29

数理解析研究所講究録
第2011巻 2016年 29-35

AN ITERATIVE ALGORITHM FOR GENERALIZED MIXEDEQUILIBRIUM PROBLEMS, MONOTONE MAPPINGS AND
PSEUDOCONTRACTIVE MAPPINGSJONG SOO JUNG
DEPARTMENT OF MATHEMATICS, DONG‐A UNIVERSITY

ABSTRACT. In this paper, we introduce a new iterative algorithm for finding a common
element of the set of solutions of a generalized mixed equilibrium problem related to acontinuous monotone mapping, the set of solutions of a variational inequality problem foracontinuous monotone mapping, and the set of fixed points of a continuous pseudocon‐

tractive mapping in Hilbert spaces. Weak convergence for the proposed iterative algorithm
is proved.

1. INTRODUCTION
Let H be a real Hilbert space with inner product

C be a nonempty closed convex subset of H.

\rangle and induced norm \Vert.

and letLet B : C\rightarrow H be a nonlinear mapping, let$\varphi$ : C\rightarrow \mathbb{R} be a function, and let $\Theta$ be a bifunction of C\times C into \mathbb{R} , where \mathbb{R} is the set ofreal numbers.
The generalized mixed equilibrium problem

that

(1.1)
was

(for short, GMEP) of finding x\in C such$\Theta$(x, y)+\{Bx, y-x\rangle+ $\varphi$(y)- $\varphi$(x)\geq 0, \forall y\in C,

introduced by Peng and Yao [17]denoted by GMEP ( $\Theta$, $\varphi$, B)(also see [24]). The set of solutions of the GMEP is.

The GMEP is very general in the sense that it includes, as special cases, the generalized
equilibrium problem (for short, GEP) in case that $\varphi$=0 in (1.1) ([20]), the mixed equilib‐
rium problem (for short, MEP) in case that B=0 in (1.1) ([5, 22
the equilibrium problem(for short, EP) in case that B=0 and $\varphi$=0 in (1.1) ([2, 8, 9]) and others. In particular, if$\varphi$=0 and $\Theta$(x, y)=0 for all x, y\in C in (1.1), the GMEP reduces the following variational
inequality problem (for short, VIP) of finding x\in C such that

{Bx, y-x)\geq 0,\forall y\in C.

The set of solutions of the VIP is denoted by VI(C, B).

A mapping F of C into H is called monotone if\langle x-y, Fx-Fy\rangle\geq 0, \forall x, y\in C.
A mapping F of C into H is called a inverse‐strongly monotone (see [10]) if there exists a

positive real number $\alpha$ such that\{x-y Fx—Fy )\geq $\alpha$\Vert Fx-Fy\Vert^{2},,\forall x, y\in C.

Recall that a mapping T:C\rightarrow H is said to be pseudocontractive if\{x-y Tx‐Ty)\leq\Vert x-y\Vert^{2},,

2010 Mathematics Subject Classification.

Primary 47\mathrm{H}05 ; Secondary 47\mathrm{H}09, 47\mathrm{H}10, 47\mathrm{J}05, 47\mathrm{J}20, 47\mathrm{J}25.
Key words and phrases. Generalized mixed equilibrium problem, Fixed point, Continuous pseudocontrac‐
tive mapping, Continuous monotone mapping, Variational inequality, Metric projection, Weak convergence.
The results presented in this lecture are collected mainly from the work [13] by the author of this report.

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JONG SOO JUNG

and T is said to be k ‐strictly pseudocontractive if there exists a constant k\in[0 , 1) such that{ x-y Tx‐,Ty} \leq\Vert x-y\Vert^{2}-k\Vert(I-T)x-(I-T)y\Vert^{2},\forall x, y\in C.

where I is the identity mapping. A mapping T of C into itself is called nonexpansive if \Vert Tx-

Ty\Vert\leq\Vert x-y\forall x, y\in C Obviously, the class of k‐strictly pseudocontractive mappings.

includes the class of nonexpansive mappings as a subclass, and the class of pseudocontractive

mappings includes the class of strictly pseudocontractive mappings as a subclass. Moreover,this inclusion is strict due to an example in [7] (see, also Example 5.7.1 and Example 5.7.2
in [1]).

Recently, many authors have introduced some iterative algorithms for finding a common
element of the set of the solutions of the GMEP, the GEP, the MEP, the EP, and the

VIP combined with the fixed point problem for strictly pseudocontractrive mappings; see[3, 4, 6, 12, 15] and the references therein.
In 2007, Tada and Takahashi [19] introduced an iterative algorithm for finding a com‐element of the set of solutions of the EP and the set of fixed points of a nonexpansive
mapping, and proved weak convergence of the sequence generated by the proposed iterativealgorithm. In 2008, Moudafi [16] proposed an iterative algorithm for finding a common ele‐ment of the set of solutions of the GEP related to an $\alpha$-\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}‐strongly monotone mapping
B and the set of fixed points of a nonexpansive mapping, and obtained weak convergenceof the sequence generated by the proposed iterative algorithm. In 2009, Ceng et al. [4]presented an iterative algorithm for finding a common element of the set of solutions of the
EP and the set of fixed points of a k ‐strictly pseudocontractive mapping, and showed weak
convergence of the sequence generated by the proposed iterative algorithm. In 2012, Jungmon[12] considered an iterative algorithm for finding a common element of the set of solutionsof the GMEP related to $\alpha$-\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}‐strongly monotone mapping B , the set of solutions of
the VIP for $\beta$-\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e}‐strongly monotone mapping F and and the set of fixed points of ak ‐strictly pseudocontractive mapping, and established weak convergence of the sequence
generated by the proposed iterative algorithm.
On the other hand, in 2003, Takahashi and Toyoda [21] proposed an iterative algorithm
for finding a common element of the set of solutions of the VIP for $\alpha$-\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e} ‐strongly
monotone mapping F and the set of fixed points of a nonexpansive mapping, and proved
weak convergence of the sequence generated by the proposed iterative algorithm.
In this paper, motivated and inspired by the above mentioned results, we introduce a
new iterative algorithm for finding a common element of the set of solutions of the GMEP
related to a continuous monotone mapping B the set of solutions of the \mathrm{V}\mathrm{l}\mathrm{P} for a continuous
monotone mapping F and the set of fixed points of a continuous pseudocontractive mapping
T in a Hilbert space. We prove weak convergence of the sequence generated by the proposed
iterative algorithm to a common element of three sets. As direct consequences, we obtain
the results for the GEP related to a continuous monotone mapping B the MEP and theEP, combined with the VIP for a continuous monotone mapping F and the fixed point
problem for a continuous pseudocontractive mapping T Our results extend, improve, anddevelop some recent results in the literature.
,,.

2. PRELIMINARIES AND LEMMAS

In the following,
we

wedenote by Fix (T) the set of fixed points of the mapping T , anddenote the strong convergence and the weak convergence of \{x_{n}\} to x by x_{n}\rightarrow x and

x_{n}\rightarrow x , respectively.
Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H
H , we have

\Vert $\lambda$ x+(1- $\lambda$)y\Vert^{2}= $\lambda$\Vert x\Vert^{2}+(1- $\lambda$)\Vert y\Vert^{2}- $\lambda$(1- $\lambda$)\Vert x-y\Vert^{2}

.

In

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AN ITERATIVE ALGORITHMS FOR GENERALIZED MIXED EQUILIBRIUM PROBLEMS

for all x, y\in H and $\lambda$\in \mathbb{R}.

For every point x\in H , there exists a unique nearest point in

C denoted by P_{C}(x) , such that,

\Vert x-P_{C}(x)\Vert\leq\Vert x-y\Verty\in C. P_{C} is called the metric projection of H onto C. P_{C}(x) is characterized byfor allthe property:(2.1)

u=P_{C}(x)\Leftrightarrow\langle x-u, u-y)\geq 0 for all x\in H, y\in C.

It is also well known that H satisfies the Opial condition, that is, for any sequence \{x_{n}\}with x_{n}\rightarrow x , the inequality\displaystyle \lim_{n\rightar ow}\inf_{\infty}\Vert x_{n}-x\Vert<\lim \mathrm{i}t\mathrm{f}\mathrm{f}n\rightar ow\infty\Vert x_{n}-y\Vertholds for every

y\in H with y\neq x.
For solving the GMEP, the GEP, the MEP, and the EP for a bifunction $\Theta$ : C\times C\rightarrow \mathbb{R},let us assume that $\Theta$ satisfies the following conditions:(A1) $\Theta$(x, x)=0 for all x\in C ;(A2) $\Theta$ is monotone, that is, $\Theta$(x, y)+ $\Theta$(y, x)\leq 0 for all x, y\in C ;(A3) for each x, y, z\in C,\displaystyle \lim_{t\downarrow}\sup_{0} $\Theta$(tz+(1-t)x, y)\leq $\Theta$(x,y);(A4) for each x\in C, y\mapsto $\Theta$(x, y) is convex and lower semicontinuous.
We can prove the following lemma by using the same method as in [14, 24], and so weomit its proof.
Lemma 2. 1.LetC be a nonempty closed convex subset of H. Let $\Theta$ be a bifunction form

C\times C to \mathbb{R} satisfies (\mathrm{A}1)-(\mathrm{A}4) , and let $\varphi$ : C\rightarrow \mathbb{R} be a proper lower semicontinuous andconvexfunction. Let B : C\rightarrow H be a continuous monotone mapping. Then, for r>0 andx\in H , there eststs u\in C such that$\Theta$(u, y)+\displaystyle \{Bu, y-u)+ $\varphi$(y)- $\varphi$(u)+\frac{1}{r}\{y-u, u-x\rangle\geq 0, \forall y\in C.
Defineamapping K_{r}:H\rightarrow C as follows:

K_{r}x=\{u\in C $\Theta$(u,y)+\langle Bu, y-u\rangle+ $\varphi$(y)- $\varphi$(u)+\displaystyle \frac{1}{r}\{y-u, u-x\rangle\geq 0, \foral y\in C\}:

for all x\in H and r>0 Then, the following hold:.

(i)K_{r}(x)\neq\emptyset ;(ii) K_{r} is single‐valued;(iii) K_{r} is firmly nonexpansive, that is, for any x, y\in H,
For each x\in H,\Vert K_{r}x-K_{r}y\Vert^{2}\leq\langle K_{r}x-K_{r}y, x-y\rangle ;Fix (K_{r})=GMEP( $\Theta$, $\varphi$, B) ;(iv)
(v) GMEP ( $\Theta$, $\varphi$, B) is closed and convex.
We also need the following lemmas for the proof of our main results.
Lemma 2.2([18]). Let H be a real Hilbert space, let \{$\alpha$_{n}\} be a sequence of real numbers

such that 0<a\leq$\alpha$_{n}\leq b<1 for all n\geq 1 , and let \{v_{n}\} and \{w_{n}\} be sequences in H suchthat, for some c\displaystyle \lim_{n\rightar ow}\sup_{\infty}\Vert v_{n}\Vert\leq c, \displaystyle \lim_{n\rightar ow}\sup_{\infty}\Vert w_{n}\Vert\leq c,and\displaystyle \lim_{n\rightar ow}\sup_{\infty}\Vert$\alpha$_{n}v_{n}+(1-$\alpha$_{n})w_{n}\Vert=\mathrm{c}.

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JONG SOO JUNG

Then \displaystyle \lim_{n\rightarrow\infty}\Vert v_{n}-w_{n}\Vert=0.
Lemma 2.3([21]). Let C be a nonempty closed convex subset of a real Hilbert spaces H,

and let \{x_{n}\} be a sequence in H.

If\Vert x_{n+1}-x\Vert\leq\Vert x_{n}-x\forall x\in C and Vn \geq 1,then \{P_{C}x_{n}\} converges strongly to some z\in C , where P_{C} stands for the metric projectionof H onto C.

The following lemmas are Lemma 2.3 and Lemma 2.4 of Zegeye [23], respectively.
Lemma 2.4 ([23]). Let C be a closed convex subset of a real Hilbert space H. Let F:C\rightarrow
H be a continuous monotone mapping.

Then, for r>0 and x\in H there exists z\in C such,that\displaystyle \{Fz, y-z\}+\frac{1}{r}\langle y-z, z-x\rangle\geq 0, \foral y\in C.

For r>0 and x\in H , define F_{r}:H\rightarrow C by

F_{r}x=\{z\in C { y-z)+\displaystyle \frac{1}{r}\{y-z, z-x\rangle\geq 0, \foral y\in C\}.
:Fz ,Then the following hold:(i) F_{r} is single‐valued;(ii) F_{r} is firmly nonexpansive, that is,\Vert F_{r}x-F_{r}y\Vert^{2}\leq\{F_{r}x-F_{r}y, x-y\rangle, \forall x, y\in H ;(iii) Fix (F_{r})=VI(C, F) ;(iv) VI(C, F) is a closed convex subset of C.
Lemma 2.5 ([23]). Let C be a closed convex subset of a real Hilbert space H. Let T:C\rightarrow
H beacontinuous pseudocontractive mapping.

Then, for r>0 and x\in H,

z\in C such that\langle Tz
For r>0 and x\in H ,,y-z\displaystyle \rangle-\frac{1}{r}\{y-z, (1+r)z-x\rangle\leq 0, \forall y\in C.

define T_{r} : H\rightarrow C by

T_{r}x=\{z\in C { y-z)-\displaystyle \frac{1}{r}\{y-z, (1+r)z-x)\leq 0, \forall y\in C\}.
:Tz ,Then the following hold:(i) T_{r} is single‐valued;(ii) T_{r} is firmly nonexpansive, that is,\Vert T_{r}x-T_{r}y\Vert^{2}\leq\langle T_{r}x-T_{r}y, x-y\rangle, \forall x, y\in H ;(iii) Fix (T_{r})=Fix(T) ;(iv) Fix (T) is a closed convex subset of C.
3. MAIN RESULTS

Throughout the rest of this paper, we always assume the following:
H is a real Hilbert space;

C is a nonempty closed convex subset of H ;$\Theta$ is a bifunction form C\times C to \mathbb{R} satisfies (\mathrm{A}1)-(\mathrm{A}4) ;$\varphi$ : C\rightarrow \mathbb{R} is a proper lower semicontinuous and convex function;
B:C\rightarrow H is a continuous monotone mapping;

GMEP( $\Theta$, $\varphi$, B) is the set of solutions of the GMEP related to B :there exists

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AN ITERATIVE ALGORITHMS FOR GENERALIZED MIXEDEQUILIBRIUM PROBLEMS

K_{r_{n}} : H\rightarrow C is a mapping defined by

K_{r_{n}}x=\{u\in C $\Theta$(u, y)+\{Bu, y-u\rangle+ $\varphi$(y)- $\varphi$(u)+\displaystyle \frac{1}{r_{n} \langle y-u, u-x\}\geq 0, \foral y\in C\}:

for all x\in H and for r_{n}\in(0, \infty) and \displaystyle \lim\inf_{n\rightarrow\infty}r_{n}>0 ;F:C\rightarrow H is a continuous monotone mapping;

VI(C, F) is the set of solutions of the VIP for F ;\bullet

T:C\rightarrow C is a continuous pseudocontractive mapping;

F_{r_{n}} : H\rightarrow C is a mapping defined by

F_{r_{n}}x=\{z\in C \langle Fz y-z\displaystyle \}+\frac{1}{r_{n} \langle y-z, z-x)\geq 0, \foral y\in C\}:,

for all x\in H and for r_{n}\in(0, \infty) and \displaystyle \lim\inf_{n\rightarrow\infty}r_{n}>0 ;T_{r_{n}} : H\rightarrow C is a mapping defined by

T_{r_{n}}x=\{z\in C \langle Tz y-z\displaystyle \rangle-\frac{1}{r_{n} \langle y-z, (1+r_{n})z-x)\leq 0, \foral y\in C\}:,

for all x\in H and for r_{n}\in(0, \infty) and \displaystyle \lim\inf_{n\rightarrow\infty}r_{n}>0 ;\bullet$\Omega$_{1} :=GMEP( $\Theta$, $\varphi$, B)\cap VI(C, F)\cap Fix(T)\neq\emptyset.

By Lemma 2.1, Lemma 2.4 and Lemma 2.5, we note that K_{r_{n}}, F_{r_{n}} and T_{r_{n}} are nonexpansive,
and Fix (K_{r_{n}})=GMEP( $\Theta$, $\varphi$, B) , Fix (F_{r_{n}})=VI(C, F) and Fix (T_{r_{n}})=Fix(T).

Now, we propose a new iterative algorithm for finding a common point of the set ofsolutions of the GMEP related to a continuous monotone mapping B , the set of solutionsof the VIP for a continuous monotone mapping F , and the set of fixed points of a continuous

pseudocontractive mapping T.
Algorithm 3.1. For an arbitrarily chosen x_{1}\in C let the iterative sequences \{x_{n}\} and\{u_{n}\} be generated by,\left{\begin{ar y}{l $\Theta$(u_{n},y)+\{Bu_n},y-u_{n}\+$\varphi$(y)-$\varphi$(u_{n})\ +\frac{1}r_{n}\langey-u_{n},u_{n}-x_{n})\geq0,\foraly\inC,\ x_{n+1}=$\alph$_{n}x_{n}+(1-$\alph$_{n})T_{r n}F_{r n}K_{r n}x_{n},\foraln\geq1, \end{ar y}\right.

(3.1)

where \{$\alpha$_{n}\}\subset[a, b] for some a,b\in(0,1) and r_{n}\in(0, \infty) and \displaystyle \lim\inf_{n\rightarrow\infty}r_{n}>0.

Theorem 3.1. The sequences \{x_{n}\} and \{u_{n}\} generated by Algorithm 3.1 converge weakly
to z\in$\Omega$_{1} , where z=\displaystyle \lim_{n\rightarrow\infty}P_{$\Omega$_{1} (xn).

If we take C\equiv H in Theorem 3.1, then we obtain the following result.

Corollary 3.2. Let $\Omega$_{2} :=GMEP( $\Theta$, $\varphi$, B)\cap F^{-1}(0)\cap Fix(T) The sequences \{x_{n}\} and\{u_{n}\} generated by Algorithm 3.1 converge weakly to z\in$\Omega$_{2} where z=\displaystyle \lim_{n\rightar ow\infty}P_{$\Omega$_{2} (xn).
,

Proof.

Since D(F)=H ,Theorem 3.1.

wehave VI(H, F)=F^{-1}(0).

Thus the result follows from

Now, in order to obtain direct consequences of Theorem 3.1, we recall special cases of
the GMEP again.
If $\varphi$=0 in (1.1), then the GMEP reduces the following generalized equilibrium problem(for short, GEP) of finding x\in C such that$\Theta$(x, y)+\langle Bx, y-x)\geq 0, \forall y\in C.
The set of solutions of the GEP is denoted by GEP( $\Theta$, B).

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JONG SOO JUNG

If B=0 in (1.1), then the GMEP reduces the following mixed equilibrium problem (forshort, MEP) of finding x\in C such that$\Theta$(x, y)+ $\varphi$(y)- $\varphi$(x)\geq 0, \forall y\in C.
The set of solutions of the MEP is denoted by MEP( $\Theta$, $\varphi$).

If B=0 and $\varphi$=0 in (1.1), then the GMEP reduces the following equilibrium problem(for short, EP) of finding x\in C such that$\Theta$(x, y)\geq 0, \forall y\in C.
The set of solutions of the EP is denoted by EP( $\Theta$).

If we take $\varphi$\equiv 0 in Theorem 3.1, then we obtain the following result.

Corollary 3.3. Let $\Omega$_{3} :=GEP( $\Theta$, B)\cap VI(C, F)\cap Fix(T)\neq\emptyset
be sequences generated by x_{1}\in C and,and let \{x_{n}\} and \{u_{n}\}\left\{ begin{ar ay}{l $\Theta$(u_{n},y)+\{Bu_{n},y-u_{n}\rangle+\frac{1}{r_n}\{y-u_{n},u_{n}-x_{n}\rangle\geq0,&\foral y\inC,\ x_{n+1}=$\alpha$_{n}x_{n}+(1-$\alpha$_{n})T_{r n}F_{r n}u_{n},\foral n\geq1.& \end{ar ay}\right.
Then \{x_{n}\} and \{u_{n}\} converge weakly to z\in$\Omega$_{3} , where z=\displaystyle \lim_{n\rightar ow\infty}P_{$\Omega$_{3} (xn).
If we take B\equiv 0 in Theorem 3.1, we get the following result.

Corollary 3.4. Let $\Omega$_{7}:=MEP( $\Theta$, $\varphi$)\cap VI(C, F)\cap Fix(T)\neq\emptyset
be sequences generated by x_{1}\in C and,and let \{x_{n}\} and \{u_{n}\}\left\{ begin{ar ay}{l $\Theta$(u_{n},y)+$\varphi$(y)-$\varphi$(u_{n})+\frac{1}r_{n}\langley-u_{n},u_{n}-x_{n})\geq0,&\foral y\inC,\ x_{n+1}=$\alpha$_{n}x_{n}+(1-$\alpha$_{n})T_{r n}F_{r n}u_{n},\foral n\geq1.& \end{ar ay}\right.
Then \{x_{n}\} and \{u_{n}\} converge weakly to z\in$\Omega$_{7} , where z=\displaystyle \lim_{n\rightar ow\infty}P_{$\Omega$_{7} (xn).
It we take B\equiv 0 and $\varphi$\equiv 0 in Theorem 3.1, we obtain the following result.

Corollary 3.5. it Let $\Omega$_{5} :=EP( $\Theta$)\cap VI(C, F)\cap Fix(T)\neq\emptyset and let \{x_{n}\} and \{u_{n}\} besequences generated by x_{1}\in C and,\left\{ begin{ar y}{l $\Theta$(u_{n},y)+\frac{1}r_{n}\langley-u_{n},u_{n}-x_{n}\rangle\geq0,\foral y\inC,\ x_{n+1}=$\alpha$_{n}x_{n}+(1-$\alpha$_{n})T_{r n}F_{r n}u_{n},\foral n\geq1. \end{ar y}\right.
Then \{x_{n}\} and \{u_{n}\} converge weakly to z\in$\Omega$_{5} , where z=\displaystyle \lim_{n\rightar ow\infty}P_{$\Omega$_{5} (xn).
Remark 3.1.

1) For finding a common element of GMEP ( $\Theta$, $\varphi$, B)\cap VI(C, F)\cap Fix(T),where B is a continuous monotone mapping, F is a continuous monotone mapping, and T isa continuous

pseudocontractive mapping, Theorem 3.1 is a new ones different from previous
by several authors. Consequently, in the sense that our convergenceis for the more general class of continuous monotone and continuous pseudocontractive
mappings, our results improve, develop and complement the corresponding results, whichwere obtained recently by several authors in references; for example, see[4, 12, 16, 19, 21]
those introduced

and references therein.

ACKNOWLEDGMENTS
This research was supported by the Basic Science Research Program through the National

Research Foundation of Korea (NRF) funded by the Ministry of Education, Science andTechnology (2014010491).

35
AN ITERATIVE ALGORITHMS FOR GENERALIZED MIXEDEQUILIBRIUM PROBLEMSREFERENCES[1]

R. P. Agarwal, D. ORegan and D. R. Sahu, Fixed Point

Theory for Lipschitntan‐type Mappings with
Applications, Springer (2009).
[2] E. Blum and W. Oettli, FVvm optimization and variational inequalities to equilibrium problems, Math.
Student, 63 (1994), 123‐145.
[3] A. Bnouhachem, An iterative algorthm for system of generalized equilbreum problems and fixed point
problem, Fixed Point Theory Appl. 20142014:235 doi:lO.1186/1687‐1812‐2014‐235.
[4] L.‐C.Ceng, S. Al‐Homidan, Q. H. Ansari and J.‐C.Yao, An iterative scheme for equilibnum problems
and fixred point problems of strect pseudo‐contraction mappings, J. Comput. Appl. Math. 223 (2009),
967‐974.

[5] L.‐C. Ceng and J.‐C. Yao, A hybrid iterative scheme for m $\iota$ xed equilibreum problems and fixed point
problems, J. Comput. Appl. Math. 214 (2008), 186‐201.
[6] T. Chamnarnpan and P. Kumam, A new iterative method for a common solution of a fixed points forpseudo‐contractive mappings and varzational inequalities, Fixed Point Theory Appl. 2012:67 (2012)
doi:10.1186/1687‐1812‐2012‐67.
[7] C. E. Chidume and S. Mutangadura, An expample on the Mann iteration method for Lipschitz pseudo‐
contractions, Proc. Amer. Math. Soc. 129 (2001), 2359‐2363.
[8] P. I. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex
Anal. 6 (2005), 117‐136.
[9] S. D. Flam and A. S. Antipin, Equilibrium programming vsing proximal‐like algorithm, Math. Program.
78 (1997), 29‐41.
[10] H. Iiduka and W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse‐
strongly monotone mappings, Nonlinear Anal. 61 (2005), 341‐350.
[11] J. S. Jung, it A new iteration method for nonexpansive mappings and monotone mappings in Hilbert
spaces, J. Inequal. Appl. 2010 (2010), Article ID 251761, 16 pages, \mathrm{d}\mathrm{o}\mathrm{i}:10.1155/2010/251761.
[12] J. S. Jung, Weak converg ence theorems for strictly pseudocontractive mappings and general‐ized mixed equilbntum problems, J. Appl. Math. 2012 (2012), Article ID 384108, 18 pages,
doi:lO.1155/2012/384108.
[13] J. S. Jung, Weak convergence theorems for generalized mixed equilibrium problems, monotone mappingsand pseudocontractive mappings, J. Korean Math. Soc. 52 (2015) no. 6, 1176−1194[14] P. Katchang, T. Jitpeera and P. Kumam, Strong convergence theorems for solving generalized mixed
equilibrium problems and general system of variational inequalities by the hybr $\iota$ d method, Nonlinear
Anal. Hybrid Systems, 4 (2010), 838‐852.

[15] P. Kumam, U. Hamphries and P. Katchang, Common solutions of generalized mixed equilbrium prob‐
lems_{J} variational inclusions and common fixed points for nonexpansive semigroups and strictly pseudo‐
contractive mappings, J. Appl. Math. 2011 (2011) Atricle ID 953903, 27 pages.
[16] A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems, J. Non‐linear Convex Anal. 9 (2008), no. 1, 37‐43.
[17] J.‐W. Peng and J.‐C. Yao, A new Hybrid‐extragradient method for generalized mixed equilibnum prob‐
lems_{J} fixed point problems and variational inequality problems, Taiwan. J. Math. 12 (2008), 1401‐1432.
[18] J. Schu, Weak and strong convergence to fixed ponts of asymptotically nonexpansive mappings, Bull.

Aust. Math. Soc. 43 (1991), 153‐159.
[19] A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping andan equilibrium problem, J. Optim. Theory Appl. 133 (2007), 359‐370.
[20] S. Takahashi and W. Takahashi, Strong convergence theorem for a generalized equilibrium problem anda nonempansive mapping in a Habert space, Nonlinear Anal. 69 (2008), 1025‐1033.
[21] W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone
mappings, J. Optim. Theory Appl. 118 (2003), no. 2, 417‐428.
[22] Y. H. Yao, M. Aslam Noor, S. Zainab and Y.‐C. Liou, Mixed equihbntm problems and optimization
problems, J. Math. Anal. Appl. 354 (2009), 319‐329.
[23] H. Zegeye, An iterative approximation method for a common fixed point of two pseudocontractive map‐

pings, Interational Scholarly Reserach Network ISRN Math. Anal. 2011 Article ID 621901, 14 pages.

[24] S. Zhang, Generalized mixed equihbrium problems in Banach spaces, Appl. Math. Mech. (English Ed.)
30 (2009), 1105‐1112.
DEPARTMENT OF MATHEMATICS, DONG‐A UNIVERSITY, BUSAN 49315, KOREA
E‐mail address: jungjs@dau.ac.kr