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Volume 2011, Article ID 392741, 24 pagesdoi:10.1155/2011/392741 Research Article Strong Convergence of a New Iterative Method for Infinite Family of Generalized Equilibrium and Fixed-Poi

Volume 2011, Article ID 392741, 24 pages

doi:10.1155/2011/392741

Research Article

Strong Convergence of a New Iterative Method

for Infinite Family of Generalized Equilibrium and Fixed-Point Problems of Nonexpansive Mappings

in Hilbert Spaces

Shenghua Wang1, 2 and Baohua Guo1, 2

1 National Engineering Laboratory for Biomass Power Generation Equipment,

North China Electric Power University, Baoding 071003, China

2 Department of Mathematics and Physics, North China Electric Power University, Baoding 071003, China

Correspondence should be addressed to Shenghua Wang,sheng-huawang@hotmail.com

Received 15 October 2010; Accepted 18 November 2010

Academic Editor: Qamrul Hasan Ansari

Copyrightq 2011 S Wang and B Guo This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited

We introduce an iterative algorithm for finding a common element of the set of solutions of aninfinite family of equilibrium problems and the set of fixed points of a finite family of nonexpansivemappings in a Hilbert space We prove some strong convergence theorems for the proposediterative scheme to a fixed point of the family of nonexpansive mappings, which is the uniquesolution of a variational inequality As an application, we use the result of this paper to solve amultiobjective optimization problem Our result extends and improves the ones of Colao et al

2008 and some others

1 Introduction

Let H be a real Hilbert space and T be a mapping of H into itself T is said to be nonexpansive

if

If there exists a point u ∈ H such that Tu  u, then the point u is called a fixed point of T The set of fixed points of T is denoted by FT It is well known that FT is closed convex and also nonempty if T has a bounded trajectorysee 1

Let f : H → H be a mapping If there exists a constant 0 ≤ κ < 1 such that

then f is called a contraction with the constant κ Recall that an operator A : H → H is called

to be strongly positive with coefficient γ > 0 if

Recently, Marino and Xu3 introduced a new iterative scheme from an arbitrary point

x0∈ H by the viscosity approximation method as follows:

Let{Tn} N

n1be a finite family of nonexpansive mappings of H into itself In 2007, Yao

4 defined the mappings

Then he proved that the iterative scheme1.10 strongly converges to the unique solution x∗

of the variational inequality:

Let A : C → H be a nonlinear mapping Let EPG, A denote the set of all solutions to the

following equilibrium problem:

EP cG, A x ∈ C : Gx, y

In the case of A ≡ 0, EPG, A is deduced to EP In the case of G ≡ 0, EPG, A is also denoted

by VIC, A

In 2007, S Takahashi and W Takahashi 6 introduced a viscosity approximation

method for finding a common element of EP G and FT from an arbitrary initial element

and proved that the iterative schemes{xn} and {un} converge strongly to the unique solution

z of the variational inequality:

where h is a potential function for γf.

Very recently, for finding a common element of the set of a finite family ofnonexpansive mappings and the set of solutions of an equilibrium problem, by combiningthe schemes1.11 and 1.17, Colao et al 5 proposed the following explicit scheme:

The equilibrium problems have been considered by many authors; see, for example,

6,8 19 and the reference therein But, in these references, the authors only considered atmost finite family of equilibrium problems and few of authors investigate the infinite family

of equilibrium problems in a Hilbert space or Banach space In this paper, we consider a newiterative scheme for obtaining a common element in the solution set of an infinite family

of generalized equilibrium problems and in the common fixed-point set of a finite family

of nonexpansive mappings in a Hilbert space Let {Tn} N

n1 N ≥ 1 be a finite family of nonexpansive mappings of H into itself, be {Gn} : C × C → R be an infinite family of

bifunctions, and be{An} : C → H be an infinite family of kn-inverse-strongly monotonemappings Let{rn} be a sequence such that rn ⊂ r, 2kn with r > 0 for each n ≥ 1 Define the mapping Tri : H → C by

i1F Ti∞i1EP Gi , A i / ∅ For an arbitrary initial point x1 ∈ H, we

define the iterative scheme{xn} by

2 Preliminaries

Let C be a closed convex subset of a Hilbert space H For any point x ∈ H, there exists a unique nearest point in C, denoted by P C x, such that

Then PC is called the metric projection of H onto C It is well known that PCis a nonexpansive

mapping of H onto C and satisfies the following:

for all x, y ∈ C However, A is called an α-inverse-strongly monotone mapping if there exists

a positive real number α such that

Hence, if λ ∈ 0, 2α, then I − λA is a nonexpansive mapping of C into H.

If there exists u ∈ C such that

for all v ∈ C, then u is called the solution of this variational inequality The set of all solutions

of the variational inequality is denoted by VIC, A

In this paper, we need the following lemmas

Lemma 2.1 see 21 Given x ∈ H and y ∈ C Then PC x  y if and only if there holds the

Therefore, the following lemma naturally holds

Lemma 2.3 Let H be a real Hilbert space The following identity holds:

x  y2

Lemma 2.4 see 3 Assume that A is a strongly positive linear bounded operator on a Hilbert

Lemma 2.5 see 2 Assume that {an} is a sequence of nonnegative numbers such that

Lemma 2.6 see 23 Let C be a nonempty closed convex subset of a Hilbert space H and let G :

C × C → R be a bifunction which satisfies the following:

A1 Gx, x  0 for all x ∈ C;

A2 G is monotone, that is, Gx, y  Gy, x ≤ 0 for all x, y ∈ C;

A3 For each x, y, z ∈ C,

lim

t↓0 G

tz  1 − tx, y≤ Gx, y

A4 For each x ∈ C, y → Gx, y is convex and lower semicontinuous.

4 EPG is closed and convex.

It is easy to see that if there exists some point v ∈ C such that v  TrI − rAv, where

v  Tr I − rAv, one has

Hence, v ∈ EPG, A.

Let C be a nonempty convex subset of a Banach space Let {Ti} N

i1be a finite family of

nonexpansive mappings of C into itself and λ1, λ2, , λ Nbe real numbers such that 0≤ λi≤ 1

for each i  1, 2, , N Define a mapping W of C into itself as follows:

Lemma 2.7 see 26 Let C be a nonempty closed convex subset of a Banach space Let

i1F Ti / ∅ and let

λ1, λ2, , λ N be real numbers such that 0 < λ i < 1 for each i  1, 2, , N − 1 and 0 < λN ≤ 1 Let

W be the W-mapping of C generated by T1, T2, , T N and λ1, λ2, , λ N Then F W N

W-mappings generated by T1, T2, , T N and λ1, λ2, , λ N and T1, T2, , T N and λ n,1 , λ n,2 , , λ n,N ,

lim

3 Main Results

Now, we give our main results in this paper

Theorem 3.1 Let H be a Hilbert space and C be a nonempty closed convex subset of H Let f : H →

H be a contraction with coefficient 0 < κ < 1, A, B : H → H be strongly positive linear bounded

n1 : H →

H N ≥ 1 be a finite family of nonexpansive mappings, {Gn} : C × C → R be an infinite family

i1F Ti ∩ ∞

i1EP Gi , A i / ∅ Let {n} and {δn} be two sequences in 0, 1, {λn,i} N

i1be asequence in a, b with 0 < a ≤ b < 1, {rn}

Take a fixed number γ > 0 with 0 < γ − γκ < 1 Assume that

E1 limn→ ∞ n 0 and ∞n1 n ∞;

E2 limn→ ∞|λn 1,i − λn,i|  0 for each i  1, 2, , N;

E3 0 ≤ δn  n ≤ 1 for all n ≥ 1;

Then the sequence {xn} defined by 1.23 converges strongly to x∗∈ Ω, which is the unique solution

of the variational inequality:1.24, that is,

generality, that  n < 1 − δn B  A −1for all n ≥ 1 Noting that A and B are both the linear

bounded self-adjoint operators, one has

Moreover, it follows from3.7, f1/ B  > 1 and E4 that

δ n < 1

Next, we proceed the proof with following steps

Let p ∈ Ω Lemma 2.6 shows that every T ri is firmly nonexpansive and hence

nonexpansive Since r < r i < 2k i , I −ri A i is nonexpansive for each i ≥ 1 Therefore, TriI −ri A i

is nonexpansive for each i ≥ 1 Noting that {αn} is strictly decreasing, α0 1, we have

It follows from n ∈ 0, 1 and 0 < γ − γκ < 1 that 0 < nγ − γκ < 1 Therefore, by the simple

It follows from the definition of Wnthat

Un 1,N−i z n − Un,N −i z n

 λn 1,N−i T N −i U n 1,N−i−1 z n  1 − λn 1,N−i zn − λn,N −i T N −i U n,N −i−1 z n − 1 − λn,N −i zn

≤ λn 1,N−i TN −i U n 1,N−i−1 z n − TN −i U n,N −i−1 z n

 |λn 1,N−i − λn,N −i | TN −i U n,N −i−1 z n  |λn 1,N−i − λn,N −i | zn

≤ Un 1,N−i−1 z n − Un,N −i−1 z n   zn  TN −i U n,N −i−1 z n |λn 1,N−i − λn,N −i|

≤ Un 1,N−i−1 z n − Un,N −i−1 z n  M1|λn 1,N−i − λn,N −i|

Hence, by3.4 and 3.18, we get

where M2 supn {γ fxn−1  Bxn−1  BWn−1z n−1  AWn−1z n−1 }

Set M3  min{β − B , γ − γκ} It follows from 0 ≤ γ − γκ < 1 and β > B due to

xn1− xn ≤ 1 − δn  nM3 xn − xn−1  δn  nM3

×

1

Then it follows from3.21 that

Notice that, for any x∈ Ω,

and hence, for each i≥ 1,

 Tri I − ri A ixn − Tri I − ri A ix 2

≤ Tri I − ri A ixn − Tri I − ri A ix, I − ri A ixn − I − ri A ix

 Tri I − ri A ixn − x, xn − x  riTri I − ri A ixn − x, Ai x − Ai x n

By using3.8, 3.9, 3.35, Lemmas2.3and2.4, we havenoting that δn < 1/β

This shows that for, each i≥ 1,

Since{αn} is strictly decreasing, δn → 0, n → 0, Ai x n − Ai x → 0 and xn − xn1 → 0, we

have, for each i≥ 1,

To prove this, we pick a subsequence{xnj } of {xn} such that

Without loss of generality, we may further assume that xnj )x Obviously, to proveStep 5,

we only need to prove that )x ∈ Ω.

Indeed, for each i ≥ 1, since xn − Tri I − ri A ixn → 0, xnj → )x and Tri I − ri A i is

nonexpansive, by demiclosed principle of nonexpansive mapping we have

)x ∈ FTri I − ri A i  EPGi , A i, i ≥ 1. 3.43

Assume that λ nm,k → λk ∈ 0, 1 for each k  1, 2, , N Let W be the W-mapping generated by T1, , T N and λ1, , λ N Then, byLemma 2.8, we have

Moreover, it follows fromLemma 2.7thatN

n1F Ti  FW Assume that )x /∈ FW.

Then )x / W )x Since )x ∈ FTri I − ri A i for each i ≥ 1, byStep 3,3.44 and Opial’s property

of the Hilbert space H, we have

By using Lemmas2.3and2.4, we have

which implies that

ByTheorem 3.1, we have the following direct corollaries

Corollary 3.2 Let H be a Hilbert space and C be a nonempty closed convex subset of H Let f :

H → H be a contraction with coefficient 0 < κ < 1, A : H → H be strongly positive linear

n1: H → H N ≥ 1 be a finite family of

i1F Ti ∩ EPG, A / ∅ Let {εn} and {δn} be two sequences in 0, 1, {λn,i} N

i1be a sequence in a, b with 0 < a ≤ b < 1, r be a number in

0, 2α, and {αn} be a sequence 0, 1 Take a fixed number γ > 0 with 0 < γ − γκ < 1 Assume that

E1 limn→ ∞ n  0 and ∞n1ε n  ∞;

E2 limn→ ∞|λn 1,i − λn,i|  0 for each i  1, 2, , N;

E3 0 ≤ δn  n ≤ 1 for all n ≥ 1;

E4 {δn} ⊂ 0, min{c, 1/β, 2 B  B 2−ββ − B 2− 2 B 2 8β B /4β B } with

c < 1;

E5 ∞n1|n1− n| < ∞, ∞n1|αn1− αn| < ∞, ∞n1|δn1− δn| < ∞.

of the variational inequality:

x∗ PΩ

which was used by many others for obtaining xn1− xn → 0 as n → ∞ see 4,5,28 Theproof method of)x ∈∞

i1EP Gi , A i is simple and different with ones of others.

4 Applications for Multiobjective Optimization Problem

In this section, we study a kind of multiobjective optimization problem by using the result

of this paper That is, we will give an iterative algorithm of solution for the followingmultiobjective optimization problem with the nonempty set of solutions:

min h1x,

where h1x and h2x are both the convex and lower semicontinuous functions defined on a closed convex subset of C of a Hilbert space H.

We denote by A the set of solutions of the problem4.1 and assume that A / ∅ Also,

we denote the sets of solutions of the following two optimization problems by A1 and A2,respectively,

Now, let G1and G2be two bifunctions from C × C to R defined by

respectively It is easy to see that EP G1  A1 and EP G2  A2, where EP Gi denotes the

set of solutions of the equilibrium problem:

G i



x, y

respectively In addition, it is easy to see that G1and G2satisfy the conditionsA1–A4 Let

{αn} be a sequence in 0,1 and r1, r2∈ 0, 1 Define a sequence {xn} by

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