- Research
- Open Access

# Some extragradient methods for common solutions of generalized equilibrium problems and fixed points of nonexpansive mappings

- Jian-Wen Peng
^{1}Email author

**2011**:12

https://doi.org/10.1186/1687-1812-2011-12

© Peng; licensee Springer. 2011

**Received:**30 November 2010**Accepted:**29 June 2011**Published:**29 June 2011

## Abstract

In this article, we introduce some new iterative schemes based on the extragradient method (and the hybrid method) for finding a common element of the set of solutions of a generalized equilibrium problem, and the set of fixed points of a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for a monotone, Lipschitz-continuous mapping in Hilbert spaces. We obtain some strong convergence theorems and weak convergence theorems. The results in this article generalize, improve, and unify some well-known convergence theorems in the literature.

## Keywords

- Generalized equilibrium problem
- Extragradient method
- Hybrid method
- Nonex-pansive mapping
- Strong convergence
- Weak convergence

## 1. Introduction

*H*be a real Hilbert space with inner product 〈.,.〉 and induced norm ||·||. Let

*C*be a nonempty closed convex subset of

*H*. Let

*F*be a bifunction from

*C × C*to

*R*and let

*B*:

*C*→

*H*be a nonlinear mapping, where

*R*is the set of real numbers. Moudafi [1], Moudafi and Thera [2], Peng and Yao [3, 4], Takahashi and Takahashi [5] considered the following generalized equilibrium problem:

*GEP*(

*F*,

*B*). If

*B*= 0, the generalized equilibrium problem (1.1) becomes the equilibrium problem for

*F*:

*C × C*→

*R*, which is to find

*x*∈

*C*such that

The set of solutions of (1.2) is denoted by *EP*(*F*).

The problem (1.1) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games, and others; see for instance [1–7].

We denote the set of fixed points of *S* by Fix(*S*).

*A*:

*C*→

*H*be monotone and

*k*-Lipschitz-continuous. The variational inequality problem is to find

*x*∈

*C*such that

for all *y* ∈ *C*. The set of solutions of the variational inequality problem is denoted by *V I*(*C*, *A*).

Several algorithms have been proposed for finding the solution of problem (1.1). Moudafi [1] introduced an iterative scheme for finding a common element of the set of solutions of problem (1.1) and the set of fixed points of a nonexpansive mapping in a Hilbert space, and proved a weak convergence theorem. Moudafi and Thera [2] introduced an auxiliary scheme for finding a solution of problem (1.1) in a Hilbert space and obtained a weak convergence theorem. Peng and Yao [3, 4] introduced some iterative schemes for finding a common element of the set of solutions of problem (1.1), the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for a monotone, Lipschitz-continuous mapping and obtain both strong convergence theorems, and weak convergence theorems for the sequences generated by the corresponding processes in Hilbert spaces. Takahashi and Takahashi [5] introduced an iterative scheme for finding a common element of the set of solutions of problem (1.1) and the set of fixed points of a nonexpansive mapping in a Hilbert space, and proved a strong convergence theorem.

Some methods also have been proposed to solve the problem (1.2); see, for instance, [8–19] and the references therein. Takahashi and Takahashi [9] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a non-expansive mapping, and proved a strong convergence theorem in a Hilbert space. Su et al. [10] introduced and researched an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for an *α*-inverse-strongly monotone mapping in a Hilbert space. Tada and Takahashi [11] introduced two iterative schemes for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping in a Hilbert space, and obtained both strong convergence and weak convergence theorems. Plubtieng and Punpaeng [12] introduced an iterative processes based on the extragradient method for finding the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the set of solutions of variational inequality problem for an *α*-inverse-strongly monotone mapping. Chang et al. [13] introduced an iterative processes based on the extragradient method for finding the common element of the set of solutions of an equilibrium problem, the set of common fixed point for a family of infinitely nonexpansive mappings and the set of solutions of variational inequality problem for an *α*-inverse-strongly monotone mapping. Yao et al. [14] and Ceng and Yao [15] introduced some iterative viscosity approximation schemes for finding the common element of the set of solutions of problem (1.2) and the set of fixed points of a family of infinitely nonexpansive mappings in a Hilbert space. Colao et al. [16] introduced an iterative viscosity approximation scheme for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a family of finitely nonexpansive mappings in a Hilbert space. We observe that the algorithms in [13–16] involves the *W*-mapping generated by a family of infinitely (finitely) nonexpansive mappings which is an effective tool in nonlinear analysis (see [20, 21]). However, the *W*-mapping generated by a family of infinitely (finitely) nonexpansive mappings is too completed to use for finding the common element of the set of solutions of problem (1.2) and the set of fixed points of a family of infinitely (finitely) nonexpansive mappings. It is natural to raise and to give an answer to the following question: Can one construct algorithms for finding a common element of the set of solutions of a generalized equilibrium problem (an equilibrium problem), the common set of fixed points of a family of infinitely nonexpansive mappings and the set of solutions of a variational inequality without the *W*-mapping generated by a family of infinitely (finitely) nonexpansive mappings? In this article, we will give a positive answer to this question.

Recently, OHaraa et al. [22] introduced and researched an iterative approach for finding a nearest point of infinitely many nonexpansive mappings in a Hilbert spaces without using the *W*-mapping generated by a family of infinitely (finitely) nonexpansive mappings. Inspired by the ideas in [1–6, 8–16, 22] and the references therein, we introduce some new iterative schemes based on the extragradient method (and the hybrid method) for finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a family of infinitely nonexpansive mappings, and the set of solutions of the variational inequality for a monotone, Lipschitz--continuous mapping without using the *W*-mapping generated by a family of infinitely (finitely) nonexpansive mappings. We obtain both strong convergence theorems and weak convergence theorems for the sequences generated by the corresponding processes. The results in this article generalize, improve, and unify some well-known convergence theorems in the literature.

## 2. Preliminaries

*H*be a real Hilbert space with inner product 〈·,·〉 and norm ||·||. Let

*C*be a nonempty closed convex subset of

*H*. Let symbols → and ⇀ denote strong and weak convergences, respectively. In a real Hilbert space

*H*, it is well known that

for all *x*, *y* ∈ *H* and *λ* ∈ [0, 1].

*x*∈

*H*, there exists the unique nearest point in

*C*, denoted by

*P*

_{ C }(

*x*), such that ||

*x - P*

_{ C }(

*x*)|| ≤ ||

*x - y*|| for all

*y*∈

*C*. The mapping

*P*

_{ C }is called the metric projection of

*H*onto

*C*. We know that

*P*

_{ C }is a nonexpansive mapping from

*H*onto

*C*. It is also known that

*P*

_{ C }

*x*∈

*C*and

for all *x* ∈ *H* and *y* ∈ *C*.

for all *x* ∈ *H* and *y* ∈ *C*.

*x*,

*y*∈

*C*. A mapping

*A*of

*C*into

*H*is called

*α*-inverse-strongly monotone if there exists a positive real number

*α*such that

*x*,

*y*∈

*C*. A mapping

*A*:

*C*→

*H*is called

*k*-Lipschitz-continuous if there exists a positive real number

*k*such that

for all *x*, *y* ∈ *C*. It is easy to see that if *A* is *α*-inverse-strongly monotone, then *A* is monotone and Lipschitz-continuous. The converse is not true in general. The class of *α*-inverse-strongly monotone mappings does not contain some important classes of mappings even in a finite-dimensional case. For example, if the matrix in the corresponding linear complementarity problem is positively semidefinite, but not positively definite, then the mapping *A* will be monotone and Lipschitz-continuous, but not *α*-inverse-strongly monotone (see [23]).

*A*be a monotone mapping of

*C*into

*H*. In the context of the variational inequality problem, the characterization of projection (2.1) implies the following:

*H*satisfies the Opial's condition [24], i.e., for any sequence {

*x*

_{ n }} ⊂

*H*with

*x*

_{ n }⇀

*x*, the inequality

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

*T*:

*H*→ 2

^{ H }is called monotone if for all

*x*,

*y*∈

*H*,

*f*∈

*Tx*and

*g*∈

*Ty*imply 〈

*x - y*,

*f - g*〉 ≥ 0. A monotone mapping

*T*:

*H*→ 2

^{ H }is maximal if its graph

*G*(

*T*) of

*T*is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping

*T*is maximal if and only if for (

*x*,

*f*) ∈

*H*×

*H*, 〈

*x - y*,

*f - g*〉 ≥ 0 for every (

*y*,

*g*) ∈

*G*(

*T*) implies

*f*∈

*Tx*. Let A be a monotone,

*k*-Lipschitz-continuous mapping of

*C*into

*H*and

*N*

_{ C }

*v*be normal cone to

*C*at

*v*∈

*C*, i.e.,

*N*

_{ C }

*v*= {

*w*∈

*H*: 〈

*v - u*,

*w*〉 ≥ 0, ∀

*u*∈

*C*}. Define

Then, *T* is maximal monotone and 0 ∈ *Tv* if and only if *v* ∈ *V I*(*C*, *A*) (see [25]).

For solving the equilibrium problem, let us assume that the bifunction *F* satisfies the following condition:

(A1) *F*(*x*, *x*) = 0 for all *x* ∈ *C*;

(A2) *F* is monotone, i.e., *F*(*x*, *y*) + *F*(*y*, *x*) ≤ 0 for any *x*, *y* ∈ *C*;

(A4) for each *x* ∈ *C*, *y* ↦ *F*(*x*, *y*) is convex and lower semicontinuous.

We recall some lemmas which will be needed in the rest of this article.

**Lemma 2.1**.[7] Let

*C*be a nonempty closed convex subset of

*H*, let

*F*be a bifunction from

*C × C*to

*R*satisfying (A1)-(A4). Let

*r >*0 and

*x*∈

*H*. Then, there exists

*z*∈

*C*such that

**Lemma 2.2**.[8] Let

*C*be a nonempty closed convex subset of

*H*, let

*F*be a bi-function from

*C × C*to

*R*satisfying (A1)-(A4). For

*r >*0 and

*x*∈

*H*, define a mapping

*Tr*:

*H*→

*C*as follows:

## 3. The main results

We first show a strong convergence of an iterative algorithm based on extragradient and hybrid methods which solves the problem of finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a family of infinitely nonexpansive mappings, and the set of solutions of the variational inequality for a monotone, Lipschitz-continuous mapping in a Hilbert space.

**Theorem 3.1**. Let

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

*H*. Let

*F*be a bifunction from

*C × C*to

*R*satisfying (A1)-(A4). Let

*A*be a monotone and

*k*-Lipschitz-continuous mapping of

*C*into

*H*and

*B*be an

*α*-inverse-strongly monotone mapping of

*C*into

*H*. Let

*S*

_{1},

*S*

_{2},... be a family of infinitely nonexpansive mappings of

*C*into itself such that . Assume that for all

*i*∈ {1, 2,...} and for any bounded subset

*K*of

*C*, thenthere holds

*n*= 1, 2,... where {

*λ*

_{ n }} ⊂ [

*a*,

*b*] for some , {

*r*

_{ n }} ⊂ [

*d*,

*e*] for some

*d*,

*e*∈ (0, 2

*α*), and {

*α*

_{ n }}, {

*β*

_{ n }}, {

*γ*

_{ n }} are three sequences in [0, 1] satisfying the conditions:

- (i)
*α*_{ n }+*β*_{ n }≤ 1 for all*n*∈*N*; - (ii)
;

- (iii)
;

- (iv)
and for all

*n*∈*N*;

Then, {*x*_{
n
} }, {*u*_{
n
} }, {*y*_{
n
} } and {*z*_{
n
} } converge strongly to *w* = *P*_{Ω}(*x*).

**Proof**. It is obvious that

*C*

_{ n }is closed, and

*Q*

_{ n }is closed and convex for every

*n*= 1, 2,.... Since

*C*

_{ n }is convex for every

*n*= 1, 2,.... It is easy to see that 〈

*x*

_{ n }

*- z*,

*x - x*

_{ n }〉 ≥ 0 for all

*z*∈

*Q*

_{ n }and by (2.1), . Let

*t*

_{ n }=

*P*

_{ C }(

*u*

_{ n }-

*λ*

_{ n }

*Ay*

_{ n }) for every

*n*= 1, 2,.... Let

*u*∈ Ω and let >be a sequence of mappings defined as in Lemma 2.2. Then . From and the

*α*-inverse strongly monotonicity of

*B*, we have

*y*

_{ n }= (1

*- γ*

_{ n })

*u*

_{ n }+

*γ*

_{ n }

*P*

_{ C }(

*u*

_{ n }

*- λ*

_{ n }

*Au*

_{ n }) and

*A*is

*k*-Lipschitz-continuous, we have

*z*

_{ n }= (1

*- α*

_{ n }

*- β*

_{ n })

*x*

_{ n }+

*α*

_{ n }

*y*

_{ n }+

*β*

_{ n }

*S*

_{ n }

*t*

_{ n }and

*u*=

*S*

_{ n }

*u*, we have

for every *n* = 1, 2,... and hence *u* ∈ *C*_{
n
} . So, Ω ⊂ *C*_{
n
} for every *n* = 1, 2,.... Next, let us show by mathematical induction that *x*_{
n
} is well defined and Ω ⊂ *C*_{
n
} ∩ *Q*_{
n
} for every *n* = 1, 2,.... For *n* = 1 we have *x*_{1} = *x* ∈ *C* and *Q*_{1} = *C*. Hence, we obtain Ω ⊂*C*_{1} ∩ *Q*_{1}. Suppose that *x*_{
k
} is given and Ω ⊂ *C*_{
k
} ∩ *Q*_{
k
} for some *k* ∈ *N*. Since Ω is nonempty, *C*_{
k
} ∩ *Q*_{
k
} is a nonempty closed convex subset of *H*. Hence, there exists a unique element *x*_{k+1}∈ *C*_{
k
}∩ *Q*_{
k
}such that
. It is also obvious that there holds 〈*x*_{k+1}- *z*, *x* - *x*_{k+1}〉 ≥ 0 for every *z* ∈ *C*_{
k
}∩ *Q*_{
k
}. Since Ω ⊂ *C*_{
k
} ∩ *Q*_{
k
} , we have 〈*x*_{k+1}- *z*, *x* - *x*_{k+1}〉 ≥ 0 for every *z* ∈ Ω and hence Ω ⊂ *Q*_{k+1}. Therefore, we obtain Ω ⊂ *C*_{k+1}∩ *Q*_{k+1}.

*n*= 1, 2,.... Therefore, {

*x*

_{ n }} is bounded. From (3.2) to (3.5) and the lipschitz continuity of

*A*, we also obtain that {

*u*

_{ n }}, {

*y*

_{ n }}, {

*Au*

_{ n }}, {

*t*

_{ n }} and {

*z*

_{ n }} are bounded. Since

*x*

_{n+1}∈

*C*

_{ n }∩

*Q*

_{ n }⊂

*C*

_{ n }and , we have

for every *n* = 1, 2,.... It follows from (3.6) that lim_{n→∞}||*x*_{
n
}- *x*|| exists.

*x*

_{n+1}∈

*C*

_{ n }, we have ||

*z*

_{ n }-

*x*

_{n+1}||

^{2}≤ ||

*x*

_{ n }-

*x*

_{n+1}||

^{2}+ (3 - 3

*γ*

_{ n }+

*α*

_{ n })

*b*

^{2}||

*Au*

_{ n }||

^{2}and hence it follows from lim

_{n→∞}

*γ*

_{ n }= 1 and lim

_{n→∞}

*α*

_{ n }= 0 that lim

_{n→∞}||

*z*

_{ n }-

*x*

_{n+1}|| = 0. Since

for every *n* = 1, 2,..., we have ||*x*_{
n
} *- z*_{
n
} || → 0.

_{n→∞}

*γ*

_{ n }= 1 and lim

_{n→∞}

*α*

_{ n }= 0, {

*x*

_{ n }}, {

*y*

_{ n }}, {

*Au*

_{ n }}, and {

*z*

_{ n }} are bounded, we have

Thus, lim_{n→∞}||*t*_{
n
} *- u*||^{2}*-* ||*x*_{
n
} *- u*||^{2} = 0.

The assumptions on *γ*_{
n
} and *λ*_{
n
} imply that
and
. Consequently, lim_{n→∞}||*u*_{
n
} *- y*_{
n
} || = lim_{n→∞}||*t*_{
n
} *- y*_{
n
} || = 0. Since *A* is Lipschitz-continuous, we have lim_{n→∞}||*At*_{
n
} *- Ay*_{
n
} || = 0. It follows from ||*u*_{
n
} *- t*_{
n
} || ≤ ||*u*_{
n
} *- y*_{
n
} || + ||*t*_{
n
} *- y*_{
n
} || that lim_{n→∞}||*u*_{
n
} *- t*_{
n
} || = 0.

From lim_{n→∞}||*z*_{
n
} *- x*_{
n
} || = 0, lim_{n→∞}*α*_{
n
}= 0, the boundedness of {*x*_{
n
} }, {*y*_{
n
}} and lim inf_{n→∞}*β*_{
n
}> 0 we infer that lim_{n→∞}||*S*_{
n
}*t*_{
n
}- *x*_{
n
}|| = 0.

Since lim_{n→∞}*α*_{
n
}= 1, lim inf_{n→∞}*β*_{
n
}> 0, lim_{n→∞}*γ*_{
n
}= 1, ||*x*_{
n
} *- z*_{
n
} || → 0 and the sequences {*x*_{
n
} } and {*z*_{
n
} } are bounded, we obtain ||*Bx*_{
n
} *- B*_{
u
} || → 0.

Since lim_{n→∞}*α*_{
n
}= 0, lim inf_{n→∞}*β*_{
n
}> 0, lim_{n→∞}*γ*_{
n
}= 1, ||*x*_{
n
}- *z*_{
n
}|| → 0, ||*Bx*_{
n
}- *Bu*|| → 0 and the sequences {*x*_{
n
}}, {*u*_{
n
}} and {*z*_{
n
}} are bounded, we obtain ||*x*_{
n
}- *u*_{
n
}|| → 0. From ||*z*_{
n
} *- t*_{
n
} || ≤ ||*z*_{
n
} *- x*_{
n
} ||+||*x*_{
n
} *- u*_{
n
} ||+||*u*_{
n
} *- t*_{
n
} ||, we have ||*z*_{
n
} *- t*_{
n
} || → 0.

From ||*t*_{
n
} *- x*_{
n
} || ≤ ||*t*_{
n
} *- u*_{
n
} || + ||*x*_{
n
} *- u*_{
n
} ||, we also have ||*t*_{
n
} *- x*_{
n
} || → 0.

*z*

_{ n }= (1

*- α*

_{ n }-

*β*

_{ n })

*x*

_{ n }+

*α*

_{ n }

*y*

_{ n }+

*β*

_{ n }

*S*

_{ n }

*t*

_{ n }, we have

*β*

_{ n }(

*S*

_{ n }

*t*

_{ n }

*- t*

_{ n }) = (1

*- α*

_{ n }

*- β*

_{ n })(

*t*

_{ n }

*- x*

_{ n }) +

*α*

_{ n }(

*t*

_{ n }

*- y*

_{ n }) + (

*z*

_{ n }

*- t*

_{ n }). Then

As {*x*_{
n
} } is bounded, there exists a subsequence
of {*x*_{
n
} } such that *x*_{
ni
} ⇀ *w*. From ||*x*_{
n
} *- u*_{
n
} || → 0, we obtain that *u*_{
ni
} ⇀ *w*. From ||*u*_{
n
} *- t*_{
n
} || → 0, we also obtain that *t*_{
ni
} ⇀ *w*. Since {*u*_{
ni
} } ⊂ *C* and *C* is closed and convex, we obtain *w* ∈ *C*.

*t*with 0

*< t*≤ 1 and

*y*∈

*C*, let

*y*

_{ t }=

*t*

_{ y }+ (1

*- t*)

*w*. Since

*y*∈

*C*and

*w*∈

*C*, we obtain

*y*

_{ t }∈

*C*. So, from (3.10) we have

*B*, we have . Hence, from (A4), and , we have

This implies that *w* ∈ *GEP*(*F*, *B*).

*i*

_{0}∈ {1, 2,...} from the Opial condition, we have

This is a contradiction. Hence, we get .

where *N*_{
C
}*v* is the normal cone to *C* at *v* ∈ *C*. We have already mentioned that in this case the mapping *T* is maximal monotone, and 0 ∈ *Tv* if and only if *v* ∈ *V I*(*C*, *A*). Let (*v*, *g*) ∈ *G*(*T*). Then *Tv* = *Av* + *N*_{
C
}*v* and hence *g - Av* ∈ *N*_{
C
}*v*.

*v - t*,

*g - Av*〉 ≥ 0 for all

*t*∈

*C*. On the other hand, from

*t*

_{ n }=

*P*

_{ C }(

*u*

_{ n }

*- λ*

_{ n }

*Ay*

_{ n }) and

*v*∈

*C*, we have

Hence, we obtain 〈*v - w*, *g*〉 ≥ 0 as *i* → *∞*. Since *T* is maximal monotone, we have *w* ∈ *T*^{-1}0 and hence *w* ∈ *V I*(*C*, *A*). This implies that *w* ∈ Ω.

As *i* → *∞*, we obtain *-* ||*l*_{0}*- w*||^{2} ≥ 〈*l*_{0}*- w*, *x - l*_{0}〉 ≥ 0 by *l*_{0} = *P*_{Ω}*x* and *w* ∈ Ω. Hence, we have *w* = *l*_{0}. This implies that *x*_{
n
} → *l*_{0}. It is easy to see *u*_{
n
} → *l*_{0}, *y*_{
n
} → *l*_{0} and *z*_{
n
} → *l*_{0}. The proof is now complete.

By combining the arguments in the proof of Theorem 3.1 and those in the proof of Theorem 3.1 in [3], we can easily obtain the following weak convergence theorem for an iterative algorithm based on the extragradient method which solves the problem of finding a common element of the set of solutions of a generalized equilibrium problem, the set of fixed points of a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for a monotone, Lipschitz-continuous mapping in a Hilbert space.

**Theorem 3.2**. Let

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

*H*. Let

*F*be a bifunction from

*C × C*to

*R*satisfying (A1)-(A4). Let

*A*be a monotone, and

*k*-Lipschitz-continuous mapping of

*C*into

*H*and

*B*be an

*α*-inverse-strongly monotone mapping of

*C*into

*H*. Let

*S*

_{1},

*S*

_{2},... be a family of infinitely nonexpansive mappings of

*C*into itself such that . Assume that for all

*i*∈ {1, 2,...} and for any bounded subset

*K*of

*C*, thenthere holds

for every *n* = 1, 2,.... If {*λ*_{
n
} } ⊂ [*a*, *b*] for some
, {*β*_{
n
} } ⊂ [*δ*, *ε*] for some *δ*, *ε* ∈ (0, 1) and {*r*_{
n
} } ⊂ [*d*, *e*] for some *d*, *e* ∈ (0, 2*α*). Then, {*x*_{
n
} }, {*u*_{
n
} } and {*y*_{
n
} } converge weakly to *w* ∈ Ω, where *w* = lim_{n→∞}*P*_{Ω}*x*_{
n
}.

## 4. Applications

By Theorems 3.1 and 3.2, we can obtain many new and interesting convergence theorems in a real Hilbert space. We give some examples as follows:

Let *A* = 0, by Theorems 3.1 and 3.2, respectively, we obtain the following results.

**Theorem 4.1**. Let

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

*H*. Let

*F*be a bifunction from

*C×C*to

*R*satisfying (A1)-(A4). Let

*B*be an

*α*-inverse-strongly monotone mapping of

*C*into

*H*. Let

*S*

_{1},

*S*

_{2},... be a family of infinitely nonexpansive mappings of

*C*into itself such that . Assume that for all

*i*∈ {1, 2,...} and for any bounded subset

*K*of

*C*, thenthere holds

*n*= 1, 2,.... where {

*r*

_{ n }} ⊂ [

*d*,

*e*] for some

*d*,

*e*∈ (0, 2

*α*), and {

*α*

_{ n }}, {

*β*

_{ n }} are sequences in [0, 1] satisfying the conditions:

- (i)
*α*_{ n }+*β*_{ n }≤ 1 for all*n*∈*N*; - (ii)
;

- (iii)
for all

*n*∈*N*;

Then, {*x*_{
n
} }, {*u*_{
n
} }, and {*z*_{
n
} } converge strongly to *w* = *P*_{∑}(*x*).

**Theorem 4.2**. Let

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

*H*. Let

*F*be a bifunction from

*C×C*to

*R*satisfying (A1)-(A4). Let

*B*be an

*α*-inverse-strongly monotone mapping of

*C*into

*H*. Let

*S*

_{1},

*S*

_{2},... be a family of infinitely nonexpansive mappings of

*C*into itself such that . Assume that for all

*i*∈ {1, 2,...} and for any bounded subset

*K*of

*C*, thenthere holds

for every *n* = 1, 2,.... If {*β*_{
n
} } ⊂ [*δ*, *ε*] for some *δ*, *ε* ∈ (0, 1) and {*r*_{
n
} } ⊂ [*d*, *e*] for some *d*, *e* ∈ (0, 2*α*). Then, {*x*_{
n
} } and {*u*_{
n
} } converge weakly to *w* ∈ ∑, where *w* = lim_{n→∞}*P*_{∑}*x*_{
n
} .

**Theorem 4.3**. Let

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

*H*. Let

*F*be a bifunction from

*C×C*to

*R*satisfying (A1)-(A4). Let

*A*be a monotone and

*k*-Lipschitz-continuous mapping of

*C*into

*H*and

*B*be an

*α*-inverse-strongly monotone mapping of

*C*into

*H*. Let

*S*

_{1},

*S*

_{2},... be a family of infinitely nonexpansive mappings of

*C*into itself such that . Assume that for all

*i*∈ {1, 2,...} and for any bounded subset

*K*of

*C*, thenthere holds

for every *n* = 1, 2,... where {*λ*_{
n
} } ⊂ [*a*, *b*] for some
, {*r*_{
n
} } ⊂ [*d*, *e*] for some *d*, *e* ∈ (0, 2*α*), and {*β*_{
n
} } is a sequence in [0, 1] satisfying
. Then, {*x*_{
n
} }, {*u*_{
n
} }, {*y*_{
n
} }, and {*z*_{
n
} } converge strongly to *w* = *P*_{Ω}(*x*).

**Proof**. Putting *γ*_{
n
} = 1 and *α*_{
n
} = 0, by Theorem 3.1, we obtain the desired result.

Let *B* = 0, by Theorems 3.1, 3.2, and 4.3, we obtain the following results.

**Theorem 4.4**. Let

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

*H*. Let

*F*be a bifunction from

*C×C*to

*R*satisfying (A1)-(A4). Let

*A*be a monotone and

*k*-Lipschitz-continuous mapping of

*C*into

*H*. Let

*S*

_{1},

*S*

_{2},... be a family of infinitely nonexpansive mappings of

*C*into itself such that . Assume that for all

*i*∈ {1, 2,...}, and for any bounded subset

*K*of

*C*, there holds

*n*= 1, 2,.... where {

*λ*

_{ n }} ⊂ [

*a*,

*b*] for some , {

*r*

_{ n }} ⊂ [

*d*,

*+*∞) for some

*d >*0, and {

*α*

_{ n }}, {

*β*

_{ n }}, {

*γ*

_{ n }} are three sequences in [0, 1] satisfying the following conditions:

- (i)
*α*_{ n }+*β*_{ n }≤ 1 for all*n*∈*N*; - (ii)
;

- (iii)
;

- (iv)
and for all

*n*∈*N*;

Then, {*x*_{
n
} }, {*u*_{
n
} }, {*y*_{
n
} } and {*z*_{
n
} } converge strongly to *w* = *P*_{Λ}(*x*).

**Theorem 4.5**. Let

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

*H*. Let

*F*be a bifunction from

*C×C*to

*R*satisfying (A1)-(A4). Let

*A*be a monotone and

*k*-Lipschitz-continuous mapping of

*C*into

*H*. Let

*S*

_{1},

*S*

_{2},... be a family of infinitely nonexpansive mappings of

*C*into itself such that . Assume that for all

*i*∈ {1, 2,...} and for any bounded subset

*K*of

*C*, thenthere holds

for every *n* = 1, 2,.... If {*λ*_{
n
} } ⊂ [*a*, *b*] for some
,{*β*_{
n
} } ⊂ [*δ*, *ε*], for some *δ*, *ε* ∈ (0, 1) and {*r*_{
n
} } ⊂ [*d*, *+*∞] for some *d* > 0, then {*x*_{
n
} }, {*u*_{
n
} } and {*y*_{
n
} } converge weakly to *w* ∈ Λ, where *w* = lim_{n→∞}*P*_{Λ}*x*_{
n
} .

**Theorem 4.6**. Let

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

*H*. Let

*F*be a bifunction from

*C × C*to

*R*satisfying (A1)-(A4). Let

*A*be a monotone and

*k*-Lipschitz-continuous mapping of

*C*into

*H*. Let

*S*

_{1},

*S*

_{2},... be a family of infinitely nonexpansive mappings of

*C*into itself such that . Assume that for all

*i*∈ {1, 2,...} and for any bounded subset

*K*of

*C*, thenthere holds

for every *n* = 1, 2,.... where {*λ*_{
n
} } ⊂ [*a*, *b*] for some
, {*r*_{
n
} } ⊂ [*d*, *+*∞) and for some *d >* 0, and {*β*_{
n
} } is a sequence in [0, 1] satisfying
. Then, {*x*_{
n
} }, {*u*_{
n
} }, {*y*_{
n
} }, and {*z*_{
n
} } converge strongly to *w* = *P*_{Λ}(*x*).

Let *B* = 0 and *F*(*x*, *y*) = 0 for *x*, *y* ∈ *C*, by Theorems 3.1 and 4.3, we obtain the following results.

**Theorem 4.7**. Let

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

*H*. Let

*A*be a monotone and

*k*-Lipschitz-continuous mapping of

*C*into

*H*. Let

*S*

_{1},

*S*

_{2},... be a family of infinitely nonexpansive mappings of

*C*into itself such that . Assume that for all

*i*∈ {1, 2,...} and for any bounded subset

*K*of

*C*, thenthere holds

*n*= 1, 2,.... where {

*λ*

_{ n }} ⊂ [

*a*,

*b*] for some , and {

*α*

_{ n }}, {

*β*

_{ n }}, {

*γ*

_{ n }}are three sequences in [0, 1] satisfying the following conditions:

- (i)
*α*_{ n }+*β*_{ n }≤ 1 for all*n*∈*N*; - (ii)
;

- (iii)
;

- (iv)
and for all

*n*∈*N*;

Then, {*x*_{
n
} }, {*y*_{
n
} }, and {*z*_{
n
} } converge strongly to *w* = *P*_{Γ}(*x*).

**Theorem 4.8**. Let

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

*H*. Let

*A*be a monotone and

*k*-Lipschitz-continuous mapping of

*C*into

*H*. Let

*S*

_{1},

*S*

_{2},... be a family of infinitely nonexpansive mappings of

*C*into itself such that . Assume that for all

*i*∈ {1, 2,...} and for any bounded subset

*K*of

*C*, thenthere holds

for every *n* = 1, 2,.... where {*λ*_{
n
} } ⊂ [*a*, *b*] for some
, and {*β*_{
n
} } is a sequence in [0, 1] satisfying
. Then, {*x*_{
n
} }, {*y*_{
n
} }, and {*z*_{
n
} } converge strongly to *w* = *P*_{Γ}(*x*).

Let *F*(*x*, *y*) = 0 for *x*, *y* ∈ *C*, then by Theorem 3.2 and the proof of Theorem 4.7 in [3], we obtain the following result.

**Theorem 4.9**. Let

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

*H*. Let

*A*be a monotone and

*k*-Lipschitz-continuous mapping of

*C*into

*H*and

*B*be an

*α*-inverse-strongly monotone mapping of

*C*into

*H*. Let

*S*

_{1},

*S*

_{2},... be a family of infinitely nonexpansive mappings of

*C*into itself such that . Assume that for all

*i*∈ {1, 2,...} and for any bounded subset

*K*of

*C*, thenthere holds

for every *n* = 1, 2,.... if {*λ*_{
n
} } ⊂ [*a*, *b*] for some
, {*β*_{
n
} } ⊂ [*δ*, *ε*] for some *δ*, *ε* ∈ (0, 1) and {*r*_{
n
} } ⊂ [*d*, *e*] for some *d*, *e* ∈ (0, 2*α*). Then, {*x*_{
n
} } and {*u*_{
n
} } converge weakly to *w* ∈ Ξ, where *w* = lim_{n→∞}*P*_{Ξ}*x*_{
n
} .

**Remark 4.1**.

- (i)
For all

*n*≥ 1, let*S*_{ n }=*S*be a nonexpansive mapping, by Theorems 3.2, 4.2, 4.7, 4.8, and 4.9 we recover Theorem 3.1 in [5], Theorem 3.1 in [1], Theorem 5 in [26], Theorem 3.1 in [23], and Theorem 4.7 in [3]. In addition, let*A*= 0, by Theorems 4.6 and 4.5, respectively, we recover Theorems 3.1 and 4.1 in [11]. - (ii)
For all

*n*≥ 1, let*S*_{ n }=*S*be a nonexpansive mapping, by Theorems 3.1, 4.3, and 4.4, respectively, we recover Theorems 4.3, 4.4, and 4.7 in [4] with some modified conditions on*F*. - (iii)
Theorems 3.1, 3.2, 4.3-4.7 also improve the main results in [10, 12, 13] because the inverse strongly monotonicity of

*A*has been replaced by the monotonicity and Lipschitz continuity of*A*.

The following result illustrates that there are the nonexpansive mappings *S*_{1}, *S*_{2} ,... satisfying the condition (*).

**Lemma 4.1**. Let

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

*H*. Let

*T*be a nonexpansive mapping of

*C*into itself such that Fix(

*T*) ≠ ∅. If we define for

*n*∈ {1, 2,...}, and

*x*∈

*C*, then the following results hold:

- (a)
- (b)
.

- (c)

**Proof**.

- (a)
- (b)
It follows from (a) that .

Moreover, it is obvious that . Hence, .

- (c)

*S*

_{1},

*S*

_{2},... are nonexpansive mappings. It follows from (4.1) and (4.2) that for any bounded subset

*K*of

*C*, there holds

From Lemma 4.1, we know that by Theorems 3.1 and 3.2, respectively, we can obtain the following results.

**Theorem 4.10**. Let

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

*H*. Let

*F*be a bifunction from

*C × C*to

*R*satisfying (A1)-(A4). Let

*A*be a monotone and

*k*-Lipschitz-continuous mapping of

*C*into

*H*and

*B*be an

*α*-inverse-strongly monotone mapping of

*C*into

*H*. Let

*T*be a nonexpansive mapping of

*C*into itself such that Θ = Fix(

*T*)∩

*VI*(

*C*,

*A*)∩

*GEP*(

*F*,

*B*) ≠ ∅. Let {

*λ*

_{ n }} ⊂ [

*a*,

*b*] for some , {

*r*

_{ n }} ⊂ [

*d*,

*e*] and for some

*d*,

*e*∈ (0, 2

*α*), and {

*α*

_{ n }}, {

*β*

_{ n }}, and {

*γ*

_{ n }} be three sequences in [0, 1] satisfying the following conditions:

- (i)
*α*_{ n }+*β*_{ n }≤ 1 for all*n*∈*N*; - (ii)
;

- (iii)
;

- (iv)
and for all

*n*∈*N*; If we define for*n*∈ {1, 2,...}, and*x*∈*C*, then the sequences {*x*_{ n }}, {*u*_{ n }}, {*y*_{ n }}, and {*z*_{ n }} generated by algorithm (3.1) converge strongly to*w*=*P*_{Θ}(*x*).

**Theorem 4.11**. Let *C* be a nonempty closed convex subset of a real Hilbert space *H*. Let *F* be a bifunction from *C × C* to *R* satisfying (A1)-(A4). Let *A* be a monotone and *k*-Lipschitz-continuous mapping of *C* into *H* and *B* be an *α*-inverse-strongly monotone mapping of *C* into *H*, and *T* be a nonexpansive mapping of *C* into itself such that Θ = Fix(*T*)∩*VI*(*C*, *A*)∩*GEP*(*F*, *B*) ≠ ∅. Assume that {*λ*_{
n
} } ⊂ [*a*, *b*] for some
{*β*_{
n
} } ⊂ [*δ*, *ε*] for some *δ*, *ε* ∈ (0, 1), and {*r*_{
n
} } ⊂ [*d*, *e*] some *d*, *e* ∈ (0, 2*α*). If we define
for *n* ∈ {1, 2,...} and *x* ∈ *C*, then the sequences {*x*_{
n
} }, {*u*_{
n
} }, and {*y*_{
n
} } generated by algorithm (3.12) converge weakly to *w* ∈ Θ, where *w* = lim_{n→∞}*P*_{Θ}*x*_{
n
} .

## Declarations

### Acknowledgements

This research was supported by the National Natural Science Foundation of China, the Natural Science Foundation of Chongqing (Grant No. CSTC, 2009BB8240), and the Special Fund of Chongqing Key Laboratory (CSTC). The author is grateful to the referees for their detailed comments and helpful suggestions, which have improved the presentation of this article.

## Authors’ Affiliations

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