Open Access

A new iterative scheme with nonexpansive mappings for equilibrium problems

Journal of Inequalities and Applications20122012:116

https://doi.org/10.1186/1029-242X-2012-116

Received: 5 December 2011

Accepted: 28 May 2012

Published: 28 May 2012

Abstract

In this paper, we suggest a new iteration scheme for finding a common of the solution set of monotone, Lipschitz-type continuous equilibrium problems and the set of fixed points of a nonexpansive mapping. The scheme is based on both hybrid method and extragradient-type method. We obtain a strong convergence theorem for the sequences generated by these processes in a real Hilbert space. Based on this result, we also get some new and interesting results. The results in this paper generalize, extend, and improve some well-known results in the literature.

AMS 2010 Mathematics subject classification: 65 K10, 65 K15, 90 C25, 90 C33.

Keywords

Equilibrium problemsnonexpansive mappingsmonotoneLipschitz-type continuousfixed point

1 Introduction

Let be a real Hilbert space with inner product 〈·,·〉 and norm || · ||. Let C be a nonempty closed convex subset of a real Hilbert space . A mapping S : CC is a contraction with a constant δ (0, 1), if
| | S ( x ) - S ( y ) | | δ | | x - y | | , x , y C .
If δ = 1, then S is called nonexpansive on C. Fix(S) 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 problem in the sense of Blum and Oettli (see [1]) which is presented as follows:
Find x * C such that f ( x * , y ) 0 for all y C . E P ( 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 ß > 0, if
f ( x , y ) + f ( y , x ) - β | | x - y | | 2 , x , y C ;
monotone on C, if
f ( x , y ) + f ( y , x ) 0 , x , y C ;
pseudomonotone on C, if
f ( x , y ) 0 implies f ( y , x ) 0 , x , y C ;
Lipschitz-type continuous on C with constants c1 > 0 and c2 > 0 (see [2]), if
f ( x , y ) + f ( y , z ) f ( x , z ) - c 1 | | x - y | | 2 - c 2 | | y - z | | 2 , x , y , z C .

It is well-known that Problem EP(f, C) includes, as particular cases, the optimization problem, the variational inequality problem, the Nash equilibrium problem in noncooperative games, the fixed point problem, the nonlinear complementarity problem and the vector minimization problem (see [26]).

In recent years, the problem to find a common point of the solution set of problem (EP) and the set of fixed points of a nonexpansive mapping becomes an attractive field for many researchers (see [715]). An important special case of equilibrium problems is the variational inequalities (shortly (VIP)), where F : C and f(x, y) = 〈F(x), y - x〉. Various methods have been developed for finding a common point of the solution set of problem (VIP) and the set of fixed points of a nonexpansive mapping when F is monotone (see [1618]).

Motivated by fixed point techniques of Takahashi and Takahashi in [19] and an improvement set of extragradient-type iteration methods in [20], we introduce a new iteration algorithm for finding a common of the solution set of equilibrium problems with a monotone and Lipschitz-type continuous bifunction and the set of fixed points of a nonexpansive mapping. We show that all of the iterative sequences generated by this algorithm convergence strongly to the common element in a real Hilbert space.

2 Preliminaries

Let C be a nonempty closed convex subset of a Hilbert space . We write x n x to indicate that the sequence {x n } converges weakly to x as n → ∞, x n → x implies that {x n } converges strongly to x. For any x , there exists a nearest point in C, denoted by Pr C (x), such that
| | x - P r C ( x ) | | | | x - y | | , y C .
Pr C is called the metric projection of to C. It is well known that Pr C satisfies the following properties:
x - y , P r C ( x ) - P r C ( y ) | | P r C ( x ) - P r C ( y ) | | 2 , x , y H ,
(2.1)
x - P r C ( x ) , P r C ( x ) - y > 0 , x H , y C ,
(2.2)
| | x - y | | 2 | | x - P r C ( x ) | | 2 + | | y - P r C ( x ) | | 2 , x H , y C .
(2.3)

Let us assume that a bifunction f : C × C R and a nonexpansive mapping S : CC satisfy the following conditions:

A1. f is Lipschitz-type continuous on C;

A2. f is monotone on C;

A3. for each x C, f (x, ·) is subdifferentiable and convex on C;

A4. Fix(S) ∩ Sol(f, C) ≠ .

Recently, Takahashi and Takahashi in [19] first introduced an iterative scheme by the viscosity approximation method. The sequence {x k } is defined by:
x 0 H , Find u k C such that f ( u k , y ) + 1 r k y - u k , u k - x k 0 , y C , x k + 1 = α k g ( x k ) + ( 1 - α k ) S ( u k ) , k 0 ,

where C is a nonempty closed convex subset of and g is a contractive mapping of into itself. The authors showed that under certain conditions over {α k } and {r k }, sequences {x k } and {u k } converge strongly to z = PrSol(f,C)∩Fix(S)(g(x0)). Recently, iterative methods for finding a common element of the set of solutions of equilibrium problems and the set of fixed points of a nonexpansive mapping have further developed by many authors. These methods require to solve approximation auxilary equilibrium problems.

In this paper, we introduce a new iteration method for finding a common point of the set of fixed points of a nonexpansive mapping S and the set of solutions of problem EP(f, C). At each our iteration, the main steps are to solve two strongly convex problems
y k = argmin { λ k f ( x k , y ) + 1 2 | | y - x k | | 2 : y C } , t k = argmin { λ k f ( y k , y ) + 1 2 | | y - x k | | 2 : y C } ,
(2.4)
and compute the next iteration point by Mann-type fixed points
x k + 1 = α k g ( x k ) + ( 1 - α k ) S ( t k ) ,
(2.5)

where g : CC is a δ-contraction with 0 < δ < 1 2 .

To investigate the convergence of this scheme, we recall the following technical lemmas which will be used in the sequel.

Lemma 2.1 (see [21]) Let {a n } be a sequence of nonnegative real numbers such that:
a n + 1 ( 1 - α n ) a n + β n , n 0 ,

where {α n }, and {ß n } satisfy the conditions:

(i) α n (0, 1) and n = 1 α n = ;

(ii) lim sup n β n α n 0 o r n = 1 | β n | < .

Then
lim n a n = 0 .

Lemma 2.2 ([22]) Assume that S is a nonexpansive self-mapping of a nonempty closed convex subset C of a real Hilbert space . If Fix(S) ≠ Ø, then I - S is demiclosed; that is, whenever {x k } is a sequence in C weakly converging to some x ̄ C and the sequence{(I - S)(x k )} strongly converges to some y ¯ , it follows that ( I - - S ) ( x ̄ ) = ȳ . Here I is the identity operator of .

Lemma 2.3 (see [20], Lemma 3.1) Let C be a nonempty closed convex subset of a real Hilbert space . Let f : C × C R be a pseudomonotone, Lipschitz-type continuous bifunction with constants c1 > 0 and c2 > 0. For each × C, let f(x, ·) be convex and subdifferentiable on C. Suppose that the sequences {x k }, {y k }, {t k } generated by Scheme (2.4) and x* Sol(f, C). Then
| | t k - x * | | 2 | | x k - x * | | 2 - ( 1 - 2 λ k c 1 ) | | x k - y k | | 2 - ( 1 - 2 λ k c 2 ) | | y k - t k | | 2 , k 0 .

3 Main results

Now, we prove the main convergence theorem.

Theorem 3.1 Suppose that Assumptions A1-A4 are satisfied, x0 C and two positive sequences {λ k }, {a k } satisfy the following restrictions:
k = 0 | α k + 1 - α k | < , lim k α k = 0 , k = 0 α k = , k = 0 | λ k + 1 - λ k | < , { λ k } [ a , b ] f o r s o m e a , b ( 0 , 1 L ) , w h e r e L = max { 2 c 1 , 2 c 2 } .
Then the sequences {x k }, {y k } and {t k } generated by (2.4) and (2.5) converge strongly to the same point x*, where
x * = P r F i x ( S ) S o l ( f , C ) g ( x * ) .

The proof of this theorem is divided into several steps.

Step 1. Claim that
lim k | | x k - t k | | = 0 .
Proof of Step 1. For each x* Fix(S) ∩ Sol(f, C), it follows from xk+1= a k g(x k ) + (1 - a k )S(t k ), Lemma 2.3 and δ ( 0 , 1 2 ) that
| | x k + 1 - x * | | 2 = | | α k ( g ( x k ) - x * ) + ( 1 - α k ) ( S ( t k ) - S ( x * ) ) | | 2 α k | | g ( x k ) - x * | | 2 + ( 1 - α k ) | | S ( t k ) - S ( x * ) | | 2 = α k | | ( g ( x k ) - g ( x * ) ) + ( g ( x * ) - x * ) | | 2 + ( 1 - α k ) | | S ( t k ) - S ( x * ) | | 2 2 δ 2 α k | | x k - x * | | 2 + 2 α k | | g ( x * ) - x * | | 2 + ( 1 - α k ) | | t k - x * | | 2 2 δ 2 α k | | x k - x * | | 2 + 2 α k | | g ( x * ) - x * | | 2 + ( 1 - α k ) | | x k - x * | | 2 - ( 1 - α k ) ( 1 - 2 λ k c 1 ) | | x k - y k | | 2 - ( 1 - α k ) ( 1 - 2 λ k c 2 ) | | y k - t k | | 2 | | x k - x * | | 2 + 2 α k | | g ( x * ) - x * | | 2 - ( 1 - α k ) ( 1 - 2 λ k c 1 ) | | x k - y k | | 2 - ( 1 - α k ) ( 1 - 2 λ k c 2 ) | | y k - t k | | 2 .
Then, we have
( 1 - α k ) ( 1 - 2 b c 1 ) | | x k - y k | | 2 ( 1 - α k ) ( 1 - 2 λ k c 1 ) | | x k - y k | | 2 | | x k - x * | | 2 - | | x k + 1 - x * | | 2 + 2 α k | | g ( x * ) - x * | | 2 0 as k ,
and
lim k | | x k - y k | | = 0 .
(3.1)
By the similar way, also
lim k | | y k - t k | | = 0 .
Combining this, (3.1) and the inequality ||x k - t k || = ||x k - y k || + || y k - t k ||, we have
lim k | | x k - t k | | = 0 .
(3.2)
Step 2. Claim that
lim k | | x k + 1 - x k | | = 0 .
Proof of Step 2. It is easy to see that t k = argmin  { 1 2 | | t - x k | | 2 + λ k f ( y k , t ) : t C } if and only if
0 2 ( λ k f ( y k , y ) + 1 2 | | y - x k | | 2 ) ( t k ) + N C ( t k ) ,
where N C (x) is the (outward) normal cone of C at x C. This means that 0 = λ k w + t k - x k + w ̄ , where w 2f(y k , t k ) and w ̄ N C ( t k ) . By the definition of the normal cone N C we have, from this relation that
t k - x k , t - t k λ k w , t k - t t C .
Substituting t = tk+1into this inequality, we get
t k - x k , t k + 1 - t k λ k w , t k - t k + 1 .
(3.3)
Since f(x, ·) is convex on C for all x C, we have
f ( y k , t ) - f ( y k , t k ) w , t - t k t C , w 2 f ( y k , t k ) .
Using this and (3.3), we have
t k - x k , t k + 1 - t k λ k w , t k - t k + 1 λ k ( f ( y k , t k ) - f ( y k , t k + 1 ) ) .
(3.4)
By the similar way, we also have
t k + 1 - x k + 1 , t k - t k + 1 λ k + 1 ( f ( y k + 1 , t k + 1 ) - f ( y k + 1 , t k ) ) .
(3.5)
Using (3.4), (3.5) and f is Lipschitz-type continuous and monotone, we get
1 2 | | x k + 1 - x k | | 2 - 1 2 | | t k + 1 - t k | | 2 t k + 1 - t k , t k - x k - t k + 1 + x k + 1 λ k ( f ( y k , t k ) - f ( y k , t k + 1 ) ) + λ k + 1 ( f ( y k + 1 , t k + 1 ) - f ( y k + 1 , t k ) ) λ k ( - f ( t k , t k + 1 ) - c 1 | | y k - t k | | 2 - c 2 | | t k - t k + 1 | | 2 ) + λ k + 1 ( - f ( t k + 1 , t k ) - c 1 | | y k + 1 - t k + 1 | | 2 - c 2 | | t k - t k + 1 | | 2 ) ( λ k + 1 - λ k ) f ( t k , t k + 1 ) - | λ k + 1 - λ k | | f ( t k , t k + 1 ) | .
Hence
| | t k + 1 - t k | | | | x k + 1 - x k | | 2 + 2 | λ k + 1 - λ k | | f ( t k , t k + 1 ) | | | x k + 1 - x k | | + 2 | λ k + 1 - λ k | | f ( t k , t k + 1 ) |
(3.6)
Since (3.6), ak+1- a k 0 as k →∞, g is contractive on C, Lemma 2.3, Step 2 and the definition of xk+1that xk+1= a k g(x k ) + a k S(t k ), we have
| | x k + 1 - x k | | = | | α k g ( x k ) + α k S ( t k ) - α k - 1 g ( x k - 1 ) - α k - 1 S ( t k - 1 ) | | = | | ( α k - α k - 1 ) ( g ( x k - 1 ) - S ( t k - 1 ) ) + ( 1 - α k ) ( S ( t k ) - S ( t k - 1 ) ) + α k ( g ( x k ) - g ( x k - 1 ) ) | | | α k - α k - 1 | | | g ( x k - 1 ) - S ( t k - 1 ) | | + ( 1 - α k ) | | t k - t k - 1 | | + α k δ | | x k - x k - 1 | | | α k - α k - 1 | | | g ( x k - 1 ) - S ( t k - 1 ) | | + ( 1 - α k ) ( | | x k - x k - 1 | | + 2 | λ k - λ k - 1 | | f ( t k - 1 , t k ) | ) + α k δ | | x k - x k - 1 | | = ( 1 - ( 1 - δ ) α k ) | | x k - x k - 1 | | + | α k - α k - 1 | | | g ( x k - 1 ) - S ( t k - 1 ) | | + ( 1 - α k ) 2 | λ k - λ k - 1 | | f ( t k - 1 , t k ) | ( 1 - ( 1 - δ ) α k ) | | x k - x k - 1 | | + M | α k - α k - 1 | + K ( 1 - α k ) 2 | λ k - λ k - 1 | ,

where δ is contractive constant of the mapping g, M = sup{||g(xk - 1) - S(tk - 1)||: k = 0, 1, ...} and K = sup | f ( t k - 1 , t k ) | : k = 0 , 1 , , since k = 0 | α k - α k - 1 | < and k = 0 | λ k - λ k - 1 | < , in view of Lemma 2.1, we have lim k | | x k + 1 - x k | | = 0 .

Step 3. Claim that
lim k | | t k - S ( t k ) | | = 0 .
Proof of Step 3. From xk+1= a k g(x k ) + (1 - a k )S(t k ), we have
x k + 1 - x k = α k g ( x k ) + ( 1 - α k ) S ( t k ) - x k = α k ( g ( x k ) - x k ) + ( 1 - α k ) ( t k - x k ) + ( 1 - α k ) ( S ( t k ) - t k )
and hence
( 1 - α k ) | | S ( t k ) - t k | | | | x k + 1 - x k | | + α k | | g ( x k ) - x k | | + ( 1 - α k ) | | t k - x k | | .
Using this, lim k α k = 0 , Step 1 and Step 2, we have
lim k | | t k - S ( t k ) | | = 0 .
Step 4. Claim that
lim sup k x * - g ( x * ) , S ( t k ) - x * 0 .
Proof of Step 4. By Step 1, {t k } is bounded, there exists a subsequence { t k i } of {t k } such that
lim sup k x * - g ( x * ) , t k - x * = lim i x * - g ( x * ) , t k i - x * .
Since the sequence { t k i } is bounded, there exists a subsequence { t k i j } of { t k i } which converges weakly to t ̄ . Without loss of generality we suppose that the sequence { t k i } converges weakly to t ̄ such that
lim sup k x * - g ( x * ) , t k - x * = lim i x * - g ( x * ) , t k i - x * .
(3.7)
Since Lemma 2.2 and Step 3, we have
S ( t ̄ ) = t ̄ t ̄ F i x ( S ) .
(3.8)
Now we show that t ̄ S o l ( f , C ) . By Step 1, we also have
x k i t ̄ , y k i t ̄ .
Since y k is the unique solution of the strongly convex problem
min { 1 2 | | y - x k | | 2 + f ( x k , y ) : y C } ,
we have
0 2 ( λ k f ( x k , y ) + 1 2 | | y - x k | | 2 ) ( y k ) + N C ( y k ) .
This follows that
0 = λ k w + y k - x k + w k ,
where w 2f (x k , y k ) and w k N C (y k ). By the definition of the normal cone N C , we have
y k - x k , y - y k λ k w , y k - y , y C .
(3.9)
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
f ( x k , y ) - f ( x k , y k ) w , y - y k , y C .
Combining this with (3.9), we have
λ k ( f ( x k , y ) - f ( x k , y k ) ) y k - x k , y k - y , y C .
Hence
λ k j ( f ( x k j , y ) - f ( x k j , y k j ) ) y k j - x k j , y k j - y , y C .
Then, using { λ k } [ a , b ] ( 0 , 1 L ) and the continuity of f , we have
f ( t ̄ , y ) 0 , y C .
Combining this and (3.8), we obtain
t k i t ̄ F i x ( S ) S o l ( f , C ) .
By (3.7) and the definition of x*, we have
lim sup k x * - g ( x * ) , t k - x * = x * - g ( x * ) , t ̄ - x * 0 .
Using this and Step 3, we get
lim sup k x * - g ( x * ) , S ( t k ) - x * = x * - g ( x * ) , t ̄ - x * 0 .

Step 5. Claim that the sequences {x k }, {y k } and {t k } converge strongly to x*.

Proof of Step 5. Using xk+1= α k g(x k ) + (1 - a k )S(t k ) and Lemma 2.3, we have
| | x k + 1 - x * | | 2 = | | α k ( g ( x k ) - x * ) + ( 1 - α k ) ( S ( t k ) - x * ) | | 2 = α k 2 | | g ( x k ) - x * | | 2 + ( 1 - α k ) 2 | | S ( t k ) - x * | | 2 + 2 α k ( 1 - α k ) g ( x k ) - x * , S ( t k ) - x * α k 2 | | g ( x k ) - x * | | 2 + ( 1 - α k ) 2 | | x k - x * | | 2 + 2 α k ( 1 - α k ) g ( x k ) - x * , S ( t k ) - x * = α k 2 | | g ( x k ) - x * | | 2 + ( 1 - α k ) 2 | | x k - x * | | 2 + 2 α k ( 1 - α k ) g ( x k ) - g ( x * ) , S ( t k ) - x * + 2 α k ( 1 - α k ) g ( x * ) - x * , S ( t k ) - x * α k 2 | | g ( x k ) - x * | | 2 + ( 1 - α k ) 2 | | x k - x * | | 2 + 2 δ α k ( 1 - α k ) | | x k - x * | | | | ( t k ) - x * | | + 2 α k ( 1 - α k ) g ( x * ) - x * , S ( t k ) - x * α k 2 | | g ( x k ) - x * | | 2 + ( ( 1 - α k ) 2 + 2 δ α k ( 1 - α k ) ) | | x k - x * | | 2 + 2 α k ( 1 - α k ) g ( x * ) - x * , S ( t k ) - x * ( 1 - α k + 2 δ α k ) | | x k - x * | | 2 + α k 2 | | g ( x k ) - x * | | 2 + 2 α k ( 1 - α k ) max { 0 , g ( x * ) - x * , S ( t k ) - x * } = ( 1 - A k ) | | x k - x * | | 2 + B k ,
where A k and B k are defined by
A k = α k ( 1 - 2 δ ) , B k = α k 2 | | g ( x k ) - x * | | 2 + 2 α k ( 1 - α k ) max { 0 , g ( x * ) - x * , S ( t k ) - x * } .
Since lim k α k = 0 , k = 1 α k = , Step 4, we have lim sup k x * - g ( x * ) , S ( t k ) - x * 0 and hence
B k = o ( A k ) , lim k A k = 0 , k = 1 A k = .

By Lemma 2.1, we obtain that the sequence {x k } converges strongly to x*. It follows from Step 1 that the sequences {y k } and {t k } also converge strongly to the same solution x*= PrFix(S)∩Sol(f,C)g(x*).

   □

4 Applications

Let C be a nonempty closed convex subset of a real Hilbert space and F be a function from C into . 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 . V I ( F , C )
Let f : C × C R be defined by f(x, y) = 〈F(x), y - x〉. Then Problem EP(f, C) 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 ß > 0 if
F ( x ) - F ( y ) , x - y β | | x - y | | 2 , x , y C ;
monotone on C if
F ( x ) - F ( y ) , x - y 0 , x , y C ;
pseudomonotone on C if
F ( y ) , x - y 0 F ( x ) , x - y 0 , x , y C ;
Lipschitz continuous on C with constants L > 0 if
| | F ( x ) - F ( y ) | | L | | x - y | | , x , y C .
Since
y k = argmin { λ k f ( x k , y ) + 1 2 | | y - x k | | 2 : y C } = argmin { λ k F ( x k ) , y - x k + 1 2 | | y - x k | | 2 : y C } = P r C ( x k - λ k F ( x k ) ) ,

(2.4), (2.5) and Theorem 3.1, we obtain that the following convergence theorem for finding a common element of the set of fixed points of a nonexpansive mapping S 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 , F be a function from C to such that F is monotone and L-Lipschitz continuous on C, g : CC is contractive with constant δ ( 0 , 1 2 ) , S: CC be nonexpansive and positive sequences {a k } and {λ k } satisfy the following restrictions
k = 0 | α k + 1 - α k | < , lim k α k = 0 , k = 0 α k = , k = 0 | λ k + 1 - λ k | < , { λ k } [ a , b ] f o r s o m e a , b ( 0 , 1 L ) .
Then sequences {x k }, {y k } and {t k } generated by
y k = P r C ( x k - λ k F ( x k ) ) , t k = P r C ( x k - λ k F ( y k ) ) , x k + 1 = α k g ( x k ) + ( 1 - α k ) S ( t k ) ,

converge strongly to the same point x* PrFix(S)∩Sol(F,C)g(x*).

Thus, this scheme and its convergence become results proposed by Nadezhkina and Takahashi in [23]. As direct consequences of Theorem 3.1, we obtain the following corollary.

Corollary 4.2 Suppose that Assumptions A1-A3 are satisfied, Sol(f, C) ≠ Ø, x0 C and two positive sequences {λ k }, {a k } satisfy the following restrictions:
k = 0 | α k + 1 - α k | < , lim k α k = 0 , k = 0 α k = , k = 0 | λ k + 1 - λ k | < , { λ k } [ a , b ] f o r s o m e a , b ( 0 , 1 L ) , w h e r e L = max { 2 c 1 , 2 c 2 } .
Then, the sequences {x k }, {y k } and {t k } generated by
y k = a r g m i n { λ k f ( x k , y ) + 1 2 | | y - x k | | 2 : y C } , t k = a r g m i n { λ k f ( y k , y ) + 1 2 | | y - x k | | 2 : y C } x k + 1 = α k g ( x k ) + ( 1 - α k ) t k ,

where g : CC is a δ-contraction with 0 < δ < 1 2 , converge strongly to the same point x*=PrSol(f,C)g(x*).

Declarations

Acknowledgements

We are very grateful to the anonymous referees for their really helpful and constructive comments that helped us very much in improving the paper.

The work was supported by National Foundation for Science and Technology Development of Vietnam (NAFOSTED).

Authors’ Affiliations

(1)
Department of Scientific Fundamentals, Posts and Telecommunications Institute of Technology
(2)
Department of Mathematics, Haiphong university

References

  1. Blum E, Oettli W: From optimization and variational inequality to equilibrium problems. The Math Stud 1994, 63: 127–149.MathSciNetGoogle Scholar
  2. Mastroeni G: On auxiliary principle for equilibrium problems. In Nonconvex Optimization and its Applications. Edited by: Daniele P, Giannessi F, Maugeri A. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003.Google Scholar
  3. Anh PN: A logarithmic quadratic regularization method for solving pseudomonotone equilibrium problems. Acta Math Vietnamica 2009, 34: 183–200.MathSciNetGoogle Scholar
  4. Anh PN: An LQP regularization method for equilibrium problems on polyhedral. Vietnam J Math 2008, 36: 209–228.MathSciNetGoogle Scholar
  5. Anh PN, Kim JK: Outer approximation algorithms for pseudomonotone equilibrium problems. Comp Math Appl 2011, 61: 2588–2595. 10.1016/j.camwa.2011.02.052MathSciNetView ArticleGoogle Scholar
  6. Quoc TD, Anh PN, Muu LD: Dual extragradient algorithms to equilibrium Problems. J Glob Optim 2012, 52: 139–159. 10.1007/s10898-011-9693-2MathSciNetView ArticleGoogle Scholar
  7. Anh PN: Strong convergence theorems for nonexpansive mappings and Ky Fan inequalities. J Optim Theory Appl 2012. DOI 10.1007/s10957–012–0005-xGoogle Scholar
  8. Anh PN, Kim JK, Nam JM: Strong convergence of an extragradient method for equilibrium problems and fixed point problems. J Korean Math Soc 2012, 49: 187–200. 10.4134/JKMS.2012.49.1.187MathSciNetView ArticleGoogle Scholar
  9. Anh PN, Son DX: A new iterative scheme for pseudomonotone equilibrium problems and a finite family of pseudocontractions. J Appl Math Inform 2011, 29: 1179–1191.MathSciNetGoogle Scholar
  10. Ceng LC, Schaible S, Yao JC: Implicit iteration scheme with perturbed mapping for equilibrium problems and fixed point problems of finitely many nonexpansive mappings. J Optim Theory Appl 2008, 139: 403–418. 10.1007/s10957-008-9361-yMathSciNetView ArticleGoogle Scholar
  11. Chen R, Shen X, Cui S: Weak and strong convergence theorems for equilibrium problems and countable strict pseudocontractions mappings in Hilbert space. J Ineq Appl 2010. DOI:10.1155/2010/474813Google Scholar
  12. Wang S, Cho YJ, Qin X: A new iterative method for solving equilibrium problems and fixed point problems for an infinite family of nonexpansive mappings. Fix Point Theory Appl 2010. DOI: 10.1155/2010/165098Google Scholar
  13. Wangkeeree R: An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings. Fix Point Theory Appl 2008. DOI:10.1155/2008/134148Google Scholar
  14. Yao Y, Liou YC, Jao JC: Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings. Fix Point Theory Appl 2007. DOI:10.1155/2007/64363Google Scholar
  15. Yao Y, Liou YC, Wu YJ: An extragradient method for mixed equilibrium problems and fixed point problems. Fix Point Theory Appl 2009. DOI: 10.1155/2009/632819Google Scholar
  16. Ceng LC, Petrusel A, Lee C, Wong MM: Two extragradient approximation methods for variational inequalities and fixed point problems of strict pseudo-contractions. Taiwanese J Math 2009, 13: 607–632.MathSciNetGoogle Scholar
  17. Takahashi S, Toyoda M: Weakly convergence theorems for nonexpansive mappings and monotone mappings. J Optim Theory Appl 2003, 118: 417–428. 10.1023/A:1025407607560MathSciNetView ArticleGoogle Scholar
  18. Zeng LC, Yao JC: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwanese J Math 2010, 10: 1293–1303.MathSciNetGoogle Scholar
  19. Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J Math Anal Appl 2007, 331: 506–515. 10.1016/j.jmaa.2006.08.036MathSciNetView ArticleGoogle Scholar
  20. Anh PN: A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization 2012. DOI:10.1080/02331934.2011.607497Google Scholar
  21. Xu HK: Viscosity approximation methods for nonexpansive mappings. J Math Anal Appl 2004, 298: 279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView ArticleGoogle Scholar
  22. Goebel K, Kirk WA: Topics on metric fixed point theory. Cambridge University Press, Cambridge, England; 1990.View ArticleGoogle Scholar
  23. Nadezhkina N, Takahashi W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J Optim Theory Appl 2006, 128: 191–201. 10.1007/s10957-005-7564-zMathSciNetView ArticleGoogle Scholar

Copyright

© PN and DD; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.