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Hindawi Publishing Corporation

Fixed Point Theory and Applications

Volume 2008, Article ID 714939, 15 pages

doi:10.1155/2008/714939

Research Article

Viscosity Approximation Methods for Generalized

Mixed Equilibrium Problems and Fixed Points of

a Sequence of Nonexpansive Mappings

Wei-You Zeng,1Nan-Jing Huang,1and Chang-Wen Zhao2

1Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

2College of Business and Management, Sichuan University, Chengdu, Sichuan 610064, China

Correspondence should be addressed to Nan-Jing Huang, nanjinghuang@hotmail.com

Received 17 July 2008; Accepted 11 November 2008

Recommended by Wataru Takahashi

We introduce an iterative scheme by the viscosity approximation method for finding a common

element of the set of common solutions for generalized mixed equilibrium problems and the set

of common fixed points of a sequence of nonexpansive mappings in Hilbert spaces. We show a

strong convergence theorem under some suitable conditions.

Copyright q 2008 Wei-You Zeng 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.

1. Introduction

Equilibrium problems theory provides us with a unified, natural, innovative, and general

framework to study a wide class of problems arising in finance, economics, network analysis,

transportation, elasticity, and optimization, which has been extended and generalized in

many directions using novel and innovative techniques; see ?1–8?. Inspired and motivated

by the research and activities going in this fascinating area, we introduce and consider a

new class of equilibrium problems, which is known as the generalized mixed equilibrium

problems.

Let C be a nonempty closed convex subset of a real Hilbert space H and T : C → 2Ha

multivalued mapping. Let ϕ : C × C → R be a real-valued function and Φ : H × C × C → R

an equilibrium-like function, that is,

Φ?w,u,v? ? Φ?w,v,u? ? 0,

∀?w,u,v? ∈ H × C × C.

?1.1?

We consider the problem of finding u ∈ C and w ∈ T?u? such that

Φ?w,u,v? ? ϕ?v,u? − ϕ?u,u? ≥ 0,

∀v ∈ C,

?1.2?

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2 Fixed Point Theory and Applications

which is called the generalized mixed equilibrium problem ?for short, GMEP?. If T is a single-

valued mapping, then problem ?1.2? is equivalent to finding u ∈ C such that

Φ?T?u?,u,v?? ϕ?v,u? − ϕ?u,u? ≥ 0,

∀v ∈ C.

?1.3?

We denote Ω for the set of solutions of GMEP ?1.2?. This class is a quite general

and unifying one and includes several classes of equilibrium problems and variational

inequalities as special cases. In recent years, several numerical techniques including

projection, resolvent, and auxiliary principle have been developed and analyzed for solving

variational inequalities. It is well known that projection- and resolvent-type methods cannot

be extended for equilibrium problems. To overcome this drawback, one usually uses the

auxiliary principle technique. Glowinski et al. ?9? have used this technique to study the

existence of a solution of mixed variational inequalities. The viscosity approximation method

is one of the important methods for approximation fixed points of nonexpansive type

mappings. It was first discussed by Moudafi ?10?. Recently, Hirstoaga ?11? and S. Takahashi

and W. Takahashi ?12? applied viscosity approximation technique for finding a common

element of set of solutions of an equilibrium problem ?EP? and set of fixed points of a

nonexpansive mapping. Very recently, Yao et al. ?13? introduced and studied an iteration

process for finding a common element of the set of solutions of the EP and the set of common

fixed points of infinitely many nonexpansive mappings in H. Let {Tn}∞

nonexpansive mappings of C into itself and let {λn}∞

in ?0,1?. For any n ≥ 1, define a mapping Snof C into itself as follows:

n?1be a sequence of

n?1be a sequence of nonnegative numbers

Un,n?1? I,

Un,n? λnTnUn,n?1??1 − λn

Un,n−1? λn−1Tn−1Un,n??1 − λn−1

...

Un,k? λkTkUn,k?1??1 − λk

Un,k−1? λk−1Tk−1Un,k??1 − λk−1

...

Un,2? λ2T2Un,3??1 − λ2

Sn? Un,1? λ1T1Un,2??1 − λ1

?I,

?I,

?I,

?I,

?I,

?I.

?1.4?

Such a mapping Snis called the S-mapping generated by Tn,Tn−1,...,T1and λn,λn−1,...,λ1,

see ?14?.

The purpose of this paper is to develop an iterative algorithm for finding a common

element of set of solutions of GMEP ?1.2? and set of common fixed points of a sequence of

nonexpansive mappings in Hilbert spaces. The result presented in this paper improves and

extends the main result of S. Takahashi and W. Takahashi ?12?.

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Wei-You Zeng et al.3

2. Preliminaries

Let H be a real Hilbert space with inner product ?·,·? and norm ?·?, and let C be a closed

convex subset of H. Then, for any x ∈ H, there exists a unique nearest point in C, denoted by

PC?x?, such that

??x − PC?x???≤ ?x − y?,

∀y ∈ C.

?2.1?

PC is called metric projection of H onto C. It is well known that PC is nonexpansive.

Furthermore, for x ∈ H and u ∈ C,

u ? PC?x? ⇐⇒ ?x − u,u − y? ≥ 0,

∀y ∈ C.

?2.2?

We denote by F?T? the set of fixed points of a self-mapping T on C, that is, F?T? ? {x ∈

C : Tx ? x}. It is well known that if C ⊂ H is nonempty, bounded, closed, and convex and T

is nonexpansive, then F?T? is nonempty; see ?15?. Let {Tn}∞

mappings of C into itself, where C is a nonempty closed convex subset of a real Hilbert space

H. Given a sequence {λn}∞

?1.4?. Then we have the following lemmas which are important to prove our results.

n?1be a sequence of nonexpansive

n?1in ?0,1?, we define a sequence {Sn}∞

n?1of self-mappings on C by

Lemma 2.1 ?see ?14??. Let C be a nonempty closed convex subset of a real Hilbert space H. Let

{Tn}∞

{λn}∞

limn→∞Un,kx exists.

n?1be a sequence of nonexpansive mappings of C into itself such that?∞

n?1F?Tn?/ ?∅, and let

n?1be a sequence in ?0,b? for some b ∈ ?0,1?. Then, for every x ∈ C and k ∈ N the limit

Using Lemma 2.1, one can define mapping S of C into itself as follows:

Sx ? lim

n→∞Snx ? lim

n→∞Un,1x,

?2.3?

for every x ∈ C. Such a mapping S is called the S-mapping generated by T1,T2,... and

λ1,λ2,.... Throughout this paper, we will assume that 0 < λn ≤ b < 1 for every n ≥ 1.

Since Snis nonexpansive, S : C → C is also nonexpansive.

Lemma 2.2 ?see ?14??. Let C be a nonempty closed convex subset of a real Hilbert space H. Let

{Tn}∞

{λn}∞

Let C be a convex subset of a real Hilbert space H and κ : C → R a Fr´ echet differential

function. Then κ is said to be η-convex strongly convex if there exists a constant μ > 0 such

that

n?1be a sequence of nonexpansive mappings of C into itself such that?∞

n?1F?Tn?/ ?∅, and let

n?1be a sequence in ?0,b? for some b ∈ ?0,1?. Then, F?S? ??∞

n?1F?Tn?.

κ?y? − κ?x? −?κ??x?,η?y,x??≥μ

2?x − y?2,

∀x,y ∈ C.

?2.4?

If μ ? 0, then κ is said to be η-convex. In particular, if η?y,x? ? y − x for all y,x ∈ C, then κ is

said to be strongly convex.

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4 Fixed Point Theory and Applications

Let C be a nonempty subset of a real Hilbert space H. A bifunction ϕ?·,·? : C × C → R

is said to be skew-symmetric if

ϕ?u,v? ? ϕ?v,u? − ϕ?u,u? − ϕ?v,v? ≤ 0,

∀u,v ∈ C.

?2.5?

If the skew-symmetric bifunction ϕ?·,·? is linear in both arguments, then

ϕ?u,u? ≥ 0,

∀u ∈ C.

?2.6?

We denote ? for weak convergence and → for strong convergence. A function ψ : C×C → R

is called weakly sequentially continuous at ?x0,y0? ∈ C × C, if ψ?xn,yn? → ψ?x0,y0? as

n → ∞ for each sequence {?xn,yn?} in C × C converging weakly to ?x0,y0?. The function

ψ?·,·? is called weakly sequentially continuous on C×C if it is weakly sequentially continuous

at each point of C × C.

LetCB?X?denotethesetofnonemptyclosedboundedsubsetsofX.ForA,B ∈ CB?X?,

define the Hausdorff metric H as follows:

?

b∈B

H?A,B? ? maxsup

a∈A

inf

d?a,b?,sup

b∈B

inf

a∈A

d?b,a?

?

.

?2.7?

Lemma 2.3 ?see ?16??. Let A,B ∈ CB?X? and a ∈ A. Then for ? > 1, there must exist a point b ∈ B

such that d?a,b? ≤ ?H?A,B?.

Let C be a nonempty closed convex subset of a real Hilbert space H and T : C → 2H

a multivalued mapping. For x ∈ C, let w ∈ T?x?. Let ϕ : C ×C → R be a real-valued function

satisfying the following:

?ϕ1? ϕ?·,·? is skew symmetric;

?ϕ2? for each fixed y ∈ C, ϕ?·,y? is convex and upper semicontinuous;

?ϕ3? ϕ?·,·? is weakly continuous on C × C.

Let κ : C → R be a differentiable functional with Fr´ echet derivative κ??x? at x satisfying the

following:

?κ1? κ?is sequentially continuous from the weak topology to the strong topology;

?κ2? κ?is Lipschitz continuous with Lipschitz constant ν > 0.

Let η : C × C → H be a function satisfying the following:

?η1? η?x,y? ? η?y,x? ? 0 for all x,y ∈ C;

?η2? η?·,·? is affine in the first coordinate variable;

?η3? for each fixed y ∈ C, x ?→ η?y,x? is sequentially continuous from the weak topology

to the weak topology.

Let us consider the equilibrium-like function Φ : H×C×C → R which satisfies the following

conditions with respect to the multivalued mapping T : C → 2H:

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Wei-You Zeng et al.5

?Φ1? for each fixed v ∈ C, ?w,u? ?→ Φ?w,u,v? is an upper semicontinuous function

from H × C to R, that is, wn → w and un → u imply limsupn→∞Φ?wn,un,v? ≤

Φ?w,u,v?;

?Φ2? for each fixed ?w,v? ∈ H × C, u ?→ Φ?w,u,v? is a concave function;

?Φ3? for each fixed ?w,u? ∈ H × C, v ?→ Φ?w,u,v? is a convex function.

Let r be a positive parameter. For a given element x ∈ C and wx∈ T?x?, consider the

following auxiliary problem for GMEP?1.2?: find u ∈ C such that

Φ?wx,u,v? ? ϕ?v,u? − ϕ?u,u? ?1

r

?κ??u? − κ??x?,η?v,u??≥ 0,

∀v ∈ C.

?2.8?

It is easy to see that if u ? x, then u is a solution of GMEP?1.2?.

Lemma 2.4 ?see ?6??. Let C be a nonempty closed convex bounded subset of a real Hilbert space H

and ϕ : C×C → R a real-valued function satisfying the conditions ?ϕ1?–?ϕ3?. Let T : C → 2Hbe a

multivalued mapping and Φ : H ×C×C → R the equilibrium-like function satisfying the conditions

?Φ1?–?Φ3?. Assume that η : C×C → H is a Lipschitz function with Lipschitz constant λ > 0 which

satisfies the conditions ?η1?–?η3?. Let κ : C → R be an η-strongly convex function with constant

μ > 0 which satisfies the conditions ?κ1? and ?κ2?. For each x ∈ C, let wx∈ T?x?. For r > 0, define a

mapping Tr: C → C by

?

Tr?x? ?

u ∈ C : Φ?wx,u,v? ? ϕ?v,u? − ϕ?u,u? ?1

r

?κ??u? − κ??x?,η?v,u??≥ 0, ∀v ∈ C

?

?2.9?

.

Then one has the following:

?a? the auxiliary problem ?2.8? has a unique solution;

?b? Tris single valued;

?c? if λν/μ and Φ?w1,Tr?x1?,Tr?x2?? ? Φ?w2,Tr?x2?,Tr?x1?? ≤ 0 for all x1,x2∈ C and all

w1∈ T?x1?, w2∈ T?x2?, it follows that Tris nonexpansive;

?d? F?Tr? ? Ω;

?e? Ω is closed and convex.

We also need the following lemmas for our main results.

Lemma 2.5 ?see ?17??. Let {an}, {bn}, and {cn} be three sequences of nonnegative numbers such

that

an?1≤ bnan? cn,

∀n ? 1,2,....

?2.10?

If bn≥ 1,?∞

Lemma 2.6. Let {an} and {cn} be sequences of nonnegative numbers such that

n?1?bn− 1? < ∞, and?∞

n?1cn< ∞, then limn→∞anexists.

an?1≤ Θan? cn,

∀n ? 1,2,....

?2.11?

If Θ ∈ ?0,1? and?∞

n?1cn< ∞, then limn→∞an? 0.

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6 Fixed Point Theory and Applications

Proof. It is easy to see that inequality ?2.11? is equivalent to

an?1≤ Θbnan? cn,

∀n ? 1,2,...,

?2.12?

where Θ ∈ ?0,1?, bn? 1 and?∞

n?1cn< ∞. It follows that

an?1≤ bnan? cn

∀n ? 1,2,....

?2.13?

Note that Lemma 2.5 implies that limn→∞anexists. Suppose limn→∞an? d for some d > 0.

It is obvious that limn→∞cn ? 0 and so inequality ?2.12? implies that d ≤ Θd, which is a

contradiction. Thus, limn→∞an? d ? 0. This completes the proof.

Lemma 2.7 ?see ?6??. Let {xn} be a sequence in a normed space ?X,?·?? such that

??xn?1− xn?2

where Θ ∈ ?0,1?, and {bn} and {cn} are sequences satisfy the following conditions:

?i? bn≥ 1 for all n ? 1,2,... and?∞

??≤ Θ??xn− xn?1

??bn? cn,

∀n ? 1,2,...,

?2.14?

n?1?bn− 1? < ∞;

n?1cn< ∞.

?ii? cn≥ 0 for all n ? 1,2,... and?∞

Then {xn} is a Cauchy sequence.

Lemma 2.8 ?see ?18??. Let {an} be a sequence of nonnegative real numbers such that

an?1≤?1 − λn

where {λn}, {σn} and {ξn} are sequences of real numbers satisfying the following conditions:

?i? {λn} ⊂ ?0,1?, limn→∞λn? 0 and?∞

?ii? limsupn→∞σn≤ 0;

?iii? ξn≥ 0 for all n ? 1,2,... and?∞

Then, limn→∞an? 0.

?an? λnσn? ξn,

∀n ? 1,2,...,

?2.15?

n?1λn? ∞;

n?1ξn< ∞.

3. Iterative algorithm and convergence theorem

Let C be a nonempty closed convex subset of a real Hilbert space H, T : C → CB?H? a

multivalued mapping, f : C → C a contraction mapping with constant α ∈ ?0,1?, and

Sn : C → C an S-mapping generated by T1,T2,... and λ1,λ2,..., where sequence {Tn} is

nonexpansive. Let {αn} be a sequence in ?0,1? and {rn} a sequence in ?0,∞?. We can develop

Algorithm 3.1 for finding a common element of a set of fixed points of S-mapping Snand a

set of solutions of GMEP?1.2?.

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Wei-You Zeng et al.7

Algorithm 3.1. For given x1 ∈ C and w1 ∈ T?x1?, there exist sequences {xn}, {un} in C and

{wn: wn∈ T?xn?} in H such that for all n ? 1,2,...,

?

Φ?wn,un,v?? ϕ?v,un

xn?1? αnf?xn

We now prove the strong convergence of iterative sequence {xn}, {un}, and {wn}

generated by Algorithm 3.1.

??wn− wn?1

?− ϕ?un,un

??≤

1 ?1

n

?

H?T?xn

?κ??un

???1 − αn

?,T?xn?1

?,η?v,un

?Sn

??;

??1

rn

?− κ??xn

??≥ 0,

∀v ∈ C;

?un

?.

?3.1?

Theorem 3.2. Let C be a nonempty closed convex bounded subset of a real Hilbert space H, T :

C → CB?H? a multivalued H-Lipschitz continuous mapping with constant L > 0, f : C → C a

contraction mapping with constant α ∈ ?0,1?. Let ϕ : C×C → R be a real-valued function satisfying

the conditions ?ϕ1?–?ϕ3? and let Φ : H × C × C → R be an equilibrium-like function satisfying

conditions ?Φ1?–?Φ3? and ?Φ4?:

?Φ4? Φ?w,Tr?x?,Ts?y?? ? Φ? ? w,Ts?y?,Tr?x?? ≤ −γ?Tr?x? − Ts?y??2for all x,y ∈ C and

Assume that η : C × C → H is a Lipschitz function with Lipschitz constant λ > 0 which satisfies

the conditions ?η1?∼?η3?. Let κ : C → R be an η-strongly convex function with constant μ > 0

which satisfies conditions ?κ1? and ?κ2? with λν/μ ≤ 1. Let Sn: C → C be an S-mapping generated

by T1,T2,... and λ1,λ2,... and?∞

in ?0,∞? satisfying the following conditions:

?C1? limn→∞αn? 0,?∞

?C3??∞

strongly to w∗∈ T?x∗?, where x∗? P∩∞

Proof. It is easy to see from ?Φ4? that

r,s ∈ ?0,∞?, where γ > 0, w ∈ T?x? and ? w ∈ T?y?.

n?1F?Tn? ∩ Ω/ ?∅, where sequence {Tn} is nonexpansive. Let {xn},

{un} and {wn} be sequences generated by Algorithm 3.1, where {αn} is a sequence in ?0,1? and {rn}

n?1αn? ∞ and?∞

n?1|αn− αn?1| < ∞;

?C2? liminfn→∞rn> 0 and?∞

Then the sequences {xn} and {un} converge strongly to x∗∈?∞

n?1|rn− rn?1| < ∞;

n?1?1 − αn?εn< ∞ where εn? supx∈C?Sn?x? − Sn?1?x??.

n?1F?Tn? ∩ Ω, and {wn} converges

n?1F?Tn?∩Ωf?x∗?.

Φ?w,Tr?x?,Ts?y??? Φ?? w,Ts?y?,Tr?x??≤ 0?3.2?

for all x,y ∈ C and r,s ∈ ?0,∞?, where γ > 0, w ∈ T?x?, and ? w ∈ T?y?. All the conclusions

Let Q ? P∩∞

??Qf?x? − Qf?y???≤??f?x? − f?y???< α?x − y?,

Hence there exists a unique element q ∈ C such that q ? Qf?q?. Noting that f?q? ∈ C and

Qf?q? ∈?∞

?a?–?e? of Lemma 2.4 hold.

n?1F?Tn?∩Ω. Then Qf is a contraction of C into itself. In fact,

∀x,y ∈ C.

?3.3?

n?1F?Tn? ∩ Ω, we get that q ∈?∞

n?1F?Tn? ∩ Ω.

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8 Fixed Point Theory and Applications

Now, we prove that ?xn− xn?1? → 0 and ?un− un?1? → 0 as n → ∞. Observe that

??xn?1− xn

?????αnf?xn

???1 − αn

?− αnf?xn−1

?Sn

?Sn−1

?− f?xn−1

???Sn

?Sn

?un

?? αnf?xn−1

?−?1 − αn

?un−1

??????αn− αn−1

?− αn−1f?xn−1

?−?1 − αn−1

?Sn−1

?un−1

???

???αnf?xn

?− αn−1f?xn−1

?Sn−1

?−?1 − αn−1

?????f?xn−1

?

??1 − αn

??1 − αn

??f?xn

?un

?un−1

?Sn−1

?

?un−1

???

≤ αn

??????Sn−1

?un−1

????

??1 − αn

≤ ααn

??1 − αn

≤ ααn

?un

??? 2??αn− αn−1

?− Sn−1

?un−1

???

??xn− xn−1

??diam?C?

????Sn

?un

??? 2??αn− αn−1

?− Sn−1

?un

??????Sn−1

?un

?− Sn−1

?un−1

????

??xn− xn−1

??diam?C? ??1 − αn

????un− un−1

??? εn−1

?.

?3.4?

Noting that un? Trnxnand un?1? Trn?1xn?1, it follows from ?3.1? that

Φ?wn,un,v?? ϕ?v,un

Φ?wn?1,un?1,v??ϕ?v,un?1

?− ϕ?un,un

??1

1

rn?1

rn

?κ??un

?κ??un?1

?− κ??xn

?−κ??xn?1

?,η?v,un

?,η?v,un?1

??≥ 0,

??≥ 0

?3.5?

?−ϕ?un?1,un?1

??

∀v ∈ C.

?3.6?

Putting v ? un?1in ?3.5? and v ? unin ?3.6?, respectively, we have

Φ?wn,un,un?1

Φ?wn?1,un?1,un

?? ϕ?un?1,un

?? ϕ?un,un?1

?− ϕ?un,un

?− ϕ?un?1,un?1

??1

??

rn

?κ??un

1

rn?1

?− κ??xn

?− κ??xn?1

?,η?un?1,un

?,η?un,un?1

??≥ 0,

?κ??un?1

??≥ 0.

?3.7?

Adding up those inequalities, we obtain from ?2.5?, ?η1?, and ?Φ4? that

−1

rn

?κ??un

?− κ??xn

?,η?un,un?1

???

1

rn?1

?κ??un?1

?− κ??xn?1

?,η?un,un?1

??≥ γ??un− un?1

??2.

?3.8?

Page 9

Wei-You Zeng et al.9

It follows that

γrn

??un− un?1

κ??un

≤?κ??un

?

??2

≤

?

?− κ??xn

?− κ??un?1

κ??un?1

≤ −μ??un− un?1

≤ −μ??un− un?1

?−

?,η?un?1,un

?− κ??xn?1

??2?

??2? λν

rn

rn?1

?κ??un?1

?− κ??xn?1

??,η?un?1,un

??

??

?

?? κ??xn?1

???κ??xn?1

???xn?1− xn

?− κ??xn

?− κ??xn

???

?−

????1 −

rn?1

rn

rn?1

?κ??un?1

rn

rn?1

?− κ??xn?1

?− κ??xn?1

??

??,η?un?1,un

???

??,

??

????

????

??κ??un?1

???η?un?1,un

???

??rn?1− rn

??

??un?1− xn?1

???un?1− un

?3.9?

since η and κ?are Lipschitz continuous wiht Lipschitz constants λ and ν, respectively. Noting

that liminfn→∞rn> 0, without loss of generality, we assume that there exists a real number

r > 0 such that rn≥ r > 0 for all n ? 1,2,.... Thus,

γr??un?1− un

which implies that

??≤ −μ??un?1− un

??? λν

???xn?1− xn

???

??rn?1− rn

??

r

??un?1− xn?1

??

?

,

?3.10?

?

1 ?γr

μ

???un?1− un

??≤??xn?1− xn

???

??rn?1− rn

??

r

diam?C?,

?3.11?

and hence

??un?1− un

??≤ δ??xn?1− xn

???

??rn?1− rn

??

r

δdiam?C?,

?3.12?

where δ ? 1/?1?γr/μ? ∈ ?0,1?. Set Θ :? max{α,δ} ∈ ?0,1?. Combining ?3.4? and ?3.12? yields

??xn?1− xn

??≤?ααn??1 − αn

?

?δ???xn− xn−1

????1 − αn

????1 − αn

????1 − αn

r

?

?εn−1

?

?

2??αn− αn−1

?δ

??rn− rn−1

2??αn− αn−1

??

diam?C?

≤ Θ??xn− xn−1

?εn−1?

???

??rn− rn−1

??

r

?

diam?C?.

?3.13?

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10 Fixed Point Theory and Applications

From conditions ?C1? and ?C3?,

∞

?

n?1

?1 − αn?1

?εn?

∞

?

∞

?

n?1

??1 − αn

??1 − αn

?εn??αn− αn?1

?εn???αn− αn?1

?εn

??sup

?

≤

n?1

n∈N

εn

?

< ∞.

?3.14?

Set an:? ?xn− xn−1? and

cn:??1 − αn

?εn−1?

?

2??αn− αn−1

???

??rn− rn−1

??

r

?

diam?C?.

?3.15?

Then Lemmas 2.6 and 2.7 imply that limn→∞?xn?1− xn? ? 0 and {xn} is a Cauchy sequence

in C. Hence from ?3.12?, we get

lim

n→∞

??un?1− un

??? 0.

?3.16?

We know from ?C3? that limn→∞εn? 0. It follows that

??xn?1− Sn?1

?un?1

???≤??xn?1− Sn

≤ αn

≤ αndiam?C? ???un?1− un

?un

??????Sn

?un

??? εn.

?− Sn

?un?1

??????Sn

?un?1

?− Sn?1

?un?1

???

??f?xn

?− Sn

?un

??????un?1− un

??? εn

?3.17?

Thus, limn→∞?xn− Sn?un?? ? 0.

Next, we prove that there exists x∗∈ C, such that xn → x∗, un → x∗, and wn → ? w as

n → ∞, where ? w ∈ T?x∗?.

Let p ∈?∞

n?1F?Tn? ∩ Ω. Then

??un− p??2???Trn

?xn

?xn

?− Trn?p???2

≤?Trn

≤?un− p,xn− p?

≤1

2

?− Trn?p?,xn− p?

???un− p??2???xn− p??2−??un− xn

??2?,

?3.18?

and so

??un− p??2≤??xn− p??2−??un− xn

??2≤??xn− p??2.

?3.19?

Page 11

Wei-You Zeng et al. 11

By the convexity of ?·?, we have

??xn?1− p??2≤ αn

??f?xn

?− p??2??1 − αn

???Sn

?un

?− p??2

??2.

≤ αn

≤ αndiam?C?2???xn− p??2−??un− xn

??f?xn

?− p??2??1 − αn

???un− p??2

?3.20?

It follows that

??un− xn

??2≤ αndiam?C?2????xn− p??2−??xn?1− p??2?

≤ αndiam?C?2? 2??xn− xn?1

≤ αndiam?C?2????xn− p?????xn?1− p?????xn− xn?1

??

??diam?C?.

?3.21?

This implies that

lim

n→∞

??un− xn

??? 0.

?3.22?

Since {xn} is a Cauchy sequence in C, there exists an element x∗∈ C such that limn→∞xn? x∗.

Now limn→∞?un− xn? ? 0 implies that limn→∞un? x∗. From ?3.1?, we have

?

≤ 2H?T?xn

??wn− wn?1

??≤

1 ?1

n

?

H?T?xn

?,T?xn?1

?,T?xn?1

??

??

≤ 2L??xn− xn?1

??

?3.23?

and for m > n ≥ 1,

??wm− wn

m−1

?

??≤

???

m−1

?

m−1

?

i?n

??wi− wi?1

ai?1≤

??≤ 2L

?Θai? ci

m−1

?

?

i?n

??xi− xi?1

??,

?3.24?

i?n

??xi− xi?1

i?n

m−1

?

m−1

?

i?n

? Θ

m−1

?

m−1

?

m−1

?

i?n

ai?

i?n

ci

? Θ

i?n

ai?1? Θ?an− am

??

m−1

?

i?n

ci

≤ Θ

i?n

ai?1? Θan?

m−1

?

i?n

ci.

?3.25?

Page 12

12 Fixed Point Theory and Applications

Thus,

m−1

?

i?n

??xi− xi?1

??≤

Θ

1 − Θ

??xn− xn−1

???

?m−1

i?nci

1 − Θ.

?3.26?

By ?3.24? and ?3.26?, we have

lim

m,n→∞

??wm− wn

??? 0.

?3.27?

It follows that {wn} is a Cauchy sequence in H and so there exists an element ? w in H such

d?? w,T?x∗???

≤??? w − wn

that limn→∞wn? ? w:

inf

b∈T?x∗?d?? w,b?

≤??? w − wn

≤??? w − wn

??? d?wn,T?x∗??

??? L??xn− x∗??−→ 0

??? H?T?xn

?,T?x∗??

as n −→ ∞,

?3.28?

that is, d? ? w,T?x∗?? ? 0. We conclude that ? w ∈ T?x∗? as T?x∗? ∈ CB?H?.

??x∗− Sn

It follows that

?x∗???≤??x∗− un

?????un− xn

?????xn− Sn

?un

?un

??????Sn

?un

?− Sn

as n −→ ∞

?x∗??

≤ 2??x∗− un

?????un− xn

?????xn− Sn

???−→ 0

?3.29?

and so x∗? limn→∞Sn?x∗? ? S?x∗?, that is, x ∈ F?S? ??∞

n?1F?Tn?. Since xn → x∗and

un → x∗, we know that κ??un? − κ??xn? → 0. From ?3.1? and ?Φ1?, we have

Φ?? w,x∗,v?? ϕ?v,x∗?− ϕ?x∗,x∗?≥ 0,

n?1F?Tn? ∩ Ω.

?3.30?

that is, x∗∈ Ω. Thus, x∗∈?∞

we have

Since q ? Qf?q?, we have ?f?q?−q,p−q? ≤ 0 for all p ∈?∞

n?1F?Tn?∩Ω. From xn → x∗,

lim

n→∞

?f?q? − q,xn− q???f?q? − q,x∗− q?≤ 0?3.31?

Page 13

Wei-You Zeng et al. 13

and so

??xn?1− q??2????1 − αn

??Sn

?un

?un

?− q?? αn

?− q??2? 2αn

?f?xn

?f?xn

?f?xn

??xn− q????xn?1− q??? 2αn

?− q???2

≤?1 − αn?2??Sn

?− q,xn?1− q?

?− f?q? ? f?q? − q,xn?1− q?

≤?1 − αn

≤?1 − αn

≤?1 − αn

?2??un− q??2? 2αn

?2??un− q??2? ααn

?2??un− q??2? 2ααn

?f?q? − q,xn?1− q?

?f?q? − q,xn?1− q?.

???xn− q??2???xn?1− q??2?? 2αn

?3.32?

It follows from ?3.19? that

??xn?1− q??2≤

?1 − αn

?

?

?2? ααn

1 − ααn

1 −2αn?1 − α?

1 − ααn

αn

1 − αsup

??xn− q??2?

???xn− q??2?2αn?1 − α?

??xn− q??2?

2αn

1 − ααn

?f?q? − q,xn?1− q?

≤

1 − ααn

?f?q? − q,xn?1− q??

×

n∈N

1

1 − α

.

?3.33?

Set

λn:?2αn?1 − α?

1 − ααn

1 − αsup

,ξn:? 0,

?f?q? − q,xn?1− q?.

σn:?

αn

n∈N

??xn− q??2?

1

1 − α

?3.34?

Then, limn→∞λn ? 0,?∞

Remark 3.3. Theorem 3.2 improves and extends the main results of S. Takahashi and W.

Takahashi ?12?.

n?1λn ? ∞, and limsupn→∞σn ≤ 0. It follows from Lemma 2.8 that

limn→∞xn? q and so x∗? q. This completes the proof.

We now give some applications of Theorem 3.2. If the set-valued mapping T in

Theorem 3.2 is single-valued, then we have the following corollary.

Corollary 3.4. Let C be a nonempty closed convex bounded subset of a real Hilbert space H, T : C →

H a Lipschitz continuous mapping with constant L > 0, f : C → C a contraction mapping with

constant α ∈ ?0,1?. Let ϕ : C×C → R be a real-valued function satisfying the conditions ?ϕ1?–?ϕ3?

and let Φ : H ×C×C → R be an equilibrium-like function satisfying the conditions ?Φ1?–?Φ3? and

?Φ4??:

?Φ4??Φ?T?x?,Tr?x?,Ts?y???Φ?T?y?,Ts?y?,Tr?x?? ≤ −γ?Tr?x?−Ts?y??2for all x,y ∈ C and

r,s ∈ ?0,∞?.

Page 14

14 Fixed Point Theory and Applications

Assume that η : C ×C → H is a Lipschitz function with Lipschitz constant λ > 0 which satisfies the

conditions ?η1?∼?η3?. Let κ : C → R be an η-strongly convex function with constant μ > 0 which

satisfies the conditions ?κ1? and ?κ2? with λν/μ ≤ 1. Let Sn: C → C be an S-mapping generated

by T1,T2,... and λ1,λ2,... and?∞

n?1F?Tn? ∩ Ω/ ?∅, where sequence {Tn} is nonexpansive. Let {xn},

{un}, and {wn} be sequences generated by

Φ?T?xn

?,un,v?? ϕ?v,un

?− ϕ?un,un

??1

rn

?κ??un

?Sn

?− κ??xn

?un

?,η?v,un

n ? 1,2,...,

??≥ 0,

∀v ∈ C,

xn?1? αnf?xn

???1 − αn

?,

?3.35?

where {αn} is a sequence in ?0,1? and {rn} in ?0,∞? satisfying conditions ?C1?–?C3?. Then the

sequences {xn} and {un} converge strongly to x∗∈?∞

Corollary 3.5. Let C be a nonempty closed convex bounded subset of a real Hilbert space H, T :

C → CB?H? a multivalued H-Lipschitz continuous mapping with constant L > 0, f : C → C a

contraction mapping with constant α ∈ ?0,1?. Let ϕ : C×C → R be a real-valued function satisfying

the conditions ?ϕ1?–?ϕ3? and let Φ : H × C × C → R be an equilibrium-like function satisfying the

conditions ?Φ1?–?Φ4? and Ω/ ?∅. Assume that η : C×C → H is a Lipschitz function with Lipschitz

constant λ > 0 which satisfies the conditions ?η1?∼?η3?. Let κ : C → R be an η-strongly convex

function with constant μ > 0 which satisfies the conditions ?κ1? and ?κ2? with λν/μ ≤ 1. Let {xn},

{un}, and {wn} be sequences generated by

n?1F?Tn? ∩ Ω, where x∗? P∩∞

n?1F?Tn?∩Ωf?x∗?.

wn∈ T?xn

?,

?− ϕ?un,un

xn?1? αnf?xn

?wn− wn?1? ≤

?

1 ?1

n

?

H?T?xn

?− κ??xn

?un,

?,T?xn?1

?,η?v,un

n ? 1,2,...,

??,

Φ?wn,un,v?? ϕ?v,un

??1

???1 − αn

rn

?κ??un

??≥ 0,

∀v ∈ C,

?3.36?

where {αn} is a sequence in ?0,1? and {rn} in ?0,∞? satisfying conditions ?C1? and ?C2?. Then the

sequences {xn} and {un} converge strongly to x∗∈ Ω, and {wn} converges strongly to w∗∈ T?x∗?,

where x∗? PΩf?x∗?.

Proof. Let Tn ? I in Theorem 3.2 for n ? 1,2,..., where I is an identity mapping. Then

Sn? I for n ? 1,2,.... Thus, the condition ?C3? is satisfied. Now Corollary 3.5 follows from

Theorem 3.2. This completes the proof.

Acknowledgments

The authors would like to thank the referees very much for their valuable comments and

suggestions. This work was supported by the National Natural Science Foundation of China

?10671135? and Specialized Research Fund for the Doctoral Program of Higher Education

?20060610005?.

Page 15

Wei-You Zeng et al. 15

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