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# A relaxed hybrid steepest descent method for common solutions of generalized mixed equilibrium problems and fixed point problems

- Nawitcha Onjai-uea
^{1, 3}, - Chaichana Jaiboon
^{2, 3}Email author and - Poom Kumam
^{1, 3}

**2011**:32

https://doi.org/10.1186/1687-1812-2011-32

© Onjai-uea et al; licensee Springer. 2011

**Received:**13 January 2011**Accepted:**11 August 2011**Published:**11 August 2011

## Abstract

In the setting of Hilbert spaces, we introduce a relaxed hybrid steepest descent method for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of a variational inequality for an inverse strongly monotone mapping and the set of solutions of generalized mixed equilibrium problems. We prove the strong convergence of the method to the unique solution of a suitable variational inequality. The results obtained in this article improve and extend the corresponding results.

**AMS (2000) Subject Classification:** 46C05; 47H09; 47H10.

## Keywords

- relaxed hybrid steepest descent method
- inverse strongly monotone mappings
- nonexpansive mappings
- generalized mixed equilibrium problem

## 1. Introduction

Let *H* be a real Hilbert space, *C* be a nonempty closed convex subset of *H* and let *P*_{
C
} be the metric projection of *H* onto the closed convex subset *C*. Let *S* : *C* → *C* be a *nonexpansive* mapping, that is, ||*Sx* - *Sy*|| ≤ ||*x* - *y*|| for all *x*, *y* ∈ *C*. We denote by *F*(*S*) the set fixed point of *S*. If *C* ⊂ *H* is nonempty, bounded, closed and convex and *S* is a nonexpansive mapping of *C* into itself, then *F*(*S*) is nonempty; see, for example, [1, 2]. A mapping *f* : *C* → *C* is a *contraction* on *C* if there exists a constant *η* ∈ (0, 1) such that ||*f*(*x*) - *f*(*y*)|| ≤ *η*||*x* - *y*|| for all *x*, *y* ∈ *C*. In addition, let *D* : *C* → *H* be a nonlinear mapping, *φ* : *C* → ℝ ∪ {+∞} be a real-valued function and let *F* : *C* × *C* → ℝ be a bifunction such that *C* ∩ dom *φ* ≠ ∅, where ℝ is the set of real numbers and dom *φ* = {*x* ∈ *C* : *φ*(*x*) *<* +∞}.

We find that if *x* is a solution of a problem (1.1), then *x* ∈ dom *φ*.

If *D* = 0, then the problem (1.1) is reduced into the *mixed equilibrium problem* which is denoted by MEP(*F*, *φ*).

If *φ* = 0, then the problem (1.1) is reduced into the *generalized equilibrium problem* which is denoted by GEP(*F*, *D*).

If *D* = 0 and *φ* = 0, then the problem (1.1) is reduced into the *equilibrium problem* which is denoted by EP(*F*).

If *F* = 0 and *φ* = 0, then the problem (1.1) is reduced into the *variational inequality problem* which is denoted by VI(*C*, *D*).

The generalized mixed equilibrium problems include, as special cases, some optimization problems, fixed point problems, variational inequality problems, Nash equilibrium problems in noncooperative games, equilibrium problem, Numerous problems in physics, economics and others. Some methods have been proposed to solve the problem (1.1); see, for instance, [3, 4] and the references therein.

**Definition 1.1**. Let

*B*:

*C*→

*H*be nonlinear mappings. Then,

*B*is called

- (1)
*monotone*if 〈*Bx*-*By*,*x*-*y*〉 ≥ 0, ∀*x*,*y*∈*C*, - (2)
- (3)
A set-valued mapping

*Q*:*H*→ 2^{ H }is called*monotone*if for all*x*,*y*∈*H*,*f*∈*Qx*and*g*∈*Qy*imply 〈*x*-*y*,*f*-*g*〉 ≥ 0. A monotone mapping*Q*:*H*→ 2^{ H }is called*maximal*if the graph*G*(*Q*) of*Q*is not properly contained in the graph of any other monotone mapping. It is well known that a monotone mapping*Q*is maximal if and only if for (*x*,*f*) Î*H*×*H*, 〈*x*-*y*,*f*-*g*〉 ≥ 0 for every (*y*,*g*) Î*G*(*Q*) implies*f*Î*Qx*.

*H*:

where *F* is the fixed point set of a nonexpansive mapping *S* defined on *H* and *b* is a given point in *H*.

*viscosity approximation method*:

*x*

_{ n }} generated by (1.2) converges strongly to the unique solution of the variational inequality

where *h* is a potential function for *γf*.

*ξ*-inverse-strongly monotone mapping, Takahashi and Toyoda [6] introduced the following iterative scheme:

where *B* is a *ξ*-inverse-strongly monotone mapping, {*γ*_{
n
} } is a sequence in (0, 1), and {*α*_{
n
} } is a sequence in (0, 2*ξ*). They showed that if *F*(*S*) ∩ VI(*C*, *B*) is nonempty, then the sequence {*x*_{
n
} } generated by (1.3) converges weakly to some *z* ∈ *F*(*S*) ∩ VI(*C*, *B*).

The method of the steepest descent, also known as The Gradient Descent, is the simplest of the gradient methods. By means of simple optimization algorithm, this popular method can find the local minimum of a function. It is a method that is widely popular among mathematicians and physicists due to its easy concept.

*F*(

*S*) ∩ VI(

*C*,

*B*), let

*S*:

*H*→

*H*be nonexpansive mappings, Yamada [7] introduced the following iterative scheme called the

*hybrid steepest descent method*:

where *x*_{1} = *x* ∈ *H*, {*α*_{
n
} } ⊂ (0, 1), *B* : *H* → *H* is a strongly monotone and Lipschitz continuous mapping and *μ* is a positive real number. He proved that the sequence {*x*_{
n
} } generated by (1.4) converged strongly to the unique solution of the *F*(*S*) ∩ VI(*C*, *B*).

*F*(

*S*) ∩ VI(

*C*,

*B*) ∩ EP(

*F*), Su et al. [8] introduced the following iterative scheme by the viscosity approximation method in Hilbert spaces:

*x*

_{1}∈

*H*

where *α*_{
n
} ⊂ [0, 1) and *r*_{
n
} ⊂ (0, ∞) satisfy some appropriate conditions. Furthermore, they prove {*x*_{
n
} } and {*u*_{
n
} } converge strongly to the same point *z* ∈ *F*(*S*) ∩ VI(*C*, *B*) ∩ EP(*F*), where *z* = *P*_{F(S)∩VI(C,B) ∩ EP(F)}*f*(*z*).

*F*(

*S*) ∩ GEP(

*F*,

*D*), let

*C*be a nonempty closed convex subset of a real Hilbert space

*H*. Let

*D*be a

*β*-inverse-strongly monotone mapping of

*C*into

*H*, and let

*S*be a nonexpansive mapping of

*C*into itself, Takahashi and Takahashi [9] introduced the following iterative scheme:

where {*α*_{
n
} } ⊂ [0, 1], {*γ*_{
n
} } ⊂ [0, 1] and {*r*_{
n
} } ⊂ [0, 2*β*] satisfy some parameters controlling conditions. They proved that the sequence {*x*_{
n
} } defined by (1.6) converges strongly to a common element of *F*(*S*) ∩ GEP(*F*, *D*).

Recently, Chantarangsi et al. [10] introduced a new iterative algorithm using a viscosity hybrid steepest descent method for solving a common solution of a generalized mixed equilibrium problem, the set of fixed points of a nonexpansive mapping and the set of solutions of variational inequality problem in a real Hilbert space. Jaiboon [11] suggests and analyzes an iterative scheme based on the hybrid steepest descent method for finding a common element of the set of solutions of a system of equilibrium problems, the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problems for inverse strongly monotone mappings in Hilbert spaces.

In this article, motivated and inspired by the studies mentioned above, we introduce an iterative scheme using a relaxed hybrid steepest descent method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of a nonexpansive mapping and the set of solutions of variational inequality problems for inverse strongly monotone mapping in a real Hilbert space. Our results improve and extend the corresponding results of Jung [12] and some others.

## 2. Preliminaries

Throughout this article, we always assume *H* to be a real Hilbert space, and let *C* be a nonempty closed convex subset of *H*. For a sequence {*x*_{
n
} }, the notation of *x*_{
n
} ⇀ *x* and *x*_{
n
} → *x* means that the sequence {*x*_{
n
} } converges weakly and strongly to *x*, respectively.

Such a mapping *P*_{
C
} from *H* onto *C* is called the metric projection.

The following known lemmas will be used in the proof of our main results.

**Lemma 2.1**.

*Let H be a real Hilbert spaces H. Then, the following identities hold:*

- (i)
*for each x*∈*H and x** ∈*C, x** =*P*_{ C }*x*⇔ 〈*x*-*x**,*y*-*x**〉 ≤ 0, ∀*y*∈*C*; - (ii)
*P*_{ C }:*H*→*C is nonexpansive, that is*, ||*P*_{ C }*x*-*P*_{ C }*y*|| ≤ ||*x*-*y*||, ∀*x*,*y*∈*H*; - (iii)
*P*_{ C }*is firmly nonexpansive, that is*, ||*P*_{ C }*x*-*P*_{ C }*y*||^{2}≤ 〈*P*_{ C }*x*-*P*_{ C }*y*,*x*-*y*〉, ∀*x*,*y*∈*H*; - (iv)
||

*tx*+ (1 -*t*)*y*||^{2}=*t*||*x*||^{2}+ (1 -*t*)||*y*||^{2}-*t*(1 -*t*)||*x*-*y*||^{2}, ∀*t*∈ [0, 1], ∀*x*,*y*∈*H*; - (v)
||

*x*+*y*||^{2}≤ ||*x*||^{2}+ 2〈*y*,*x*+*y*〉.

**Lemma 2.2**. [2]

*Let H be a Hilbert space, let C be a nonempty closed convex subset of H, and let B be a mapping of C into H. Let x** ∈

*C. Then, for λ >*0,

*where P*_{
C
} *is the metric projection of H onto C*.

**Lemma 2.3**. [2]*Let H be a Hilbert space, and let C be a nonempty closed convex subset of H. Let β >* 0, *and let A* : *C* → *H be β-inverse strongly monotone. If* 0 *<* ϱ ≤ 2*β, then I* -ϱ*A is a nonexpansive mapping of C into H, where I is the identity mapping on H*.

**Lemma 2.4**. *Let H be a real Hilbert space, let C be a nonempty closed convex subset of H, let S* : *C* → *C be a nonexpansive mapping, and let B* : *C* → *H be a ξ-inverse strongly monotone. If* 0 *< α*_{
n
} ≤ 2*ξ, then S* - *α*_{
n
}*BS is a nonexpansive mapping in H*.

Hence, *S* - *α*_{
n
}*BS* is a nonexpansive mapping of *C* into *H*. □

**Lemma 2.5**. [13]

*Let B be a monotone mapping of C into H and let N*

_{ C }

*w*

_{1}

*be the normal cone to C at w*

_{1}∈

*C, that is, N*

_{ C }

*w*

_{1}= {

*w*∈

*H*: 〈

*w*

_{1}-

*w*

_{2},

*w*〉 ≥ 0, ∀

*w*

_{2}∈

*C*}

*and define a mapping Q on C by*

*Then, Q is maximal monotone and* 0 ∈ *Qw*_{1}*if and only if w*_{1} ∈ *VI*(*C*, *B*).

**Lemma 2.6**. [14]

*Each Hilbert space H satisfies Opial's condition, that is, for any sequence*{

*x*

_{ n }} ⊂

*H with x*

_{ n }⇀

*x, the inequality*

*holds for each y* ∈ *H with y* ≠ *x*.

**Lemma 2.7**. [5]

*Let C be a nonempty closed convex subset of H and let f be a contraction of H into itself with coefficient η*∈ (0, 1)

*and A be a strongly positive linear-bounded operator on H with coefficient*.

*Then, for*,

*That is, A* - *γ f is strongly monotone with coefficient*
.

**Lemma 2.8**. [5]*Assume A to be a strongly positive linear-bounded operator on H with coefficient*
*and* 0 *< ρ* ≤ ||*A*||^{-1}. *Then*,
.

For solving the generalized mixed equilibrium problem and the mixed equilibrium problem, let us give the following assumptions for the bifunction *F*, the function *φ* and the set *C*:

(H1) *F*(*x*, *x*) = 0, ∀*x* ∈ *C*;

(H2) *F* is monotone, that is, *F*(*x*, *y*) + *F*(*y*, *x*) ≤ 0 ∀*x*, *y* ∈ *C*;

(H3) for each *y* ∈ *C*, *x* α *F*(*x*, *y*) is weakly upper semicontinuous;

(H4) for each *x* ∈ *C*, *y* α *F*(*x*, *y*) is convex;

(H5) for each *x* ∈ *C*, *y* α *F*(*x*, *y*) is lower semicontinuous;

*x*∈

*H*and

*λ >*0, there exist abounded subset

*G*

_{ x }⊆

*C*and

*y*

_{ x }∈

*C*such that for any

*z*∈

*C*\

*n G*

_{ x },

(B2) *C* is a bounded set.

**Lemma 2.9**. [15]

*Let C be a nonempty closed convex subset of H. Let F*:

*C*×

*C*→ ℝ

*be a bifunction satisfies (H1)*-

*(H5), and let*

*φ*:

*C*→ ℝ∪{+∞}

*be a proper lower semi continuous and convex function. Assume that either (B1) or (B2) holds. For λ*> 0

*and x*∈

*H, define a mapping*

*as follows:*

*Then, the following properties hold:*

- (i)
- (ii)
- (iii)
- (iv)
- (v)
*MEP*(*F*,*φ*)*is closed and convex*.

*Then*, lim_{n →∞}*a*_{
n
}= 0.

## 3. Main results

In this section, we are in a position to state and prove our main results.

**Theorem 3.1**.

*Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be bifunction from C*×

*C to*ℝ

*satisfying (H1)*-

*(H5), and let*

*φ*:

*C*→ ℝ ∪ {+∞}

*be a proper lower semicontinuous and convex function with either (B1) or (B2). Let B*,

*D be two ξ*,

*β-inverse strongly monotone mapping of C into H, respectively, and let S*:

*C*→

*C be a nonexpansive mapping. Let f*:

*C*→

*C be a contraction mapping with η*∈ (0, 1),

*and let A be a strongly positive linear-bounded operator with*

*and*.

*Assume that*Θ :=

*F*(

*S*) ∩

*VI*(

*C*,

*B*) ∩

*GMEP*(

*F*,

*φ*,

*D*) ≠ ∅.

*Let*{

*x*

_{ n }}, {

*y*

_{ n }}

*and*{

*u*

_{ n }}

*be sequences generated by the following iterative algorithm:*

*where* {*δn*} *and* {*β*_{
n
} } *are two sequences in* (0, 1) *satisfying the following conditions:*

(C1) lim_{n →∞}*β*_{
n
}= 0 *and*
,

(C2) {*δ*_{
n
} } ⊂ [0, *b*], *for some b* ∈ (0, 1) *and* lim_{n →∞}|*δ*_{n+1}- *δ*_{
n
} | = 0,

(C3) {*λ*_{
n
} } ⊂ [*c*, *d*] ⊂ (0, 2*β*) *and* lim_{n →∞}|*λ*_{n+1}- *λ*_{
n
} | = 0,

(C4) {*α*_{
n
} } ⊂ [*e*, *g*] ⊂ (0, 2*ξ*) *and* lim_{n →∞}|*α*_{n+1}- *α*_{
n
} | = 0.

*Then*, {

*x*

_{ n }}

*converges strongly to z*∈ Θ,

*which is the unique solution of the variational inequality*

*Proof*. We may assume, in view of *β*_{
n
} → 0 as *n* → ∞, that *β*_{
n
} ∈ (0, ||*A*||^{-1}). By Lemma 2.8, we obtain
, ∀ _{
n
} ∈ ℕ.

We divide the proof of Theorem 3.1 into six steps.

**Step 1**. We claim that the sequence {*x*_{
n
} } is bounded.

*z*

_{ n }=

*PC*(

*Su*

_{ n }-

*α*

_{ n }

*BSu*

_{ n }) and

*S*-

*α*

_{ n }

*BS*be a nonexpansive mapping. Then, we have from Lemma 2.4 that

*w*

_{ n }=

*P*

_{ C }(

*Sy*

_{ n }-

*α*

_{ n }

*BSy*

_{ n }) in (3.4). Then, we can prove that

This shows that {*x*_{
n
} } is bounded. Hence, {*u*_{
n
} }, {*z*_{
n
} }, {*y*_{
n
} }, {*w*_{
n
} }, {*BSu*_{
n
} }, {*BSy*_{
n
} }, {*Az*_{
n
} } and {*f*(*x*_{
n
} )} are also bounded.

**Step 2**. We claim that lim_{n→∞}||*x*_{n+1}- *x*_{
n
} || = 0.

*c*∈ ℝ such that

*λ*

_{ n }

*> c >*0, ∀

*n*≥ 1. Then, we have

*w*

_{ n }=

*P*

_{ C }(

*Sy*

_{ n }-

*α*

_{ n }

*BSy*

_{ n }) and

*S*-

*α*

_{ n }

*BS*is nonexpansive mapping, we have

**Step 3**. We claim that lim_{n→∞}||*Sw*_{
n
} - *w*_{
n
} || = 0.

*x*

_{n+1}-

*x*

_{ n }|| → 0, ||

*Dx*

_{ n }-

*D*

_{ p }|| → 0,

*β*

_{ n }→ 0 as

*n*→ ∞ and the condition on

*λ*

_{ n }implies that

Next, we will show that ||*x*_{
n
} - *y*_{
n
} || → 0 as *n* → ∞.

We consider *x*_{n+1}- *y*_{
n
} = *δ*_{
n
} (*w*_{
n
} - *y*_{
n
} ) = *δn*(*w*_{
n
} - *z*_{
n
} + *z*_{
n
} - *y*_{
n
} ).

**Step 4**. We prove that the mapping *P*_{Θ}(γ*f* + (*I* - *A*)) has a unique fixed point.

*P*

_{Θ}(γ

*f*+ (

*I*-

*A*)) is a contraction of

*C*into itself. Therefore, by the Banach Contraction Mapping Principle, it has a unique fixed point, say

*z*∈

*C*, that is,

**Step 5**. We claim that *q* ∈ *F*(*S*) ∩ VI(*C*, *B*) ∩ GMEP(*F*, *φ*, *D*).

First, we show that *q* ∈ *F*(*S*).

This is a contradiction. Thus, we have *q* ∈ *F*(*S*).

Next, we prove that *q* ∈ GMEP(*F*, *φ*, *D*).

*y*

_{ t }=

*t*

_{ y }+ (1 -

*t*)

*q*for all

*t*∈ (0, 1] and

*y*∈

*C*. Since

*y*∈

*C*and

*q*∈

*C*, we obtain

*y*

_{ t }∈

*C*. Hence, from (3.49), we have

This implies that *q* ∈ GMEP(*F*, *φ*, *D*).

Finally, we prove that *q* ∈ VI(*C*, *B*).

*Q*is maximal monotone. Let (

*q*

_{1},

*q*

_{2}) ∈

*G*(

*Q*). Since

*q*

_{2}-

*Bq*

_{1}∈

*N*

_{ C }

*q*

_{1}and

*w*

_{ n }∈

*C*, we have 〈

*q*

_{1}-

*w*

_{ n },

*q*

_{2}

*- Bq*

_{1}〉 ≥ 0. On the other hand, from

*w*

_{ n }=

*P*

_{ C }(

*Sy*

_{ n }-

*α*

_{ n }

*BSy*

_{ n }), we have

*Q*is maximal monotone, we obtain that

*q*∈

*Q*

^{-1}0, and hence

*q*∈ VI(

*C*,

*B*). This implies

*q*∈ Θ. Since

*z*=

*P*

_{Θ}(

*γf*+ (

*I*-

*A*))(

*z*), we have

**Step 6**. Finally, we claim that *x*_{
n
} → *z*, where *z* = *P*_{Θ}(*γf* + (*I* - *A*))(*z*).

where *K* is an appropriate constant such that *K* ≥ sup_{n≥1}{||*x*_{
n
} - *z*||^{2}}.

Therefore, applying Lemma 2.10 to (3.58), we get that {*x*_{
n
} } converges strongly to *z* ∈ Θ.

This completes the proof. □

**Corollary 3.2**.

*Let C be a nonempty closed convex subset of a real Hilbert space H, let B be ξ-inverse-strongly monotone mapping of C into H, and let S*:

*C*→

*C be a nonexpansive mapping. Let f*:

*C*→

*C be a contraction mapping with η*∈ (0, 1),

*and let A be a strongly positive linear-bounded operator with*

*and*.

*Assume that*Θ :=

*F*(

*S*) ∩

*VI*(

*C*,

*B*) ≠ ∅.

*Let*{

*x*

_{ n }}

*and*{

*y*

_{ n }}

*be sequence generated by the following iterative algorithm:*

*where* {*δ*_{
n
} } *and* {*β*_{
n
} } *are two sequences in* (0, 1) *satisfying the following conditions:*

(C1) lim_{n → ∞}*β*_{
n
}= 0 *and*
,

(C2) {*δ*_{
n
} } ⊂ [0, *b*], *for some b* ∈ (0, 1) *and* lim_{n → ∞}|*δ*_{n+1}- *δ*_{
n
} | = 0,

(C3) {*α*_{
n
} } ⊂ [*e*, *g*] ⊂ (0, 2*ξ*) *and* lim_{n → ∞}|*α*_{n+1}- *α*_{
n
} | = 0.

*Then*, {

*x*

_{ n }}

*converges strongly to z*∈ Θ,

*which is the unique solution of the variational inequality*

*Proof*. Put *F*(*x*, *y*) = *φ* = *D* = 0 for all *x*, *y* ∈ *C* and *λ*_{
n
} = 1 for all *n* ≥ 1 in Theorem 3.1, we get *u*_{
n
} = *x*_{
n
} . Hence, {*x*_{
n
} } converges strongly to *z* ∈ Θ, which is the unique solution of the variational inequality (3.59). □

**Corollary 3.3**. [12]

*Let C be a nonempty closed convex subset of a real Hilbert space H and let F be bifunction from C*×

*C to*ℝ

*satisfying (H1)-(H5). Let S*:

*C*→

*C be a nonexpansive mapping and let f*:

*C*→

*C be a contraction mapping with η*∈ (0, 1).

*Assume that*Θ :=

*F*(

*S*) ∩

*EP*(

*F*) ≠ ∅.

*Let*{

*x*

_{ n }}, {

*y*

_{ n }}

*and*{

*u*

_{ n }}

*be sequence generated by the following iterative algorithm:*

*where* {*δ*_{
n
} } *and* {*β*_{
n
} } *are two sequences in* (0, 1) *and* {*λ*_{
n
} } ⊂ (0, ∞) *satisfying the following conditions:*

(C1) lim_{n → ∞}*β*_{
n
}= 0 *and*
,

(C2) {*δ*_{
n
} } ⊂ [0, *b*], *for some b* ∈ (0, 1) *and* lim_{n → ∞}|*δ*_{n+1}- *δ*_{
n
} | = 0,

(C3) lim_{n → ∞}|*λ*_{n+1}- *λ*_{
n
} | = 0.

*Then*, {*x*_{
n
} } *converges strongly to z* ∈ Θ.

*Proof*. Put *φ* = *D* = 0, *γ* = 1, *A* = *I* and *α*_{
n
} = 0 in Theorem 3.1. Then, we have *P*_{
C
} (*Su*_{
n
} ) = *Su*_{
n
} and *P*_{
C
} (*Sy*_{
n
} ) = *Sy*_{
n
} . Hence, {*x*_{
n
} } generated by (3.60) converges strongly to *z* ∈ Θ. □

## Declarations

### 6. Acknowledgements

This research was partially supported by the Research Fund, Rajamangala University of Technology Rattanakosin. The first author was supported by the 'Centre of Excellence in Mathematics', the Commission on High Education, Thailand for Ph.D. program at King Mongkuts University of Technology Thonburi (KMUTT). The second author was supported by Rajamangala University of Technology Rattanakosin Research and Development Institute, the Thailand Research Fund and the Commission on Higher Education under Grant No. MRG5480206. The third author was supported by the NRU-CSEC Project No. 54000267. Helpful comments by anonymous referees are also acknowledged.

## Authors’ Affiliations

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