𝑛,𝐽𝑥+𝑥2+2𝜌𝑥𝑛,𝑥=𝑥+2𝜌𝛼𝑛22𝑥𝑛,𝐽𝑥𝜌𝑥+𝑥2𝑥+2𝜌𝛼𝑛22𝐽𝑥𝜌𝑥𝑥𝑛+𝑥2=𝑥+2𝜌𝛼𝑛𝐽𝑥𝜌𝑥2+𝑥2𝐽𝑥𝜌𝑥2+2𝜌𝛼.(3.7) Since 𝑥𝑛=Π𝑓𝐻𝑛𝑥, it follows from (3.7) that 𝐺𝑥(𝑢,𝐽𝑥)𝐺𝑛𝑥,𝐽𝑥𝑛𝐽𝑥𝜌𝑥2+𝑥2𝐽𝑥𝜌𝑥2+2𝜌𝛼,𝑢𝐹,(3.8) which implies that {𝑥𝑛} is bounded and so is {𝐺(𝑥𝑛,𝐽𝑥)}. By the fact that 𝑥𝑛+1𝐻𝑛+1𝐻𝑛 and Lemma 2.6, we obtain 𝜙𝑥𝑛+1,𝑥𝑛𝑥+𝐺𝑛𝑥,𝐽𝑥𝐺𝑛+1,𝐽𝑥.(3.9) It is obvious that 𝜙𝑥𝑛+1,𝑥𝑛𝑥𝑛+1𝑥𝑛20,(3.10) and so {𝐺(𝑥𝑛,𝐽𝑥)} is nondecreasing. Therefore, we know that lim𝑛𝐺(𝑥𝑛,𝐽𝑥) exists.
(III) Third, we prove that, if 𝑥𝑛𝑘̂𝑥, then ̂𝑥𝐹, where {𝑥𝑛𝑘} is an arbitrarily weakly convergent subsequence of {𝑥𝑛}.
It follows from the definition of 𝐻𝑛+1 and 𝑥𝑛+1𝐻𝑛 that 𝜙𝑥𝑛+1,𝑢𝑛𝑥𝜙𝑛+1,𝑦𝑛𝑥𝜙𝑛+1,𝑥𝑛𝑥𝐺𝑛+1𝑥,𝐽𝑥𝐺𝑛,𝐽𝑥.(3.11) Taking lim𝑛 in (3.11), we get lim𝑛𝜙𝑥𝑛+1,𝑢𝑛=lim𝑛𝜙𝑥𝑛+1,𝑦𝑛=lim𝑛𝜙𝑥𝑛+1,𝑥𝑛=0.(3.12) Applying Lemma 2.3, we obtain lim𝑛𝑥𝑛+1𝑢𝑛=lim𝑛𝑥𝑛+1𝑦𝑛=lim𝑛𝑥𝑛+1𝑥𝑛=0.(3.13)
Next, we show that ̂𝑥𝐹(𝑇)=𝐹(𝑇).
From the fact that 𝐽 is uniformly norm-to-norm continuous on bounded sets, we have lim𝑛𝐽𝑥𝑛+1𝐽𝑦𝑛=lim𝑛𝐽𝑥𝑛+1𝐽𝑥𝑛=0.(3.14) Note that 𝐽𝑥𝑛+1𝐽𝑦𝑛=𝐽𝑥𝑛+1𝛼𝑛𝐽𝑥𝑛1𝛼𝑛𝐽𝑇𝑥𝑛=1𝛼𝑛𝐽𝑥𝑛+11𝛼𝑛𝐽𝑇𝑥𝑛+𝛼𝑛𝐽𝑥𝑛+1𝛼𝑛𝐽𝑥𝑛1𝛼𝑛𝐽𝑥𝑛+1𝐽𝑇𝑥𝑛𝛼𝑛𝐽𝑥𝑛+1𝐽𝑥𝑛(3.15) and thus 𝐽𝑥𝑛+1𝐽𝑇𝑥𝑛11𝛼𝑛𝐽𝑥𝑛+1𝐽𝑦𝑛+𝛼𝑛1𝛼𝑛𝐽𝑥𝑛+1𝐽𝑥𝑛11𝛼𝑛𝐽𝑥𝑛+1𝐽𝑦𝑛+𝐽𝑥𝑛+1𝐽𝑥𝑛.(3.16) From (3.14) and limsup𝑛𝛼𝑛<1, we get lim𝑛𝐽𝑥𝑛+1𝐽𝑇𝑥𝑛=0.(3.17) Since 𝐽1 is uniformly norm-to-norm continuous on bounded sets, we have lim𝑛𝑥𝑛+1𝑇𝑥𝑛=0.(3.18) Since 𝑥𝑛𝑇𝑥𝑛=𝑥𝑛𝑥𝑛+1+𝑥𝑛+1𝑇𝑥𝑛𝑥𝑛𝑥𝑛+1+𝑥𝑛+1𝑇𝑥𝑛,(3.19) we have lim𝑛𝑥𝑛𝑇𝑥𝑛=lim𝑘𝑥𝑛𝑘𝑇𝑥𝑛𝑘=0(3.20) and so ̂𝑥𝐹(𝑇)=𝐹(𝑇).(3.21) Now, we show ̂𝑥EP(𝑔). Since 𝑢𝑛𝑦𝑛𝑥𝑛+1𝑢𝑛+𝑥𝑛+1𝑦𝑛,(3.22) we get lim𝑛𝑢𝑛𝑦𝑛=0.(3.23) Since 𝐽 is a uniformly norm-to-norm continuous on bounded sets, we have lim𝑛𝐽𝑢𝑛𝐽𝑦𝑛=0.(3.24) From the assumption that 𝑟𝑛𝑎, we get lim𝑛𝐽𝑢𝑛𝐽𝑦𝑛𝑟𝑛=0.(3.25) It follows from 𝑢𝑛=𝑇𝑟𝑛𝑦𝑛 that 𝑔𝑢𝑛+1,𝑦𝑟𝑛𝐽𝑢𝑛𝐽𝑦𝑛,𝑦𝑢𝑛0,𝑦𝑀.(3.26) From (A2), we obtain 𝑦𝑢𝑛𝐽𝑢𝑛𝐽𝑦𝑛𝑟𝑛1𝑟𝑛𝐽𝑢𝑛𝐽𝑦𝑛,𝑦𝑢𝑛𝑢𝑔𝑛,𝑦𝑔𝑦,𝑢𝑛,𝑦𝑀.(3.27) Since lim𝑛𝑥𝑛+1𝑢𝑛=lim𝑛𝑥𝑛+1𝑥𝑛=0, we obtain lim𝑛𝑥𝑛𝑢𝑛=lim𝑘𝑥𝑛𝑘𝑢𝑛𝑘=0.(3.28) For any 𝑋, it follows that lim𝑘𝑢𝑛𝑘(̂𝑥)=lim𝑘𝑢𝑛𝑘𝑥𝑛𝑘𝑥+𝑛k̂𝑥=0(3.29) and so 𝑢𝑛𝑘̂𝑥. From (3.27) and (A4), we know that 𝑔(𝑦,̂𝑥)liminf𝑘𝑔𝑦,𝑢𝑛𝑘lim𝑘𝑦𝑢𝑛𝑘𝐽𝑢𝑛𝑘𝐽𝑦𝑛𝑘𝑟𝑛𝑘=0,𝑦𝑀.(3.30) Letting 𝑦𝑡=𝑡𝑦+(1𝑡)̂𝑥𝑀,0<𝑡<1,𝑦𝑀,(3.31) we have 𝑔𝑦𝑡,̂𝑥0.(3.32) It follows from (A1) that 𝑦0=𝑔𝑡,𝑦𝑡𝑦𝑡𝑔𝑡+𝑦,𝑦(1𝑡)𝑔𝑡𝑦,̂𝑥𝑡𝑔𝑡,𝑦(3.33) and thus 𝑔𝑦𝑡,𝑦0.(3.34) Taking the limit as 𝑡0 in (3.34) and from (A3), we have 𝑔(̂𝑥,𝑦)0,𝑦𝑀(3.35) and so ̂𝑥EP(𝑔).
(IV) Last, we prove that 𝑥𝑛Π𝑓𝐹𝑥.
Since 𝐹 is a closed convex set, from Lemma 2.5, we know that Π𝑓𝐹𝑥 is single-valued and denote that 𝑤=Π𝑓𝐹𝑥. Since 𝑥𝑛=Π𝑓𝐻𝑛𝑥 and 𝑤𝐹𝐻𝑛, we have 𝐺𝑥𝑛,𝐽𝑥𝐺(𝑤,𝐽𝑥),𝑛1.(3.36) For each given 𝑥, 𝐺(𝜉,𝐽𝑥) is convex and lower semicontinuous with respect to 𝜉, it is easy to see that 𝐺(𝜉,𝐽𝑥) is weakly lower semicontinuous with respect to 𝜉 and so 𝐺(̂𝑥,𝐽𝑥)liminf𝑘𝐺𝑥𝑛𝑘,𝐽𝑥limsup𝑘𝐺𝑥𝑛𝑘,𝐽𝑥𝐺(𝑤,𝐽𝑥).(3.37) From the definition of Π𝑓𝐹𝑥 and ̂𝑥𝐹, we know that ̂𝑥=𝑤 and so lim𝑘𝐺(𝑥𝑛𝑘,𝐽𝑥)=𝐺(̂𝑥,𝐽𝑥). It follows from Lemma 2.7 that lim𝑘𝑥𝑛𝑘=̂𝑥. The Kadec-Klee property of 𝑋 implies that {𝑥𝑛𝑘} converges strongly to Π𝑓𝐹𝑥. Since {𝑥𝑛𝑘} is an arbitrarily weakly convergent sequence of {𝑥𝑛}, we conclude that {𝑥𝑛} converges strongly to Π𝑓𝐹𝑥. This completes the proof.

Remark 3.2. Letting 𝑀=𝑋, EP(𝑔)=𝑋 and 𝑢𝑛=𝑦𝑛 in (3.1), then 𝐻𝑛+1={𝑧𝐻𝑛𝐺(𝑧,𝐽𝑦𝑛)𝐺(𝑧,𝐽𝑥𝑛)} and so Theorem 3.1 reduces to Theorem 4.1 of Li et al. [10].

Acknowledgments

This work was supported by the Key Program of NSFC (Grant no. 70831005) and the National Natural Science Foundation of China (11171237, 11101069). The work is also supported by HK CityU, CTTFS (9360142).

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