Open Access

An Extragradient Method for Mixed Equilibrium Problems and Fixed Point Problems

Fixed Point Theory and Applications20092009:632819

DOI: 10.1155/2009/632819

Received: 2 November 2008

Accepted: 23 May 2009

Published: 29 June 2009

Abstract

The purpose of this paper is to investigate the problem of approximating a common element of the set of fixed points of a demicontractive mapping and the set of solutions of a mixed equilibrium problem. First, we propose an extragradient method for solving the mixed equilibrium problems and the fixed point problems. Subsequently, we prove the strong convergence of the proposed algorithm under some mild assumptions.

1. Introduction

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq1_HTML.gif be a real Hilbert space and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq2_HTML.gif be a nonempty closed convex subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq3_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq4_HTML.gif be a real-valued function and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq5_HTML.gif be an equilibrium bifunction, that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq6_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq7_HTML.gif . We consider the following mixed equilibrium problem (MEP) which is to find https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq8_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ1_HTML.gif
(MEP)
In particular, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq9_HTML.gif , this problem reduces to the equilibrium problem (EP), which is to find https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq10_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ2_HTML.gif
(EP)

Denote the set of solutions of (MEP) by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq11_HTML.gif and the set of solutions of (EP) by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq12_HTML.gif . The mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and the equilibrium problems as special cases; see, for example, [15]. Some methods have been proposed to solve the equilibrium problems, see, for example, [521].

In 1997, Flåm and Antipin [15] introduced an iterative algorithm of finding the best approximation to the initial data when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq13_HTML.gif and proved a strong convergence theorem. Recently by using the viscosity approximation method S. Takahashi and W. Takahashi [8] introduced another iterative algorithm for finding a common element of the set of solutions of (EP) and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq14_HTML.gif be a nonexpansive mapping and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq15_HTML.gif be a contraction. Starting with arbitrary initial https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq16_HTML.gif , define the sequences https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq17_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq18_HTML.gif recursively by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ3_HTML.gif
(TT)
  1. S.

    Takahashi and W. Takahashi proved that the sequences https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq19_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq20_HTML.gif defined by (TT) converge strongly to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq21_HTML.gif with the following restrictions on algorithm parameters https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq22_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq23_HTML.gif :

     

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq24_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq25_HTML.gif ;

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq26_HTML.gif ;

(iii)(A1): https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq27_HTML.gif ; and (R1): https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq28_HTML.gif .

Subsequently, some iterative algorithms for equilibrium problems and fixed point problems have further developed by some authors. In particular, Zeng and Yao [16] introduced a new hybrid iterative algorithm for mixed equilibrium problems and fixed point problems and Mainge and Moudafi [22] introduced an iterative algorithm for equilibrium problems and fixed point problems.

On the other hand, for solving the equilibrium problem (EP), Moudafi [23] presented a new iterative algorithm and proved a weak convergence theorem. Ceng et al. [24] introduced another iterative algorithm for finding an element of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq29_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq30_HTML.gif be a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq31_HTML.gif -strict pseudocontraction for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq32_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq33_HTML.gif . For given https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq34_HTML.gif , let the sequences https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq35_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq36_HTML.gif be generated iteratively by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ4_HTML.gif
(CAY)

where the parameters https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq37_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq38_HTML.gif satisfy the following conditions:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq39_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq40_HTML.gif ;

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq41_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq42_HTML.gif .

Then, the sequences https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq43_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq44_HTML.gif generated by (CAY) converge weakly to an element of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq45_HTML.gif .

At this point, we should point out that all of the above results are interesting and valuable. At the same time, these results also bring us the following conjectures.

Questions

(1)Could we weaken or remove the control condition (iii) on algorithm parameters in S. Takahashi and W. Takahashi [8]?

(2)Could we construct an iterative algorithm for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq46_HTML.gif -strict pseudocontractions such that the strong convergence of the presented algorithm is guaranteed?

(3)Could we give some proof methods which are different from those in [8, 12, 16, 24].

It is our purpose in this paper that we introduce a general iterative algorithm for approximating a common element of the set of fixed points of a demicontractive mapping and the set of solutions of a mixed equilibrium problem. Subsequently, we prove the strong convergence of the proposed algorithm under some mild assumptions. Our results give positive answers to the above questions.

2. Preliminaries

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq47_HTML.gif be a real Hilbert space with inner product https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq48_HTML.gif and norm https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq49_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq50_HTML.gif be a nonempty closed convex subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq51_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq52_HTML.gif be a mapping. We use https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq53_HTML.gif to denote the set of the fixed points of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq54_HTML.gif . Recall what follows.

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq55_HTML.gif is called demicontractive if there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq56_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ5_HTML.gif
(2.1)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq57_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq58_HTML.gif , which is equivalent to
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ6_HTML.gif
(2.2)

For such case, we also say that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq59_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq60_HTML.gif -demicontractive mapping.

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq61_HTML.gif is called nonexpansive if
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ7_HTML.gif
(2.3)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq62_HTML.gif .

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq63_HTML.gif is called quasi-nonexpansive if
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ8_HTML.gif
(2.4)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq64_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq65_HTML.gif .

(iv) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq66_HTML.gif is called strictly pseudocontractive if there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq67_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ9_HTML.gif
(2.5)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq68_HTML.gif .

It is worth noting that the class of demicontractive mappings includes the class of the nonexpansive mappings, the quasi-nonexpansive mappings and the strictly pseudo-contractive mappings as special cases.

Let us also recall that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq69_HTML.gif is called demiclosed if for any sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq70_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq71_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ10_HTML.gif
(2.6)

It is well-known that the nonexpansive mappings, strictly pseudo-contractive mappings are all demiclosed. See, for example, [2527].

An operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq72_HTML.gif is said to be https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq73_HTML.gif -strongly monotone if there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq74_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ11_HTML.gif
(2.7)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq75_HTML.gif .

Now we concern the following problem: find https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq76_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ12_HTML.gif
(2.8)

In this paper, for solving problem (2.8) with an equilibrium bifunction https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq77_HTML.gif , we assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq78_HTML.gif satisfies the following conditions:

(H1) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq79_HTML.gif is monotone, that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq80_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq81_HTML.gif ;

(H2) for each fixed https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq82_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq83_HTML.gif is concave and upper semicontinuous;

(H3) for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq84_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq85_HTML.gif is convex.

A mapping https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq86_HTML.gif is called Lipschitz continuous, if there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq87_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ13_HTML.gif
(2.9)

A differentiable function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq88_HTML.gif on a convex set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq89_HTML.gif is called

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq90_HTML.gif -convex if
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ14_HTML.gif
(2.10)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq91_HTML.gif is the Frechet derivative of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq92_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq93_HTML.gif ;

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq94_HTML.gif -strongly convex if there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq95_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ15_HTML.gif
(2.11)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq96_HTML.gif be a nonempty closed convex subset of a real Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq97_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq98_HTML.gif be real-valued function and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq99_HTML.gif be an equilibrium bifunction. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq100_HTML.gif be a positive number. For a given point https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq101_HTML.gif , the auxiliary problem for (MEP) consists of finding https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq102_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ16_HTML.gif
(2.12)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq103_HTML.gif be the mapping such that for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq104_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq105_HTML.gif is the solution set of the auxiliary problem, that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq106_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ17_HTML.gif
(2.13)

We need the following important and interesting result for proving our main results.

Lemma 2.1 ([16, 28]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq107_HTML.gif be a nonempty closed convex subset of a real Hilbert space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq108_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq109_HTML.gif be a lower semicontinuous and convex functional. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq110_HTML.gif be an equilibrium bifunction satisfying conditions (H1)–(H3). Assume what follows.

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq111_HTML.gif is Lipschitz continuous with constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq112_HTML.gif such that

(a) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq113_HTML.gif ,

(b) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq114_HTML.gif is affine in the first variable,

(c)for each fixed https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq115_HTML.gif is sequentially continuous from the weak topology to the weak topology.

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq116_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq117_HTML.gif -strongly convex with constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq118_HTML.gif and its derivative https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq119_HTML.gif is sequentially continuous from the weak topology to the strong topology.

(iii)For each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq120_HTML.gif , there exist a bounded subset https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq121_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq122_HTML.gif such that for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq123_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ18_HTML.gif
(2.14)

Then there hold the following:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq124_HTML.gif is single-valued;

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq125_HTML.gif is nonexpansive if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq126_HTML.gif is Lipschitz continuous with constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq127_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq128_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ19_HTML.gif
(2.15)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq129_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq130_HTML.gif ;

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq131_HTML.gif ;

(iv) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq132_HTML.gif is closed and convex.

3. Main Results

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq133_HTML.gif be a real Hilbert space, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq134_HTML.gif be a lower semicontinuous and convex real-valued function, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq135_HTML.gif be an equilibrium bifunction. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq136_HTML.gif be a mapping and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq137_HTML.gif be a mapping. In this section, we first introduce the following new iterative algorithm.

Algorithm 3.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq138_HTML.gif be a positive parameter. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq139_HTML.gif be a sequence in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq140_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq141_HTML.gif be a sequence in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq142_HTML.gif . Define the sequences https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq143_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq144_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq145_HTML.gif by the following manner:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ20_HTML.gif
(3.1)

Now we give a strong convergence result concerning Algorithm 3.1 as follows.

Theorem 3.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq146_HTML.gif be a real Hilbert space. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq147_HTML.gif be a lower semicontinuous and convex functional. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq148_HTML.gif be an equilibrium bifunction satisfying conditions (H1)–(H3). Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq149_HTML.gif be an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq150_HTML.gif -Lipschitz continuous and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq151_HTML.gif -strongly monotone mapping and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq152_HTML.gif be a demiclosed and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq153_HTML.gif -demicontractive mapping such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq154_HTML.gif . Assume what follows.

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq155_HTML.gif is Lipschitz continuous with constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq156_HTML.gif such that

(a) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq157_HTML.gif ,

(b) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq158_HTML.gif is affine in the first variable,

(c)for each fixed https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq159_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq160_HTML.gif is sequentially continuous from the weak topology to the weak topology.

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq161_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq162_HTML.gif -strongly convex with constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq163_HTML.gif and its derivative https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq164_HTML.gif is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq165_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq166_HTML.gif .

(iii)For each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq167_HTML.gif ; there exist a bounded subset https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq168_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq169_HTML.gif such that, for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq170_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ21_HTML.gif
(3.2)

(iv) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq171_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq172_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq173_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq174_HTML.gif .

Then the sequences https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq175_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq176_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq177_HTML.gif generated by (3.1) converge strongly to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq178_HTML.gif which solves the problem (2.8) provided https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq179_HTML.gif is firmly nonexpansive.

Proof.

First, we prove that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq180_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq181_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq182_HTML.gif are all bounded. Without loss of generality, we may assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq183_HTML.gif . Given https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq184_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq185_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ22_HTML.gif
(3.3)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ23_HTML.gif
(3.4)
Take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq186_HTML.gif . From (3.1), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ24_HTML.gif
(3.5)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ25_HTML.gif
(3.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq187_HTML.gif .

Note that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq188_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq189_HTML.gif are firmly nonexpansive. Hence, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ26_HTML.gif
(3.7)
which implies that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ27_HTML.gif
(3.8)
From (2.2) and (3.1), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ28_HTML.gif
(3.9)
From (3.6)–(3.9), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ29_HTML.gif
(3.10)

This implies that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq190_HTML.gif is bounded, so are https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq191_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq192_HTML.gif .

From (3.1), we can write https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq193_HTML.gif . Thus, from (3.9), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ30_HTML.gif
(3.11)
Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq194_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq195_HTML.gif . Therefore, from (3.8) and (3.11), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ31_HTML.gif
(3.12)
We note that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq196_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq197_HTML.gif are bounded. So there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq198_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ32_HTML.gif
(3.13)
Consequently, we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ33_HTML.gif
(3.14)

Now we divide two cases to prove that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq199_HTML.gif converges strongly to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq200_HTML.gif .

Case 1.

Assume that the sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq201_HTML.gif is a monotone sequence. Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq202_HTML.gif is convergent. Setting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq203_HTML.gif .

(i)If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq204_HTML.gif , then the desired conclusion is obtained.

(ii)Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq205_HTML.gif . Clearly, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ34_HTML.gif
(3.15)
this together with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq206_HTML.gif and (3.14) implies that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ35_HTML.gif
(3.16)
that is to say
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ36_HTML.gif
(3.17)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq207_HTML.gif be a weak limit point of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq208_HTML.gif . Then there exists a subsequence of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq209_HTML.gif , still denoted by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq210_HTML.gif which weakly converges to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq211_HTML.gif . Noting that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq212_HTML.gif , we also have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ37_HTML.gif
(3.18)
Combining (3.1) and (3.17), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ38_HTML.gif
(3.19)

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq213_HTML.gif is demiclosed, then we obtain https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq214_HTML.gif .

Next we show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq215_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq216_HTML.gif , we derive
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ39_HTML.gif
(3.20)
From the monotonicity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq217_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ40_HTML.gif
(3.21)
and hence
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ41_HTML.gif
(3.22)
Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq218_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq219_HTML.gif weakly, from the weak lower semicontinuity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq220_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq221_HTML.gif in the second variable https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq222_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ42_HTML.gif
(3.23)
for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq223_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq224_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq225_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq226_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq227_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq228_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq229_HTML.gif and hence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq230_HTML.gif . From the convexity of equilibrium bifunction https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq231_HTML.gif in the second variable https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq232_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ43_HTML.gif
(3.24)
and hence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq233_HTML.gif . Then, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ44_HTML.gif
(3.25)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq234_HTML.gif and hence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq235_HTML.gif .

Therefore, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ45_HTML.gif
(3.26)
Thus, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq236_HTML.gif is a solution of problem (2.8), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ46_HTML.gif
(3.27)
Suppose that there exists another subsequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq237_HTML.gif which weakly converges to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq238_HTML.gif . It is easily checked that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq239_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ47_HTML.gif
(3.28)
Therefor, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ48_HTML.gif
(3.29)
Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq240_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq241_HTML.gif -strongly monotone, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ49_HTML.gif
(3.30)
By (3.17)–(3.30), we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ50_HTML.gif
(3.31)
From (3.12), for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq242_HTML.gif , we deduce that there exists a positive integer number https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq243_HTML.gif large enough, when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq244_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ51_HTML.gif
(3.32)
This implies that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ52_HTML.gif
(3.33)

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq245_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq246_HTML.gif is bounded, hence the last inequality is a contraction. Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq247_HTML.gif , that is to say, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq248_HTML.gif .

Case 2.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq249_HTML.gif is not a monotone sequence. Set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq250_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq251_HTML.gif be a mapping for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq252_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ53_HTML.gif
(3.34)
Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq253_HTML.gif is a nondecreasing sequence such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq254_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq255_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq256_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq257_HTML.gif . From (3.14), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ54_HTML.gif
(3.35)
thus
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ55_HTML.gif
(3.36)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ56_HTML.gif
(3.37)
Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq258_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq259_HTML.gif , from (3.12), we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ57_HTML.gif
(3.38)
which implies that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ58_HTML.gif
(3.39)
Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq260_HTML.gif is bounded, there exists a subsequence of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq261_HTML.gif , still denoted by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq262_HTML.gif which converges weakly to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq263_HTML.gif . It is easily checked that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq264_HTML.gif . Furthermore, we observe that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ59_HTML.gif
(3.40)
Hence, for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq265_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ60_HTML.gif
(3.41)
Therefore
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ61_HTML.gif
(3.42)
which implies that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ62_HTML.gif
(3.43)
Thus,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ63_HTML.gif
(3.44)
It is immediate that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ64_HTML.gif
(3.45)
Furthermore, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq266_HTML.gif , it is easily observed that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq267_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq268_HTML.gif (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq269_HTML.gif ), because https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq270_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq271_HTML.gif . As a consequence, we obtain for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq272_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ65_HTML.gif
(3.46)

Hence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq273_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq274_HTML.gif converges strongly to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq275_HTML.gif . Consequently, it easy to prove that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq276_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq277_HTML.gif converge strongly to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq278_HTML.gif . This completes the proof.

Remark 3.3.

The advantages of these results in this paper are that less restrictions on the parameters https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq279_HTML.gif are imposed.

As direct consequence of Theorem 3.2, we obtain the following.

Corollary 3.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq280_HTML.gif be a real Hilbert space. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq281_HTML.gif be a lower semicontinuous and convex functional. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq282_HTML.gif be an equilibrium bifunction satisfying conditions (H1)–(H3). Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq283_HTML.gif be an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq284_HTML.gif -Lipschitz continuous and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq285_HTML.gif -strongly monotone mapping and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq286_HTML.gif be a nonexpansive mapping such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq287_HTML.gif . Assume what follows.

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq288_HTML.gif is Lipschitz continuous with constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq289_HTML.gif such that;

(a) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq290_HTML.gif ,

(b) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq291_HTML.gif is affine in the first variable,

(c)for each fixed https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq292_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq293_HTML.gif is sequentially continuous from the weak topology to the weak topology.

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq294_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq295_HTML.gif -strongly convex with constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq296_HTML.gif and its derivative https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq297_HTML.gif is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq298_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq299_HTML.gif .

(iii)For each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq300_HTML.gif ; there exist a bounded subset https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq301_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq302_HTML.gif such that, for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq303_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ66_HTML.gif
(3.47)

(iv) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq304_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq305_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq306_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq307_HTML.gif .

Then the sequences https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq308_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq309_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq310_HTML.gif generated by (3.1) converge strongly to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq311_HTML.gif which solves the problem (2.8) provided https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq312_HTML.gif is firmly nonexpansive.

Corollary 3.5.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq313_HTML.gif be a real Hilbert space. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq314_HTML.gif be a lower semicontinuous and convex functional. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq315_HTML.gif be an equilibrium bifunction satisfying conditions (H1)–(H3). Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq316_HTML.gif be an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq317_HTML.gif -Lipschitz continuous and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq318_HTML.gif -strongly monotone mapping and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq319_HTML.gif be a strictly pseudo-contractive mapping such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq320_HTML.gif . Assume what follows.

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq321_HTML.gif is Lipschitz continuous with constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq322_HTML.gif such that

(a) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq323_HTML.gif ,

(b) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq324_HTML.gif is affine in the first variable,

(c)for each fixed https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq325_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq326_HTML.gif is sequentially continuous from the weak topology to the weak topology.

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq327_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq328_HTML.gif -strongly convex with constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq329_HTML.gif and its derivative https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq330_HTML.gif is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq331_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq332_HTML.gif .

(iii)For each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq333_HTML.gif ; there exist a bounded subset https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq334_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq335_HTML.gif such that, for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq336_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_Equ67_HTML.gif
(3.48)

(iv) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq337_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq338_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq339_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq340_HTML.gif .

Then the sequences https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq341_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq342_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq343_HTML.gif generated by (3.1) converge strongly to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq344_HTML.gif which solves the problem (2.8) provided https://static-content.springer.com/image/art%3A10.1155%2F2009%2F632819/MediaObjects/13663_2008_Article_1165_IEq345_HTML.gif is firmly nonexpansive.

Declarations

Acknowledgment

The authors are extremely grateful to the anonymous referee for his/her useful comments and suggestions. The first author was partially supposed by National Natural Science Foundation of China Grant 10771050. The second author was partially supposed by the Grant NSC 97-2221-E-230-017.

Authors’ Affiliations

(1)
Department of Mathematics, Tianjin Polytechnic University
(2)
Department of Information Management, Cheng Shiu University
(3)
Department of Applied Mathematics, Chung Yuan Christian University

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© Yonghong Yao et al. 2009

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