Weak convergence theorems for an infinite family of nonexpansive mappings and equilibrium problems (tt)

Keywords and phrases: nonexpansive mapping, pseudomonotone, Lipschitz-type continuous, equilibrium problems, fixed point.. * Corresponding author Received December 5, 2011 WEAK CONVERG

Volume 7, Number 2, 2012, Pages 113-127 Available online at http://pphmj.com/journals/jpfpta.htm Published by Pushpa Publishing House, Allahabad, INDIA

House Publishing

Pushpa

2012

©

2010 Mathematics Subject Classification: 65K10, 65K15, 90C25, 90C33.

Keywords and phrases: nonexpansive mapping, pseudomonotone, Lipschitz-type continuous, equilibrium problems, fixed point

* Corresponding author

Received December 5, 2011

WEAK CONVERGENCE THEOREMS FOR AN INFINITE FAMILY OF NONEXPANSIVE MAPPINGS AND

EQUILIBRIUM PROBLEMS

P N Anh*†, L B Long†, N V Quy and L Q Thuy

†Department of Scientific Fundamentals

Posts and Telecommunications Institute of Technology

Hanoi, Vietnam

e-mail: anhpn@ptit.edu.vn

Department of Scientific Fundamentals

Academy of Finance

Hanoi, Vietnam

Faculty of Applied Mathematics and Informatics

Hanoi University of Technology

Vietnam

Abstract

The purpose of this paper is to investigate a new iteration scheme for

finding a common element of the set of fixed points of an infinite

family of nonexpansive mappings and the solution set of a pseudomonotone and Lipschitz-type continuous equilibrium problem

The scheme is based on the extragradient-type methods and fixed

point methods We show that the iterative sequences generated by this

algorithm converge weakly to the common element in a real Hilbert

space

1 Introduction

Let H be a real Hilbert space with inner product ⋅⋅, and norm ⋅ Let

C be a closed convex subset of a real Hilbert space H and Pr be the C

projection of H onto C When { }x n is a sequence in H then , x n → x

(resp x n x) will denote strong (resp weak) convergence of the sequence

{ }x n to x as n → ∞ A mapping S :C →C is said to be nonexpansive if

( )x S( )y x y , x, y C

( )S

Fix is denoted by the set of fixed points of S Let f : C × C → R be a bifunction such that f(x, x) =0 for all x∈C We consider the equilibrium problems in the sense of Blum and Oettli [8] which are presented as follows:

Find x∗∈C such that f(x∗, y)≥ 0 for all y∈C EP(f, C)

The set of solutions of EP(f,C) is denoted by Sol(f , C) The bifunction f

is called strongly monotone on C with β > 0 if

(x, y) f(y, x) x y 2, x, y C;

monotone on C if

(x, y) f(y, x) 0, x, y C;

pseudomonotone on C if

(x, y) 0 f(y, x) 0, x, y C;

Lipschitz-type continuous on C with constants c1 > 0 and c2 > 0 if

(x, y) f(y, z) f(x, z) c1 x y 2 c2 y z 2, x, y, z C

In this paper, we are interested in the problem of finding a common element of the solution set of the equilibrium problems EP(f,C) and the set

of fixed points ∩∞k=1Fix( )S k of an infinite family of nonexpansive mappings

{ }S k , namely:

where the bifunction f and the mappings S k (k =1, 2, ) satisfy the

following conditions:

A1 f is Lipschitz-type continuous on C,

A2 f is pseudomonotone on C,

A3 f is weakly continuous on C,

A4 S is nonexpansive on C for all k ≥ ∩∞= ( )∩ ( )

, 1

k

,

∅

≠

A5 ∑∞ { ( ) ( ) }

∞

<

∈

−

1

sup

k

k

S for any bounded subset D

of C

An important special case of problem (1.1) is that f(x, y)=

( )x , y x ,

F − where F : C → H and this problem is reduced to finding a

common element of the solution set of variational inequalities and the set of

fixed points of an infinite family of nonexpansive mappings (see [6, 14, 16,

17, 22])

Motivated by the viscosity method in [16] and the approximation method

in [7] via an improvement set of extragradient methods in [2-4], we introduce

a new iteration algorithm for finding a common element of problem (1.1) At

each main iteration n, we only solve two strongly convex programs with a

pseudomonotone and Lipschitz-type continuous bifunction We show that all

of the iterative sequences generated by this algorithm converge weakly to the

common element in a real Hilbert space

This paper is organized as follows: Section 2 recalls some concepts in

equilibrium problems and fixed point problems that will be used in the sequel

and an iterative algorithm for solving problem (1.1) Section 3 investigates the convergence of the algorithms presented in Section 2 as the main results

of our paper

2 Preliminaries

In 1953, Mann [13] introduced a well-known classical iteration method

to approximate a fixed point of a nonexpansive mapping S :C →C in a real Hilbert space H This iteration is defined as

where C is nonempty closed convex subset of H and { }αk ⊂[ ]0,1 Then

{ }x k converges weakly to x∗∈Fix( )S

For finding a common fixed point of an infinite family of nonexpansive mappings { }S k , Aoyama et al [7] introduced an iterative sequence { }x k of

C defined by x0∈C and

where C is a nonempty closed convex subset of a real Hilbert space, { }αk ⊂ [ ]0,1 and ∩∞= ( )≠ ∅

i Fix S i The authors proved that the sequence { }x k

converges strongly to ∗∈∩∞= ( )

x

Recently, Yao et al [20] introduced an iterative scheme for finding a common element of the set of solutions of problem (1.1):

⎪

⎪

⎩

⎪

⎪

⎨

⎧

γ + β + α

=

∈

∀

≥

−

− +

∈

, ,

0 ,

1 , ,

1

0

k k k

k k

k k k

k k k k

k

y W x

x g x

C x x

y y x r x y f

C x

where g :C → C is contractive and W is W-mapping of k { }S k Under mild

assumptions on parameters, the authors proved that the sequences { }k

x and

{ }y k converge strongly to x where ∗,

( ) ( )( ( ))

∗

∗

∞

=

x

Methods for solving problem (1.1) have been well developed by many

researchers (see [5, 9-11, 18-21]) These methods require solving

approximation equilibrium problems with strongly monotone or monotone

and Lipschitz-type continuous bifunctions

In our scheme, the main steps are to solve two strongly convex problems

⎪

⎪

⎩

⎪⎪

⎨

⎧

⎭⎬

⎫

⎩⎨

=

⎭⎬

⎫

⎩⎨

=

, :

2

1 , min

arg

, :

2

1 , min

arg

2 2

C y x

y y

y f t

C y x

y y

x f y

k k

k k

k k

k k

(2.1)

and compute the next iteration point by Mann-type fixed points

(1 ) ( )

k k

k k

To investigate the convergence of this scheme, we recall the following

technical lemmas which will be used in the sequel

Lemma 2.1 (See [1]) Let C be a nonempty closed convex subset of a

real Hilbert space H Let f : C × C → R be a pseudomonotone,

Lipschitz-type continuous bifunction For each x∈C, let f( )x,⋅ be convex and

subdifferentiable on C Suppose that the sequences { }x k , { }y k , { }k

t are generated by scheme (2.1) and x∗∈Sol(f , C). Then

1 2

2

2

k

(1−2λ 2) − 2, ∀ ≥ 0

Lemma 2.2 [17] Let H be a real Hilbert space, { }δ be a sequence of k

real numbers such that { }δk ⊂[α,β]⊂ (0,1), c >0 and two sequences

{ }x k , { }k

y of H such that

⎪

⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

= δ

− + δ

≤

≤

∞

→

∞

→

∞

→

1

sup lim

, sup

lim

, sup

lim

c y x

c y

c x

k k

k k k

k k

k k

Then

0

∞

→

k k

Lemma 2.3 [7] Let C be a nonempty closed convex subset of a Banach

space Let { }S k be a sequence of nonexpansive mappings of C into itself and

S is a mapping of C into itself such that

⎪

⎪

⎩

⎪⎪

⎨

⎧

∈

∀

=

∞

<

∈

−

∞

→

∞

∑

,

lim

, :

sup

1

1

C x x S x

S

C x x S x S

k k i

k k

Then lim sup{ ( )− ( ) : ∈ }=0

∞

k

Lemma 2.4 (See [12]) Assume that S is a nonexpansive self-mapping of

a nonempty closed convex subset C of a real Hilbert space H If Fix( )S

,

∅

≠ then I − is demiclosed; that is, whenever S { }x k is a sequence in C weakly converging to some x ∈ and the sequence C {(I − S)( )}x k strongly converging to some , y it follows that (I −S)( )x = y. Here I is the identity operator of H

Lemma 2.5 (See [17]) Let C be a nonempty closed convex subset of a

real Hilbert space H Suppose that, for all u∈C, the sequence { }x k satisfies

0 ,

Then the sequence { ( )}k

Pr converges strongly to some x∈C

3 Convergence Results

Now, we prove the main convergence theorem

Theorem 3.1. Suppose that Assumptions A1-A5 are satisfied, x0∈C

and two positive sequences { } { }λ ,k αk satisfy the following restrictions:

⎪⎩

⎪

⎨

⎧

=

⎟

⎠

⎞

⎜

⎝

⎛

∈

⊂

λ

⊂

⊂

α

2 , 2 max ,

1 , 0 , ,

1 , 0 ,

2

1 c c L

where L

b a some for b a

d c

k

k

Then the sequences { }x k , { }k

y and { }k

t generated by (2.1) and (2.2) converge weakly to the same point ∗∈∩∞= ( )∩ ( )

( ) ( )( )

lim

k C f Sol S Fix

x

∞

→

∗ = The proof of this theorem is divided into several steps

In Steps 1 and 2, we will consider weak clusters of { }x k It follows from

Lemma 2.1 that

∗

∗

∗

1

k

x

and hence there exists

∞

The sequence { }x k is bounded and there exists a subsequence { }k j

x converges weakly to x as j →∞

Step 1. Claim that ∈∩∞= ( )

x

Proof of Step 1. By Lemma 2.1 and (3.1), we have

1

2

2 1

1−d − bc x k − y k ≤ −d − λk c x k − y k

2 1

−

as

Then

0

∞

→

k k

By the similar way, also

0

∞

→

k k

Combining this, (3.2) and the inequality x k −t k ≤ x k − y k +

,

k

y − we have

0

∞

→

k k

Since ∗∈∩∞= ( )∩ ( )

x Lemma 2.1 and (3.1), we have

( )t − x∗ ≤ t −x∗ ≤ x − x∗

and hence

sup lim S k t k x c

k

≤

∞

→ Using (3.1) and x k+1 = αk x k +(1−αk) ( )S k t k , we have

k

k k k

k

k

∞

→

∗

∗

∞

→

By Lemma 2.2, we have

∞

→

k k k k

x t

If follows from (3.1) and (3.3) that the sequence { }k

t is bounded By Assumption A5, we have

∑∞

∞

<

∈

−

1

sup

k

k k

S

Let S be a mapping of C into itself defined by

( )x S ( )x

for all x∈ and suppose that C ( )=∩∞= ( )

S Fix Then, using Lemma 2.3, (3.3) and (3.4), we obtain

( )k k

x x

k

k k k k

k

x t S t

S t S t

S x

≤

k

k k

k k

x t S t

x x S x S t

as

Then, by Lemma 2.4 and the sequence { }k j

x converges weakly to ,x we have x∈Fix( )S , i.e.,

( )

∩∞=

∈

1

i

i

S Fix x

Step 2. When x k j x as j → ∞, we show that x ∈Sol(f , C)

Proof of Step 2. Since y is the unique solution of the strongly convex k

problem

2

1

⎭⎬

⎫

⎩⎨

⎧ y − x k + f x k y y∈C

we have

2

1 ,

0 2 k f x k y y x k 2⎟ y k +N C y k

⎠

⎞

⎜

⎝

∂

∈

This follows that

,

0 =λk w+ y k − x k +w

where ( k k)

y x f

w∈∂2 , and w ∈N C( )y k By the definition of the normal

cone ,N C we imply that

,

,

x

On the other hand, since f(x k,⋅) is subdifferentiable on C, by the well

known Moreau-Rockafellar theorem, there exists w∈∂2f(x k, y k) such that

(x , y) f(x , y ) w, y y , y C

Combining this with (3.6), we have

( (f x k, y) f(x k, y k)) y k x k, y k y , y C

λ

Hence

( (f x j, y) f(x j, y j)) y j x j, y j y , y C

j

k k k k

k k

λ

Then, using { }λk ⊂[a,b]⊂ ⎜⎝⎛0, L1⎟⎠⎞, (3.2), x k j x as j → ∞ and weak

continuity of f, we have

(x, y) 0, y C

This means that x∈Sol(f , C)

Step 3. Claim that the sequences { }x k , { }k

y and { }k

t converge weakly

to the same point x where ∗,

( ) ( )( )

lim

k C f Sol S Fix

x

∞

→

∗ =

Proof of Step 3. It follows from Steps 2 and 3 that for every weak cluster

point x of the sequence { }x k satisfies ∈∩∞= ( )∩ ( )

show that { }k

x converges weakly to x Now, we assume that { n k}

x is an

another subsequence of { }x k such that x n k x

ˆ as k →∞ Then

( , ) ( ),

ˆ Sol f C Fix S

where S is defined by (3.5) We will show that xˆ = x If x ≠ ˆx, then from (3.1) and the Opial condition, it follows that

x x

k

−

=

∞

→ lim

x

x k j

j

−

=

∞

→inf lim

x

x k j

j

ˆ inf

<

∞

→

x

x k

k

ˆ lim −

=

∞

→

x

x n k

k

ˆ inf

=

∞

→

x

x n k

<

∞

→inf lim

x

x k

=

∞

→inf lim

c

= This is a contraction Thus, we have x = It implies ˆx

x

It follows from (3.2) and (3.3) that

as

,t x k → ∞

x

Setting

(f,C) Fix( )S ( )k

Sol

Then, from x∈Sol(f,C)∩Fix( )S , it implies

0 ,

0

By Lemma 2.5 and Step 1, the sequence { }z k converges strongly to z ∈

(f, C) Fix( )S

Sol ∩ Hence, we have

, 0 , − ≥

x

so, we have x = This shows that z

( ) ( )( )

lim

C f Sol S Fix

=

∞

4 Applications

Let C be a nonempty closed convex subset of a real Hilbert space H and

F be a function from C into H In this section, we consider the variational inequality problem which is presented as follows:

Find x∗∈C such that F( )x∗ , x −x∗ ≥0 for all x∈C VI(F,C)

Let f :C × C → R be defined by f(x, y) = F( )x , y − x Then problem

(f C)

EP , can be written in VI(F , C) The set of solutions of VI(F,C) is denoted by Sol(F , C). Recall that the function F is called strongly monotone

on C with β > 0 if

( )x F( )y , x y x y 2, x, y C;

monotone on C if

( )x F( )y , x y 0, x, y C;

pseudomonotone on C if

( )y , x y 0 F( )x , x y 0, x, y C;

Lipschitz continuous on C with constants L >0 if

( )x F( )y L x y , x, y C

Since

⎭⎬

⎫

⎩⎨

2

1 , min

( )

⎭⎬

⎫

⎩⎨

2

1 ,

min

( k k ( ))k ,

=

using (2.1), (2.2) and Theorem 3.1, we obtain the following convergence theorem for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings { }S i and the solution set of problem VI(F , C)

Theorem 4.1. Let C be a nonempty closed convex subset of a real Hilbert space H Let F be a function from C to H such that F is

pseudomonotone and L -Lipschitz continuous on C For each i =1, ,S i :

C

C → is nonexpansive such that ∩∞= ( )∩ ( ) ≠∅

i Fix S i Sol F C and

∑∞

k S k x S k x x D for any bounded subset D of C

If positive sequences { }α and k { }λ satisfy the following restrictions: k

⎪⎩

⎪

⎨

⎧

⎟

⎠

⎞

⎜

⎝

⎛

∈

⊂ λ

⊂

⊂ α

, 1 , 0 , ,

, 1 , 0 ,

L b

a some for b a

d c

k k

then the sequences{ } { }x k , y k and { }t k generated by

⎪

⎪

⎩

⎪

⎪

⎨

⎧

α

− + α

=

λ

−

=

λ

−

=

, ,

k k

k k k

k k

k C k

k k

k C k

t S x

x

y F x

Pr t

x F x

Pr y

converge weakly to the same point ∗∈∩∞= ( )∩ ( )

( ) ( )( )

lim

k C F Sol S Fix

x

∞

→

∗ =

Acknowledgement

The work is supported by the Vietnam National Foundation for Science Technology Development (NAFOSTED)

References

[1] P N Anh, A hybrid extragradient method extended to fixed point problems and equilibrium problems, Optimization, 2011, DOI: 10.1080/02331934.2011.607497 [2] P N Anh, A logarithmic quadratic regularization method for solving pseudo-monotone equilibrium problems, Acta Math Vietnam 34 (2009), 183-200

[3] P N Anh, An LQP regularization method for equilibrium problems on polyhedral, Vietnam J Math 36 (2008), 209-228

[4] P N Anh and J K Kim, Outer approximation algorithms for pseudomonotone equilibrium problems, Comput Math Appl 61 (2011), 2588-2595

[5] P N Anh, J K Kim and J M Nam, Strong convergence of an extragradient method for equilibrium problems and fixed point problems, J Korean Math Soc (2011), accepted

[6] P N Anh and D X Son, A new iterative scheme for pseudomonotone equilibrium problems and a finite family of pseudocontractions, J Appl Math Inform 29 (2011), 1179-1191

[7] K Aoyama, Y Kimura, W Takahashi and M Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal 67 (2007), 2350-2360

[8] E Blum and W Oettli, From optimization and variational inequality to equilibrium problems, Math Student 63 (1994), 127-149

[9] L C Ceng, S Schaible and J C Yao, Implicit iteration scheme with perturbed mapping for equilibrium problems and fixed point problems of finitely many nonexpansive mappings, J Optim Theory Appl 139 (2008), 403-418

[10] L C Ceng, A Petrusel, C Lee and M M Wong, Two extragradient approximation methods for variational inequalities and fixed point problems of strict pseudo-contractions, Taiwanese J Math 13 (2009), 607-632

[11] R Chen, X Shen and S Cui, Weak and strong convergence theorems for equilibrium problems and countable strict pseudocontractions mappings in Hilbert space, J Inequal Appl 2010, Art ID 474813, 11 pp doi:10.1155/2010/474813