Volume 2012, Article ID 859492,21pages doi:10.1155/2012/859492

*Research Article*

**Existence and Strong Convergence**

**Theorems for Generalized Mixed Equilibrium** **Problems of a Finite Family of Asymptotically** **Nonexpansive Mappings in Banach Spaces**

**Rabian Wangkeeree,**

^{1, 2}**Hossein Dehghan,**

^{3}**and Pakkapon Preechasilp**

^{1}*1**Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand*

*2**Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand*

*3**Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Gava Zang,*
*Zanjan 45137-66731, Iran*

Correspondence should be addressed to Rabian Wangkeeree,rabianw@nu.ac.th Received 5 April 2012; Accepted 10 May 2012

Academic Editor: Giuseppe Marino

Copyrightq2012 Rabian Wangkeeree 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.

We first prove the existence of solutions for a generalized mixed equilibrium problem under the new conditions imposed on the given bifunction and introduce the algorithm for solving a common element in the solution set of a generalized mixed equilibrium problem and the common fixed point set of finite family of asymptotically nonexpansive mappings. Next, the strong convergence theorems are obtained, under some appropriate conditions, in uniformly convex and smooth Banach spaces. The main results extend various results existing in the current literature.

**1. Introduction**

Let*E*be a real Banach space with the dual*E*^{∗}and*C*be a nonempty closed convex subset of
*E. We denote by*NandRthe sets of positive integers and real numbers, respectively. Also,
we denote by*J*the normalized duality mapping from*E*to 2^{E}^{∗}defined by

*Jx*

*x*^{∗}∈*E*^{∗}:x, x^{∗}x^{2}x^{∗}^{2}

*,* ∀x∈*E,* 1.1

where·,·denotes the generalized duality pairing. Recall that if*E*is smooth, then*J*is single
valued and if *E* is uniformly smooth, then *J* is uniformly norm-to-norm continuous on
bounded subsets of*E. We will still denote byJ*the single-valued duality mapping.

A mapping*S*:*C* → *Eis called nonexpansive if*Sx−Sy ≤ x−yfor all*x, y*∈*C. Also*
a mapping*S* : *C* → *Cis called asymptotically nonexpansive if there exists a sequence*{k*n*} ⊂
1,∞with *k**n* → 1 as*n* → ∞such thatS^{n}*x*−*S*^{n}*y ≤* *k**n*x−*y*for all*x, y* ∈ *C*and for
each*n*≥1. Denote by*FS*the set of fixed points of*S, that is,FS *{x∈*C*:*Sxx}. The*
following example shows that the class of asymptotically nonexpansive mappings which was
first introduced by Goebel and Kirk1is wider than the class of nonexpansive mappings.

*Example 1.1*see2. Let*B**H*be the closed unit ball in the Hilbert space*Hl*2and*S*:*B**H* →
*B**H*a mapping defined by

*Sx*1*, x*2*, x*3*, . . . *

0, x_{1}^{2}*, a*2*x*2*, a*3*x*3*, . . .*

*,* 1.2

where{a*n*}is a sequence of real numbers such that 0*< a**i**<*1 and_{∞}

*i2**a**i*1/2. Then
*Sx*−*Sy*≤2*x*−*y,* ∀x, y∈*B**H**.* 1.3

That is,*S*is Lipschitzian but not nonexpansive. Observe that
*S*^{n}*x*−*S*^{n}*y*≤2

*n*
*i2*

*a**i**x*−*y,* ∀x, y∈*B**H**, n*≥2. 1.4
Here*k**n*2_{n}

*i2**a**i* → 1 as*n* → ∞. Therefore,*S*is asymptotically nonexpansive but not non-
expansive.

A mapping*T* : *C* → *E*^{∗}is said to be relaxed*η-ξ*monotone if there exist a mapping
*η* : *C*×*C* → *E*and a function *ξ* : *E* → Rpositively homogeneous of degree*p, that is,*
*ξtz t*^{p}*ξz*for all*t >*0 and*z*∈*E*such that

*Tx*−*Ty, η*
*x, y*

≥*ξ*
*x*−*y*

*,* ∀x, y∈*C,* 1.5

where*p >*1 is a constant; see3. In the case of*ηx, y x*−*y*for all*x,y* ∈*C,T* is said to
be relaxed*ξ-monotone. In the case ofηx, y x*−*y*for all*x, y*∈*C*and*ξz kz** ^{p}*, where

*p >*1 and

*k >*0,

*T*is said to be

*p-monotone; see*4–6. In fact, in this case, if

*p*2, then

*T*is a

*k-strongly monotone mapping. Moreover, every monotone mapping is relaxedη-ξ*monotone with

*ηx, y x*−

*y*for all

*x, y*∈

*C*and

*ξ*0. The following is an example of

*η-ξ*monotone mapping which can be found in3. Let

*C*−∞,∞,

*Tx*−x, and

*η*
*x, y*

−c

*x*−*y*

*, x*≥*y,*
*c*

*x*−*y*

*,* *x < y,* 1.6

where*c >*0 is a constant. Then,*T*is relaxed*η-ξ*monotone with

*ξz *

*cz*^{2}*,* *z*≥0,

−cz^{2}*, z <*0. 1.7

A mapping*T* :*C* → *E*^{∗}is said to be*η-hemicontinuous if, for each fixedx, y*∈*C, the mapping*
*f* : 0,1 → −∞,∞defined by*ft * Tx*ty*−*x, ηy, x*is continuous at 0^{}. For
a real Banach space*E*with the dual*E*^{∗}and for*C*a nonempty closed convex subset of*E, let*
*f*:*C*×*C* → Rbe a bifunction,*ϕ*:*C* → Ra real-valued function and*T* :*C* → *E*^{∗}be a relaxed
*η-ξ*monotone mapping. Recently, Kamraksa and Wangkeeree7introduced the following
generalized mixed equilibrium problemGMEP.

Find*x*∈*C*such that*f*
*x, y*

*Tx, η*
*y, x*

*ϕ*
*y*

≥*ϕx,* ∀y∈*C.* 1.8

The set of such*x*∈*C*is denoted by GMEPf, T, that is,
GMEP

*f, T*

*x*∈*C*:*f*
*x, y*

*Tx, η*
*y, x*

*ϕ*
*y*

≥*ϕx,*∀y∈*C*

*.* 1.9

*Special Cases*

1If*T*is monotone that is*T*is relaxed*η-ξ*monotone with*ηx, y x*−*y*for all*x, y*∈*C*and
*ξ*0,1.8is reduced to the following generalized equilibrium problemGEP.

Find*x*∈*C*such that*f*
*x, y*

*Tx, y*−*x*
*ϕ*

*y*

≥*ϕx,* ∀y∈*C.* 1.10

The solution set of1.10is denoted by GEPf, that is, GEP

*f*

*x*∈*C*:*f*
*x, y*

*Tx, y*−*x*
*ϕ*

*y*

≥*ϕx,*∀y∈*C*

*.* 1.11

2In the case of*T* ≡0 and*ϕ*≡0,1.8is reduced to the following classical equilibrium
problem

Find *x*∈*C*such that*f*
*x, y*

≥0, ∀y∈*C.* 1.12

The set of all solution of1.12is denoted by EPf, that is, EP

*f*

*x*∈*C*:*f*
*x, y*

≥0,∀y∈*C*

*.* 1.13

3In the case of*f* ≡ 0,1.8is reduced to the following variational-like inequality
problem3.

Find*x*∈*C*such that *Tx, η*
*y, x*

*ϕ*
*y*

−*ϕx*≥0, ∀y∈*C.* 1.14
The generalized mixed equilibrium problemGMEP 1.8is very general in the sense
that it includes, as special cases, optimization problems, variational inequalities, minimax
problems, and Nash equilibrium problems. Using the KKM technique introduced by Kanster
et al.8and *η-ξ* monotonicity of the mapping*ϕ, Kamraksa and Wangkeeree*7obtained
the existence of solutions of generalized mixed equilibrium problem1.8in a real reflexive
Banach space.

Some methods have been proposed to solve the equilibrium problem in a Hilbert space; see, for instance, Blum and Oettli 9, Combettes and Hirstoaga 10, and Moudafi 11. On the other hand, there are several methods for approximation fixed points of a nonexpansive mapping; see, for instance,12–17. Recently, Tada and Takahashi13,16and S. Takahashi and W. Takahashi 17 obtained weak and strong convergence theorems for finding a common elements in the solution set of an equilibrium problem and the set of fixed point of a nonexpansive mapping in a Hilbert space. In particular, Tada and Takahashi16 established a strong convergence theorem for finding a common element of two sets by using the hybrid method introduced in Nakajo and Takahashi18. They also proved such a strong convergence theorem in a uniformly convex and uniformly smooth Banach space.

On the other hand, in 1953, Mann12introduced the following iterative procedure to
approximate a fixed point of a nonexpansive mapping*S*in a Hilbert space*H:*

*x*_{n1}*α**n**x**n* 1−*α**n*Sx*n**,* ∀n∈N, 1.15

where the initial point*x*0 is taken in*C*arbitrarily and{α*n*}is a sequence in0,1. However,
we note that Manns iteration process 1.15 has only weak convergence, in general; for
instance, see19–21. In 2003, Nakajo and Takahashi18proposed the following sequence
for a nonexpansive mapping*S*in a Hilbert space:

*x*0*x*∈*C,*
*y**n**α**n**x**n* 1−*α**n*Sx*n**,*
*C**n*

*z*∈*C*:*y**n*−*z*≤ x*n*−*z*
*,*
*Q**n*{z∈*C*:x*n*−*z, x*0−*x**n* ≥0},

*x*_{n1}*P**C**n*∩Q*n**x*0*,*

1.16

where 0≤*α**n*≤ *a <*1 for all*n*∈N, and*P**C**n*∩D*n* is the metric projection from*E*onto*C**n*∩*D**n*.
Then, they proved that{x*n*}converges strongly to*P*_{FT}*x*0. Recently, motivated by Nakajo and
Takahashi18and Xu22, Matsushita and Takahashi14introduced the iterative algorithm
for finding fixed points of nonexpansive mappings in a uniformly convex and smooth Banach
space:*x*0*x*∈*C*and

*C**n*co{z∈*C*:z−*Sz ≤t**n*x*n*−*Sx**n*},
*D**n*{z∈*C*:x*n*−*z, J*x−*x**n* ≥0},

*x**n1**P**C**n*∩D*n**x,* *n*≥0,

1.17

where co*D* denotes the convex closure of the set *D,* {t*n*} is a sequence in 0, 1 with
*t**n* → 0. They proved that {x*n*} generated by 1.17 converges strongly to a fixed point
of *S. Very recently, Dehghan*23 investigated iterative schemes for finding fixed point of
an asymptotically nonexpansive mapping and proved strong convergence theorems in a

uniformly convex and smooth Banach space. More precisely, he proposed the following
algorithm:*x*1 *x*∈*C,C*0*D*0*C*and

*C**n*co{z∈*C**n−1*:z−*S*^{n}*z ≤t**n*x*n*−*S*^{n}*x**n*},
*D**n* {z∈*D** _{n−1}*:x

*n*−

*z, J*x−

*x*

*n*≥0},

*x**n1**P**C**n*∩D*n**x,* *n*≥0,

1.18

where{t*n*}is a sequence in0,1with*t**n* → 0 as*n* → ∞and*S*is an asymptotically nonex-
pansive mapping. It is proved in23that{x*n*}converges strongly to a fixed point of*S.*

On the other hand, recently, Kamraksa and Wangkeeree7studied the hybrid pro- jection algorithm for finding a common element in the solution set of the GMEP and the common fixed point set of a countable family of nonexpansive mappings in a uniformly convex and smooth Banach space.

Motivated by the above mentioned results and the on-going research, we first prove
the existence results of solutions for GMEP under the new conditions imposed on the
bifunction *f. Next, we introduce the following iterative algorithm for finding a common*
element in the solution set of the GMEP and the common fixed point set of a finite family of
asymptotically nonexpansive mappings{S1*, S*2*, . . . , S**N*}in a uniformly convex and smooth
Banach space:*x*0∈*C,D*0*C*0*C, and*

*x*1*P**C*0∩D0*x*0*P**C**x*0*,*

*C*1co{z∈*C*:z−*S*1*z ≤t*1x1−*S*1*x*1},
*u*1∈*C*such that*f*

*u*1*, y*
*ϕ*

*y*

*Tu*1*, η*
*y, u*1

1

*r*1 *y*−*u*1*, Ju*1−*x*1

*,* ∀y∈*C,*
*D*1{z∈*C*:u1−*z, Jx*1−*u*1 ≥0},

*x*2*P**C*1∩D1*x*0*,*
...

*C**N*co{z∈*C**N−1*:z−*S**N**z ≤t*1x*N*−*S**N**x**N*},
*u**N*∈*C*such that*f*

*u**N**, y*
*ϕ*

*y*

*Tu**N**, η*
*y, u**N*

1

*r**N* *y*−*u**N**, J*u*N*−*x**N*

*,* ∀y∈*C,*
*D**N* {z∈*D** _{N−1}*:u

*N*−

*z, Jx*

*N*−

*u*

*N*≥0},

*x**N1**P**C**N*∩D*N**x*0*,*
*C** _{N1}*co

*z*∈*C**N* :z−*S*^{2}_{1}*z*≤*t*1x*N1*−*S*^{2}_{1}*x*_{N1}*,*
*u** _{N1}*∈

*C*such that

*f*

*u*_{N1}*, y*
*ϕ*

*y*

*Tu*_{N1}*, η*

*y, u** _{N1}*
1

*r**N1* *y*−*u**N1**, J*u*N1*−*x**N1*

*,* ∀y∈*C,*

*D**N1*{z∈*D**N*:u*N1*−*z, J*x*N1*−*u**N1* ≥0},
*x*_{N2}*P**C**N1*∩D*N1**x*0*,*

...
*C*2Nco

*z*∈*C*_{2N−1}:z−*S*^{2}_{N}*z*≤*t*1x2N−*S*^{2}_{N}*x*2N
*,*
*u*2N∈*C*such that *f*

*u*2N*, y*
*ϕ*

*y*

*Tu*2N*, η*
*y, u*2N

1

*r*2N *y*−*u*2N*, Ju*2N−*x*2N

*,* ∀y∈*C,*
*D*2N{z∈*D*2N−1:u2N−*z, Jx*2N−*u*2N ≥0},

*x*_{2N1} *P**C*2N∩D2N*x*0*,*
*C*2N1co

*z*∈*C*2N:z−S^{3}_{1}*z*≤*t*1x2N1−*S*^{3}_{1}*x*2N1
*,*
*u*_{2N1}∈*C*such that*f*

*u*2N1*, y*
*ϕ*

*y*

*Tu*2N1*, η*

*y, u*2N1
1

*r*_{2N1} *y*−*u*2N1*, Ju*2N1−*x*2N1

*,* ∀y∈*C,*
*D*_{2N1}{z∈*D*2N:u_{2N1}−*z, Jx*_{2N1}−*u*_{2N1} ≥0},

*x*2N2*P**C*_{2N1}∩D2N1*x*0*,*
...

1.19
The above algorithm is called the hybrid iterative algorithm for a finite family of asymptot-
ically nonexpansive mappings from*C*into itself. Since, for each*n* ≥ 1, it can be written as
*n* h−1Ni, where*iin*∈ {1,2, . . . , N},*hhn*≥1 is a positive integer and*hn* → ∞
as*n* → ∞. Hence the above table can be written in the following form:

*x*0∈*C,* *D*0*C*0*C,*
*C**n*co

*z*∈*C**n−1*:z−*S*^{hn}_{in}*z*≤*t**n*x*n*−*S*^{hn}_{in}*x**n*

*,* *n*≥1,
*u**n*∈*C*such that*f*

*u**n**, y*
*ϕ*

*y*

*Tu**n**, η*
*y, u**n*

1

*r**n* *y*−*u**n**, J*u*n*−*x**n*

*,* ∀y∈*C, n*≥1,
*D**n*{z∈*D**n−1*:u*n*−*z, Jx**n*−*u**n* ≥0}, *n*≥1,

*x**n1**P**C**n*∩D*n**x*0*,* *n*≥0.

1.20

Strong convergence theorems are obtained in a uniformly convex and smooth Banach space.

The results presented in this paper extend and improve the corresponding Kimura and Nakajo24, Kamraksa and Wangkeeree7, Dehghan23, and many others.

**2. Preliminaries**

Let*E*be a real Banach space and let*U*{x∈*E*:x1}be the unit sphere of*E. A Banach*
space*E*is said to be strictly convex if for any*x, y*∈*U,*

*x /y*implies*xy<*2. 2.1

It is also said to be uniformly convex if for each*ε*∈0,2, there exists*δ >*0 such that for any
*x, y*∈*U,*

*x*−*y*≥*ε*implies*xy<*21−*δ.* 2.2
It is known that a uniformly convex Banach space is reflexive and strictly convex. Define a
function*δ*:0,2 → 0,1called the modulus of convexity of*E*as follows:

*δε *inf

1−
*xy*

2

:*x, y*∈*E,*x*y*1,*x*−*y*≥*ε*

*.* 2.3

Then*E*is uniformly convex if and only if*δε>*0 for all*ε*∈0,2. A Banach space*E*is said
to be smooth if the limit

lim*t*→0

*xty*− x

*t* 2.4

exists for all*x, y* ∈*U. LetC*be a nonempty, closed, and convex subset of a reflexive, strictly
convex, and smooth Banach space*E. Then for anyx*∈*E, there exists a unique pointx*0 ∈*C*
such that

x0−*x ≤*min

*y∈C**y*−*x.* 2.5

The mapping*P**C*:*E* → *C*defined by*P**C**xx*0is called the metric projection from*E*onto*C.*

Let*x*∈*E*and*u*∈*C. The following theorem is well known.*

* Theorem 2.1. LetCbe a nonempty convex subset of a smooth Banach spaceEand letx* ∈

*Eand*

*y*∈

*C. Then the following are equivalent:*

a*y* *is a best approximation tox*:*yP**C**x,*
b*yis a solution of the variational inequality:*

*y*−*z, J*
*x*−*y*

≥0 ∀z∈*C,* 2.6

*whereJis a duality mapping andP**C**is the metric projection fromEontoC.*

It is well known that if*P**C* is a metric projection from a real Hilbert space*H*onto a
nonempty, closed, and convex subset*C, thenP**C* is nonexpansive. But, in a general Banach
space, this fact is not true.

In the sequel one will need the following lemmas.

**Lemma 2.2**see25. Let*E* *be a uniformly convex Banach space, let*{α*n*}*be a sequence of real*
*numbers such that 0* *< b* ≤ *α**n* ≤ *c <* *1 for alln* ≥ *1, and let* {x*n*} *and* {y*n*}*be sequences in* *E*
*such that lim sup*_{n}_{→ ∞}x*n* ≤*d, lim sup*_{n}_{→ ∞}y*n* ≤*dand lim*_{n}_{→ ∞}α*n**x**n* 1−*α**n*y*n**d. Then*
lim*n*→ ∞x*n*−*y**n**0.*

Dehghan23obtained the following useful result.

**Theorem 2.3**see23. Let*Cbe a bounded, closed, and convex subset of a uniformly convex Banach*
*spaceE. Then there exists a strictly increasing, convex, and continuous functionγ*:0,∞ → 0,∞
*such thatγ0 0 and*

*γ*
1

*k**m*

*S*^{m}

_{n}

*i1*

*λ**i**x**i*

−^{n}

*i1*

*λ**i**S*^{m}*x**i*

≤ max

1≤j≤k≤n

*x**j*−*x**k*− 1
*k**m*

*S*^{m}*x**j*−*S*^{m}*x**k*

2.7

*for any asymptotically nonexpansive mappingSofCintoCwith*{k*n*}, any elements*x*1*, x*2*, . . . , x**n*∈
*C, any numbersλ*1*, λ*2*, . . . , λ**n*≥*0 with*_{n}

*i1**λ**i**1 and eachm*≥*1.*

**Lemma 2.4**see26, Lemma 1.6. Let*Ebe a uniformly convex Banach space,* *Cbe a nonempty*
*closed convex subset ofEandS*:*C* → *Cbe an asymptotically nonexpansive mapping. Then*I−*S*
*is demiclosed at 0, that is, ifx**n** xand*I−*Sx**n* → *0, thenx*∈*FS.*

The following lemma can be found in7.

**Lemma 2.5**see7, Lemma 3.2. Let*Cbe a nonempty, bounded, closed, and convex subset of a*
*smooth, strictly convex, and reflexive Banach spaceE, letT* :*C* → *E*^{∗}*be anη-hemicontinuous and*
*relaxedη*−*ξmonotone mapping. Letfbe a bifunction fromC×Cto*R*satisfying (A1), (A3), and (A4)*
*and letϕbe a lower semicontinuous and convex function fromCto*R. Let*r >0 andz*∈*C. Assume*
*that*

i*ηx, y ηy, x 0 for allx, y*∈*C;*

ii*for any fixedu, v*∈*C, the mappingx*→ Tv, ηx, u*is convex and lower semicontinu-*
*ous;*

iii*ξ*:*E* → R*is weakly lower semicontinuous, that is, for any net*{x*β*}, x*β**converges toxin*
*σE, E*^{∗}*which implies thatξx*≤lim inf*ξx**β*.

*Then there existsx*0∈*Csuch that*
*f*

*x*0*, y*

*Tx*0*, η*
*y, x*0

*ϕ*
*y*

1

*r* *y*−*x*0*, Jx*0−*z*

≥*ϕx*0, ∀y∈*C.* 2.8
**Lemma 2.6**see7, Lemma 3.3. Let*Cbe a nonempty, bounded, closed, and convex subset of a*
*smooth, strictly convex, and reflexive Banach spaceE, letT* :*C* → *E*^{∗}*be anη-hemicontinuous and*
*relaxed* *η-ξ* *monotone mapping. Let* *f* *be a bifunction from* *C*×*Cto*R *satisfying (A1)–(A4) and*
*letϕbe a lower semicontinuous and convex function fromCto*R. Let*r >* *0 and define a mapping*
Φ*r* :*E* → *Cas follows:*

Φ*r*x

*z*∈*C*:*f*
*z, y*

*Tz, η*
*y, z*

*ϕ*
*y*

1

*r* *y*−*z, J*z−*x*

≥*ϕz,*∀y∈*C*

2.9

*for allx*∈*E. Assume that*

i*ηx, y ηy, x 0, for allx, y*∈*C;*

ii*for any fixedu, v*∈*C, the mappingx*→ Tv, ηx, u*is convex and lower semicontinuous*
*and the mappingx*→ Tu, ηv, x*is lower semicontinuous;*

iii*ξ*:*E* → R*is weakly lower semicontinuous;*

iv*for anyx, y*∈*C,ξx*−*y ξy*−*x*≥*0.*

*Then, the following holds:*

1 Φ*r* *is single valued;*

2Φ*r**x*−Φ*r**y, J*Φ*r**x*−*x ≤ Φ**r**x*−Φ*r**y, J*Φ*r**y*−*yfor allx, y*∈*E;*

3*FΦ**r* EPf, T;

4EPf, T*is nonempty closed and convex.*

**3. Existence of Solutions for GMEP**

In this section, we prove the existence results of solutions for GMEP under the new conditions
imposed on the bifunction*f.*

**Theorem 3.1. Let**Cbe a nonempty, bounded, closed, and convex subset of a smooth, strictly convex,*and reflexive Banach spaceE, letT* : *C* → *E*^{∗}*be anη-hemicontinuous and relaxedη-ξ* *monotone*
*mapping. Letfbe a bifunction fromC*×*Cto*R*satisfying the following conditions (A1)–(A4):*

A1*fx, x 0 for allx*∈*C;*

A2*fx, y fy, x*≤min{ξx−*y, ξy*−*x}for allx, y*∈*C;*

A3*for ally*∈*C,f·, yis weakly upper semicontinuous;*

A4*for allx*∈*C,fx,*·*is convex.*

*For anyr >0 andx*∈*E, define a mapping*Φ*r* :*E* → *Cas follows:*

Φ*r*x

*z*∈*C*:*f*
*z, y*

*Tz, η*
*y, z*

*ϕ*
*y*

1

*r* *y*−*z, J*z−*x*

≥*ϕz,*∀y∈*C*

*,*
3.1

*whereϕis a lower semicontinuous and convex function fromCto*R. Assume that
i*ηx, y ηy, x 0, for allx, y*∈*C;*

ii*for any fixedu, v*∈*C, the mappingx*→ Tv, ηx, u*is convex and lower semicontinuous*
*and the mappingx*→ Tu, ηv, x*is lower semicontinuous;*

iii*ξ*:*E* → R*is weakly lower semicontinuous.*

*Then, the following holds:*

1 Φ*r* *is single valued;*

2Φ*r**x*−Φ*r**y, J*Φ*r**x*−*x ≤ Φ**r**x*−Φ*r**y, J*Φ*r**y*−*yfor allx, y*∈*E;*

3*FΦ**r* GMEPf, T;

4GMEPf, T*is nonempty closed and convex.*

*Proof. For eachx*∈*E. It follows from Lemma*2.5thatΦ*r*xis nonempty.

1We prove thatΦ*r* is single valued. Indeed, for*x* ∈ *E*and*r >* 0, let*z*1*, z*2 ∈Φ*r**x.*

Then

*fz*1*, z*2 *Tz*2*, ηz*2*, z*1
*ϕz*2

1

*r*z1−*z*2*, J*z1−*x ≥ϕz*1,
*fz*2*, z*1 *Tz*1*, ηz*1*, z*2

*ϕz*1

1

*r*z2−*z*1*, J*z2−*x ≥ϕz*2.

3.2

Adding the two inequalities, fromiwe have

*f*z2*, z*1 *fz*1*, z*2 Tz1−*Tz*2*, ηz*2*, z*11

*r*z2−*z*1*, Jz*1−*x*−*Jz*2−*x ≥*0. 3.3
SettingΔ:min{ξz1−*z*2, ξz2−*z*1}and usingA2, we have

Δ *Tz*1−*Tz*2*, ηz*2*, z*1
1

*r*z2−*z*1*, J*z1−*x*−*Jz*2−*x ≥*0, 3.4
that is,

1

*r*z2−*z*1*, J*z1−*x*−*Jz*2−*x ≥* *Tz*2−*Tz*1*, ηz*2*, z*1

−Δ. 3.5

Since*T* is relaxed*η-ξ*monotone and*r >*0, one has

z2−*z*1*, Jz*1−*x*−*Jz*2−*x ≥rξz*2−*z*1−Δ≥0. 3.6

In3.5exchanging the position of*z*1and*z*2, we get
1

*r*z1−*z*2*, J*z2−*x*−*Jz*1−*x ≥* *Tz*1−*Tz*2*, ηz*1*, z*2

−Δ, 3.7

that is,

z1−*z*2*, Jz*2−*x*−*Jz*1−*x ≥rξz*1−*z*2−Δ≥0. 3.8

Now, adding the inequalities3.6and3.8, we have

2z2−*z*1*, Jz*1−*x*−*J*z2−*x ≥*0. 3.9

Hence,

0≤ z2−*z*1*, Jz*1−*x*−*Jz*2−*x*z2−*x*−z1−*x, Jz*1−*x*−*Jz*2−*x.* 3.10
Since*J*is monotone and*E*is strictly convex, we obtain that*z*1−*xz*2−*x*and hence*z*1*z*2.
Therefore*S**r*is single valued.

2For*x, y*∈*C, we have*

*f*

Φ*r**x,*Φ*r**y*

*T*Φ*r**x, η*

Φ*r**y,*Φ*r**x*
*ϕ*

Φ*r**y*

−*ϕΦ**r**x * 1

*r* Φ*r**y*−Φ*r**x, JΦ**r**x*−*x*

≥0,
*f*

Φ*r**y,*Φ*r**x*

*T*Φ*r**y, η*

Φ*r**x,*Φ*r**y*

*ϕΦ**r**x*−*ϕ*
Φ*r**y*

1

*r* Φ*r**x*−Φ*r**y, J*

Φ*r**y*−*y*

≥0.

3.11

SettingΛ*x,y*:min{ξΦ*r**x*−Φ*r**y, ξΦ**r**y*−Φ*r**x}*and applyingA2, we get
*T*Φ*r**x*−*TΦ**r**y, η*

Φ*r**y,*Φ*r**x*
1

*r* Φ*r**y*−Φ*r**x, JΦ**r**x*−*x*−*J*

Φ*r**y*−*y*

≥ −Λ*x,y**,* 3.12

that is, 1

*r* Φ*r**y*−Φ*r**x, JΦ**r**x*−*x*−*J*

Φ*r**y*−*y*

≥ *T*Φ*r**y*−*T*Φ*r**x, η*

Φ*r**y,*Φ*r**x*

−Λ*x,y*

≥*ξ*

Φ*r**y*−Φ*r**x*

−Λ*x,y* ≥0.

3.13

In3.13exchanging the position ofΦ*r**x*andΦ*r**y, we get*
1

*r* Φ*r**x*−Φ*r**y, J*

Φ*r**y*−*y*

−*J*Φ*r**x*−*x*

≥0. 3.14

Adding the inequalities3.13and3.14, we have 2

*r* Φ*r**y*−Φ*r**x, JΦ**r**x*−*x*−*J*

Φ*r**y*−*y*

≥0. 3.15

It follows that

Φ*r**y*−Φ*r**x, JΦ**r**x*−*x*−*J*

Φ*r**y*−*y*

≥0. 3.16

Hence

Φ*r**x*−Φ*r**y, J*Φ*r**x*−*x*

≤ Φ*r**x*−Φ*r**y, J*

Φ*r**y*−*y*

*.* 3.17

The conclusions3,4follow from Lemma2.6.

*Example 3.2. Defineξ*:R → Rand*f*:R×R → Rby

*f*
*x, y*

*x*−*y*_{2}

2 *,* *ξx x*^{2} ∀x, y∈R. 3.18
It is easy to see that *f* satisfiesA1,A3, A4, andA2:*fx, y f*y, x ≤ min{ξx−
*y, ξx*−*y},*for allx, y∈R×R.

*Remark 3.3. Theorem*3.1generalizes and improves7, Lemma 3.3in the following manners.

1The condition *fx, y fy, x* ≤ 0 has been weakened byA2that is *fx, y *
*fy, x*≤min{ξx−*y, ξy*−*x}*for all*x, y*∈*C.*

2The control condition*ξx−yξy*−x≥0 imposed on the mapping*ξ*in7, Lemma
3.3can be removed.

If*T*is monotone that is*T* is relaxed*η-ξ*monotone with*ηx, y x*−*y*for all*x, y*∈*C*
and*ξ*0, we have the following results.

**Corollary 3.4. Let**Cbe a nonempty, bounded, closed, and convex subset of a smooth, strictly convex,*and reflexive Banach spaceE. LetT* :*C* → *E*^{∗}*be a monotone mapping andf* *be a bifunction from*
*C*×*Cto*R*satisfying the following conditions (i)–(iv):*

i*fx, x 0 for allx*∈*C;*

ii*fx, y fy, x*≤*0 for allx, y*∈*C;*

iii*for ally*∈*C,f·, yis weakly upper semicontinuous;*

iv*for allx*∈*C,fx,*·*is convex.*

*For anyr >0 andx*∈*E, define a mapping*Φ*r* :*E* → *Cas follows:*

Φ*r*x

*z*∈*C*:*f*
*z, y*

*Tz, y*−*z*
*ϕ*

*y*
1

*r* *y*−*z, J*z−*x*

≥*ϕz,*∀y∈*C*

*,*
3.19

*whereϕis a lower semicontinuous and convex function fromCto*R. Then, the following holds:

1 Φ*r* *is single valued;*

2Φ*r**x*−Φ*r**y, J*Φ*r**x*−*x ≤ Φ**r**x*−Φ*r**y, J*Φ*r**y*−*yfor allx, y*∈*E;*

3*FΦ**r* GEPf;

4GEPf*is nonempty closed and convex.*

**4. Strong Convergence Theorems**

In this section, we prove the strong convergence theorem of the sequence {x*n*} defined by
1.20for solving a common element in the solution set of a generalized mixed equilibrium
problem and the common fixed point set of a finite family of asymptotically nonexpansive
mappings.

**Theorem 4.1. Let***E* *be a uniformly convex and smooth Banach space and let* *C* *be a nonempty,*
*bounded, closed, and convex subset ofE. Let* *f* *be a bifunction fromC*×*Cto*R *satisfying (A1)–*

*(A4). LetT* : *C* → *E*^{∗}*be anη-hemicontinuous and relaxedη-ξ* *monotone mapping andϕa lower*
*semicontinuous and convex function fromC* *to*R. Let, for each 1 ≤ *i* ≤ *N,S**i* : *C* → *C* *be an*
*asymptotically nonexpansive mapping with a sequence*{k*n,i*}^{∞}_{n1}*, respectively, such that* *k**n,i* → 1
*as* *n* → ∞. Assume thatΩ : _{N}

*i1**FS**i*∩GMEPf, T *is nonempty. Let* {x*n*} *be a sequence*
*generated by*1.20, where{t*n*}*and*{r*n*}*are real sequences in*0,1*satisfying lim**n*→ ∞*t**n* *0 and*
lim inf_{n→ ∞}*r**n* *>* *0. Then*{x*n*} *converges strongly, as* *n* → ∞, to *P*Ω*x*0*, whereP*Ω *is the metric*
*projection ofEonto*Ω.

*Proof. First, define the sequence*{k*n*}by*k**n*:max{k*n,i*: 1≤*i*≤*N}*and so*k**n* → 1 as*n* → ∞
and

S^{hn}_{in}*x*−*S*^{hn}_{in}*y*≤*k**n**x*−*y* ∀x, y∈*C,* 4.1
where*hn j*1 if*jN < n*≤j1N,*j* 1,2*. . . , N*and*njNin;in*∈ {1,2, . . . , N}.

Next, we rewrite the algorithm1.20as the following relation:

*x*0∈*C,* *D*0*C*0*C,*
*C**n*co

*z*∈*C** _{n−1}*:z−

*S*

^{hn}

_{in}*z*≤

*t*

*n*x

*n*−

*S*

^{hn}

_{in}*x*

*n*

*,* *n*≥0,
*D**n*{z∈*D**n−1*:Φ*r**n**x**n*−*z, Jx**n*−Φ*r**n**x**n* ≥0}, *n*≥1,

*x**n1**P**C**n*∩D*n**x*0*,* *n*≥0,

4.2

whereΦ*r* is the mapping defined by3.19. We show that the sequence{x*n*}is well defined.

It is easy to verify that *C**n* ∩*D**n* is closed and convex andΩ ⊂ *C**n* for all*n* ≥ 0. Next, we
prove thatΩ ⊂ *C**n* ∩ *D**n*. Indeed, since *D*0 *C, we also have*Ω ⊂ *C*0 ∩ *D*0. Assume that
Ω⊂*C** _{k−1}* ∩

*D*

*for*

_{k−1}*k*≥2. Utilizing Theorem3.12, we obtain

Φ*r**k**x**k*−Φ*r**k**u, JΦ**r**k**u*−*u*−*JΦ**r**k**x**k*−*x**k* ≥0, ∀u∈Ω, 4.3

which gives that

Φ*r**k**x**k*−*u, Jx**k*−Φ*r**k**x**k* ≥0, ∀u∈Ω, 4.4
henceΩ⊂ *D**k*. By the mathematical induction, we get thatΩ ⊂*C**n*∩*D**n* for each*n* ≥0 and
hence{x*n*}is well defined. Now, we show that

*n*lim→ ∞*x**n*−*x**nj*0, ∀j1,2, . . . , N. 4.5
Put*wP*_{Ω}*x*0, sinceΩ⊂*C**n*∩*D**n*and*x*_{n1}*P**C**n*∩D*n*, we have

x*n1*−*x*0 ≤ w−*x*0, ∀n≥0. 4.6

Since*x**n2*∈*D**n1* ⊂*D**n*and*x**n1**P**C**n*∩D*n**x*0, we have

x* _{n1}*−

*x*0 ≤ x

*−*

_{n2}*x*0. 4.7 Hence the sequence{x

*n*−

*x*0}is bounded and monotone increasing and hence there exists a constant

*d*such that

*n*lim→ ∞x*n*−*x*0*d.* 4.8

Moreover, by the convexity of*D**n*, we also have 1/2x*n1**x**n2*∈*D**n*and hence
x0−*x**n1* ≤x0−*x**n1**x**n2*

2

≤ 1

2x0−*x**n1*x0−*x**n2*. 4.9
This implies that

*n*lim→ ∞

1

2x0−*x** _{n1}* 1

2x0−*x** _{n2}*
lim

*n*→ ∞

x0− *x**n1**x**n2*

2

*d.* 4.10

By Lemma2.2, we have

*n*lim→ ∞x*n*−*x**n1*0. 4.11

Furthermore, we can easily see that

*n*lim→ ∞*x**n*−*x**nj*0, ∀j1,2, . . . , N. 4.12

Next, we show that

*n*lim→ ∞

x*n*−*S*^{hn−κ}_{in−κ}*x**n*0, for any*κ*∈ {1,2, . . . , N}. 4.13
Fix*κ*∈ {1,2, . . . , N}and put*mn*−*κ. Sincex**n* *P**C**n−1*∩D*n−1**x, we havex**n*∈*C** _{n−1}* ⊆ · · · ⊆

*C*

*m*. Since

*t*

*m*

*>*0, there exists

*y*1

*, . . . , y*

*P*∈

*C*and a nonnegative number

*λ*1

*, . . . , λ*

*P*with

*λ*1· · ·λ

*P*1 such that

*x**n*−^{P}

*i1*

*λ**i**y**i*

*< t**m**,* 4.14

y*i*−*S*^{hm}_{im}*y**i*≤*t**m*x*m*−*S*^{hm}_{im}*x**m*, ∀i∈ {1, . . . , P}. 4.15

By the boundedness of*C*and{k*n*}, we can put the following:

*M*sup

*x∈C*x, *uP*^{}^{N}_{i1}_{FS}_{i}_{}*x*0*,* *r*0sup

*n≥1*1*k**n*x*n*−*u.* 4.16

This together with4.14implies that

*x**n*− 1
*k**m*

*P*
*i1*

*λ**i**y**i*

≤

1− 1

*k**m*

x 1
*k**m*

*x**n*−^{P}

*i1*

*λ**i**y**i*

≤

1− 1

*k**m*

*Mt**m**,*
y*i*−*S*^{hm}_{im}*y**i*≤*t**m*x*m*−*S*^{hm}_{im}*x**m*

≤*t**m*x*m*−*S*^{hm}_{im}*ut**m*S^{hm}_{im}*u*−*S*^{hm}_{im}*x**m*

≤*t**m*x*m*−*ut**m**k**m*u−*x**m*

≤*t**m*1*k**m*x*m*−*u*

≤*t**m**r*0*,*

4.17

for all*i*∈ {1, . . . , N}. Therefore, for each*i*∈ {1, . . . , P}, we get
*y**i*− 1

*k**m**S*^{hm}_{im}*y**i*

≤y*i*−*S*^{hm}_{im}*y**i*

*S*^{hm}_{im}*y**i*− 1

*k**m**S*^{hm}_{im}*y**i*

≤*r*0*t**m*

1− 1
*k**m*

*M.*

4.18

Moreover, since each*S**i*,*i*∈ {1,2, . . . , N}, is asymptotically nonexpansive, we can obtain that

1
*k**m**S*^{hm}_{im}

_{P}

*i1*

*λ**i**y**i*

−*S*^{hm}_{im}*x**n*

≤

1
*k**m**S*^{hm}_{im}

_{P}

*i1*

*λ**i**y**i*

− 1

*k**m**S*^{hm}_{im}*x**n*

1

*k**m**S*^{hm}_{im}*x**n*−*S*^{hm}_{im}*x**n*

≤

*P*
*i1*

*λ**i**y**i*−*x**n*

1− 1

*k**m*

*M*

*t**m*

1− 1
*k**m*

*M.*

4.19

It follows from Theorem2.3and the inequalities4.17–4.19that
x*n*−*S*^{hm}_{im}*x**n*≤

*x**n*− 1
*k**m*

*P*
*i1*

*λ**i**y**i*

1

*k**m*

*P*
*i1*

*λ**i*

*y**i*−*S*^{hm}_{im}*y**i*
1

*k**m*

*P*
*i1*

*λ**i**S*^{hm}_{im}*y**i*−*S*^{hm}_{im}_{P}

*i1*

*λ**i**y**i*

1
*k**m**S*^{hm}_{im}

_{P}

*i1*

*λ**i**y**i*

−*S*^{hm}_{im}*x**n*

≤2

1− 1
*k**m*

*Mt**m*

*r*0*t**m*

*k**m*

*γ*^{−1}

1≤i≤j≤Nmax

*y**i*−*y**j*− 1
*k**m*

S^{hm}_{im}*y**i*−*S*^{hm}_{im}*y**j*

2

1− 1
*k**m*

*M*2t*m* *r*0*t**m*

*k**m*

*γ*^{−1}

1≤i≤j≤Nmax

*y**i*−*y**j*− 1
*k**m*

S^{hm}_{im}*y**i*−*S*^{hm}_{im}*y**j*

≤2

1− 1
*k**m*

*M*2t*m* *r*0*t**m*

*k**m*

*γ*^{−1}

1≤i≤j≤Nmax

*y**i*− 1

*k**m**S*^{hm}_{im}*y**i*

*y**j*− 1

*k**m**S*^{hm}_{im}*y**j*

≤2

1− 1
*k**m*

*M*2t*m* *r*0*t**m*

*k**m* *γ*^{−1}

2

1− 1
*k**m*

*M*2r0*t**m*

*.*

4.20

Since lim_{n}_{→ ∞}*k**n*1 and lim_{n}_{→ ∞}*t**n*0, it follows from the above inequality that

*n*lim→ ∞

x*n*−*S*^{hm}_{im}*x**n*0. 4.21

Hence4.13is proved. Next, we show that

*n*lim→ ∞x*n*−*S**l**x**n*0; ∀ *l*1,2, . . . , N. 4.22
From the construction of*C**n*, one can easily see that

x*n1*−*S*^{hn}_{in}*x**n1*≤*t**n*x*n*−*S*^{hn}_{in}*x**n*. 4.23

The boundedness of*C*and lim*n*→ ∞*t**n*0 implies that

*n*lim→ ∞

x*n1*−*S*^{hn}_{in}*x**n1*0. 4.24
On the other hand, since for any positive integer*n > N,n* n−*NmodN*and*n* hn−

1N*in, we have*

*n*−*N* hn−1N*in hn*−*N*−1N*in*−*N* 4.25
that is

*hn*−*N hn*−1, *in*−*N in.* 4.26

Thus,

x*n*−*S**n**x**n* ≤ x*n*−*x**n1*x*n1*−*S*^{hn}_{in}*x**n1*S^{hn}_{in}*x**n1*−S_{n}*x**n*

≤ x*n*−*x**n1*x*n1*−*S*^{hn}_{in}*x**n1*S^{hn}_{in}*x**n1*−*S**n**x**n1*S*n**x**n1*−*S**n**x**n*

≤1*k*1x*n*−*x**n1*x*n1*−*S*^{hn}_{in}*x**n1**k*1S^{hn−1}_{in}*x**n1*−*x**n1*

≤1*k*1x*n*−*x**n1*x*n1*−*S*^{hn}_{in}*x**n1*

*k*1S^{hn−1}_{in}*x**n1*−*S*^{hn−1}_{in}*x**n*S^{hn−1}_{in}*x**n*−*x**n*x*n*−*x**n1*

≤12k1x*n*−*x**n1*x*n1*−*S*^{hn}_{in}*x**n1*

*k*1S^{hn−N}_{in−N}*x**n1*−*S*^{hn−N}_{in−N}*x**n**k*1S^{hn−N}_{in−N}*x**n*−*x**n*

≤12k1x*n*−*x** _{n1}*x

*n1*−

*S*

^{hn}

_{in}*x*

_{n1}*k*1

*k*

*x*

_{n−N}*−*

_{n1}*x*

*n*

*k*1S

^{hn−N}

_{in−N}*x*

*n*−

*x*

*n*

≤12k1*k*1*k** _{n−N}*x

*n*−

*x*

*x*

_{n1}*n1*−

*S*

^{hn}

_{in}*x*

_{n1}*k*1S

^{hn−N}

_{in−N}*x*

*n*−

*x*

*n*. 4.27

Applying the facts4.11,4.13, and4.24to the above inequality, we obtain

*n*lim→ ∞x*n*−*S**n**x**n*0. 4.28

Therefore, for any*j* 1,2, . . . , N, we have

*x**n*−*S*_{nj}*x**n*≤*x**n*−*x*_{nj}*x** _{nj}*−

*S*

_{nj}*x*

_{nj}*S*

_{nj}*x*

*−*

_{nj}*S*

_{nj}*x*

*n*

≤*x**n*−*x**nj**x**nj*−*S**nj**x**nj**k*1*x**nj*−*x**n*

1*k*1*x**n*−*x*_{nj}*x** _{nj}*−

*S*

_{nj}*x*

*−→0 as*

_{nj}*n*−→ ∞,

4.29

which gives that

*n*lim→ ∞x*n*−*S**l**x**n*0; ∀l1,2, . . . , N, 4.30
as required. Since{x*n*}is bounded, there exists a subsequence{x*n**i*}of{x*n*}such that*x**n**i*

*x*∈*C. It follows from Lemma*2.4that*x*∈*FS**l*for all*l*1,2, . . . , N. That is*x*∈_{N}

*i1**FS**i*.
Next, we show that *x* ∈ GMEPf, T. By the construction of *D**n*, we see from
Theorem2.1thatΦ*r**n**x**n**P**D**n**x**n*. Since*x** _{n1}*∈

*D*

*n*, we get

x*n*−Φ*r**n**x**n* ≤ x*n*−*x**n1* −→0. 4.31

Furthermore, since lim inf*n*→ ∞*r**n**>*0, we have
1

*r**n*Jx*n*−Φ*r**n**x**n* 1

*r**n*x*n*−Φ*r**n**x**n* −→0, 4.32
as*n* → ∞. By4.32, we also haveΦ*r*_{ni}*x**n**i* *x. By the definition of* Φ*r** _{ni}*, for each

*y*∈

*C, we*obtain

*f*

Φ*r*_{ni}*x**n**i**, y*

*TΦ**r*_{ni}*x**n**i**, η*

*y,*Φ*r*_{ni}*x**n**i*

*ϕ*
*y*

1
*r**n**i*

*y*−Φ*r*_{ni}*x**n**i**, J*

Φ*r*_{ni}*x**n**i*−*x**n**i*

≥*ϕ*
Φ*r*_{ni}*x**n**i*

*.*

4.33
By A3,4.32,ii, the weakly lower semicontinuity of *ϕ*and *η-hemicontinuity ofT*, we
have

*ϕx* ≤lim inf

*i*→ ∞ *ϕ*
Φ*r*_{ni}*x**n**i*

≤lim inf

*i*→ ∞ *f*

Φ*r*_{ni}*x**n**i**, y*

lim inf

*i*→ ∞

*T*Φ*r*_{ni}*x**n**i**, η*

*y,*Φ*r*_{ni}*x**n**i*

*ϕ*

*y*

lim inf

*i*→ ∞

1
*r**n**i*

*y*−Φ*r*_{ni}*x**n**i**, J*

Φ*r*_{ni}*x**n**i*−*x**n**i*

≤*f*
*x, y*

*ϕ*
*y*

*Tx, η*
*y,x*

*.*

4.34

Hence,

*f*
*x, y*

*ϕ*
*y*

*Tx, η*
*y,x*

≥*ϕx.* 4.35

This shows that*x*∈EPf, Tand hence*x*∈Ω:_{N}

*i1**FS**i*∩GMEPf, T.

Finally, we show that*x**n* → *w* as*n* → ∞, where*w* : *P*_{Ω}*x*0. By the weakly lower
semicontinuity of the norm, it follows from4.6that

x0−*w ≤ x*0−*x ≤* lim inf

*i*→ ∞ x0−*x**n**i* ≤lim sup

*i*→ ∞ x0−*x**n**i* ≤ x0−*w.* 4.36
This shows that

*i*lim→ ∞x0−*x**n**i*x0−*w*x0−*x* 4.37

and*x* *w. SinceE*is uniformly convex, we obtain that*x*0−*x**n**i* → *x*0−*w. It follows that*
*x**n**i* → *w. So we havex**n* → *w*as*n* → ∞. This completes the proof.