- Research Article
- Open access
- Published:
Iterative Methods for Finding Common Solution of Generalized Equilibrium Problems and Variational Inequality Problems and Fixed Point Problems of a Finite Family of Nonexpansive Mappings
Fixed Point Theory and Applications volume 2010, Article number: 836714 (2010)
Abstract
We introduce a new method for a system of generalized equilibrium problems, system of variational inequality problems, and fixed point problems by using -mapping generated by a finite family of nonexpansive mappings and real numbers. Then, we prove a strong convergence theorem of the proposed iteration under some control condition. By using our main result, we obtain strong convergence theorem for finding a common element of the set of solution of a system of generalized equilibrium problems, system of variational inequality problems, and the set of common fixed points of a finite family of strictly pseudocontractive mappings.
1. Introduction
Let be a real Hilbert space, and let
be a nonempty closed convex subset of
. Let
be a nonlinear mapping, and let
be a bifunction. A mapping
of
into itself is called nonexpansive if
for all
. We denote by
the set of fixed points of
(i.e.,
). Goebel and Kirk [1] showed that
is always closed convex, and also nonempty provided
has a bounded trajectory.
A bounded linear operator on
is called strongly positive with coefficient
if there is a constant
with the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ1_HTML.gif)
The equilibrium problem for is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ2_HTML.gif)
The set of solutions of (1.2) is denoted by . Many problems in physics, optimization, and economics are seeking some elements of
, see [2, 3]. Several iterative methods have been proposed to solve the equilibrium problem, see, for instance, [2–4]. In 2005, Combettes and Hirstoaga [3] introduced an iterative scheme of finding the best approximation to the initial data when
is nonempty and proved a strong convergence theorem.
The variational inequality problem is to find a point such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ3_HTML.gif)
The set of solutions of the variational inequality is denoted by VI, and we consider the following generalized equilibrium problem.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ4_HTML.gif)
The set of such is denoted by
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ5_HTML.gif)
In the case of ,
. Numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games reduce to find element of (1.5)
A mapping of
into
is called inverse-strongly monotone, see [5], if there exists a positive real number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ6_HTML.gif)
for all .
The problem of finding a common fixed point of a family of nonexpansive mappings has been studied by many authors. The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mapping (see [6, 7]).
The ploblem of finding a common element of and the set of all common fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and importance. Many iterative methods are purposed for finding a common element of the solutions of the equilibrium problem and fixed point problem of nonexpansive mappings, see [8–10].
In 2008, S.Takahashi and W.Takahashi [11] introduced a general iterative method for finding a common element of and
. They defined
in the following way:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ7_HTML.gif)
where is an
-inverse strongly monotone mapping of
into
with positive real number
, and
,
,
, and proved strong convergence of the scheme (1.7) to
, where
in the framework of a Hilbert space, under some suitable conditions on
,
,
and bifunction
.
Very recently, in 2010, Qin, et al. [12] introduced a iterative scheme method for finding a common element of ,
and common fixed point of infinite family of nonexpansive mappings. They defined
in the following way:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ8_HTML.gif)
where is a contraction mapping and
is
-mapping generated by infinite family of nonexpansive mappings and infinite real number. Under suitable conditions of these parameters they proved strong convergence of the scheme (1.8) to
, where
.
In this paper, motivated by [11, 12], we introduced a general iterative scheme defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ9_HTML.gif)
where and
is
-mapping generated by
and
. Under suitable conditions, we proved strong convergence of
to
, and
is solution of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ10_HTML.gif)
2. Preliminaries
In this section, we collect and give some useful lemmas that will be used for our main result in the next section.
Let be closed convex subset of a real Hilbert space
, and let
be the metric projection of
onto
, that is, for
,
satisfies the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ11_HTML.gif)
The following characterizes the projection .
Lemma 2.1 (see [13]).
Given and
. Then
if and only if there holds the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ12_HTML.gif)
Lemma 2.2 (see [14]).
Let be a sequence of nonnegative real numbers satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ13_HTML.gif)
where ,
satisfy the conditions
(1),
(2).
Then .
Lemma 2.3 (see [15]).
Let be a closed convex subset of a strictly convex Banach space
. Let
be a sequence of nonexpansive mappings on C. Suppose that
is nonempty. Let
be a sequence of positive numbers with
. Then a mapping
on
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ14_HTML.gif)
for is well defined, nonexpansive, and
hold.
Lemma 2.4 (see [16]).
Let be a uniformly convex Banach space,
a nonempty closed convex subset of
, and
a nonexpansive mapping. Then
is demiclosed at zero.
Lemma 2.5 (see [17]).
Let and
be bounded sequences in a Banach space
, and let
be a sequence in
with
. Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ15_HTML.gif)
for all integer and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ16_HTML.gif)
Then .
For solving the equilibrium problem for a bifunction , let us assume that
satisfies the following conditions:
(A1)for all
;
(A2) is monotone, that is,
,
(A3) for all ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ17_HTML.gif)
(A4) for all is convex and lower semicontinuous.
The following lemma appears implicitly in [2].
Lemma 2.6 (see [2]).
Let be a nonempty closed convex subset of
, and let
be a bifunction of
into
satisfying (A1)–(A4). Let
and
. Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ18_HTML.gif)
for all .
Lemma 2.7 (see [3]).
Assume that satisfies (A1)–(A4). For
and
, define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ19_HTML.gif)
for all . Then, the following hold:
(1) is single-valued;
(2) is firmly nonexpansive, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ20_HTML.gif)
(3);
(4) is closed and convex.
In 2009, Kangtunyakarn and Suantai [18] defined a new mapping and proved their lemma as follows.
Definition 2.8.
Let be a nonempty convex subset of real Banach space. Let
be a finite family of nonexpansive mappings of
into itself. For each
, let
, where
and
. We define the mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ21_HTML.gif)
This mapping is called -mapping generated by
and
.
Lemma 2.9.
Let be a nonempty closed convex subset of strictly convex. Let
be a finite family of nonexpanxive mappings of
into itself with
, and let
,
, where
,
,
for all
,
for all
. Let
be the mapping generated by
and
. Then
.
Lemma 2.10.
Let be a nonempty closed convex subset of Banach space. Let
be a finite family of nonexpansive mappings of
into itself and
,  
, where
,
and
such that
as
for
and
Moreover, for every
, let
and
be the
-mappings generated by
and
and
and
, respectively. Then
for every
.
Lemma 2.11 (see [19]).
Let be a nonempty closed convex subset of a Hilbert space
, and let
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ22_HTML.gif)
with . Then
if and only if
.
3. Main Result
Theorem 3.1.
Let be a nonempty closed convex subset of a Hilbert space
. Let
and
be two bifunctions from
into
satisfying (A1)–(A4), respectively. Let
a
-inverse strongly monotone mapping and
be a
-inverse strongly monotone mapping. Let
be finite family of nonexpansive mappings with
, where
are defined by
,
. Let
be a contraction with the coefficient
. Let
be the S-mappings generated by
and
, where
,
and
and
. Let
be sequences generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ23_HTML.gif)
where such that
,
,
,
,
. Assume that
(i) and
,
(ii),
(iii),
(iv),
,
,
,
,
,
(v), and
, for all
.
Then the sequence ,
,
,
converge strongly to
, and
is solution of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ24_HTML.gif)
Proof.
First, we show that ,
and
are nonexpansive. Let
. Since
is
-strongly monotone and
for all
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ25_HTML.gif)
Thus is nonexpansive. By using the same proof, we obtain that
and
are nonexpansive.
We will divide our proof into 6 steps.
Step 1.
We will show that the sequence is bounded. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ26_HTML.gif)
then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ27_HTML.gif)
By Lemma 2.7, we have . By the same argument as above, we obtaine that
Let . Then
and
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ28_HTML.gif)
Again by Lemma 2.7, we have . Since
, we have
. By nonexpansiveness of
,
,
,
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ29_HTML.gif)
By induction we can prove that is bounded and so are
,
,
,
. Without of generality, assume that there exists a bounded set
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ30_HTML.gif)
Step 2.
We will show that .
Putting , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ31_HTML.gif)
From definition of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ32_HTML.gif)
By definition of , for
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ33_HTML.gif)
By (3.11), we obtain that for each ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ34_HTML.gif)
This together with the condition (iv), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ35_HTML.gif)
By (3.10), (3.13) and conditions (i), (ii), (iii), (iv), it implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ36_HTML.gif)
From Lemma 2.5, (3.9), (3.14) and condition (ii), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ37_HTML.gif)
From (3.9), we can rewrite
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ38_HTML.gif)
By (3.15), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ39_HTML.gif)
On the other hand, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ40_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ41_HTML.gif)
By (3.17) and condition (ii), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ42_HTML.gif)
Step 3.
Let ; we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ43_HTML.gif)
From definition of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ44_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ45_HTML.gif)
By (3.23), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ46_HTML.gif)
By (3.24), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ47_HTML.gif)
From conditions (i)–(iii) and (3.17), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ48_HTML.gif)
By using the same method as (3.26), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ49_HTML.gif)
By nonexpansiveness of and (3.23), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ50_HTML.gif)
By (3.28), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ51_HTML.gif)
By (3.29), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ52_HTML.gif)
From (3.17) and conditions (i)–(iii), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ53_HTML.gif)
By using the same method as (3.31), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ54_HTML.gif)
Step 4.
We will show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ55_HTML.gif)
Putting and
, we will show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ56_HTML.gif)
Let ; by (3.28), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ57_HTML.gif)
By nonexpansiveness of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ58_HTML.gif)
This implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ59_HTML.gif)
By using the same method as (3.37), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ60_HTML.gif)
Substituting (3.37) and (3.38) into (3.35), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ61_HTML.gif)
By (3.39), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ62_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ63_HTML.gif)
By conditions (i)–(iii), (3.41), (3.31), (3.32), and (3.17), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ64_HTML.gif)
By using the same method as (3.42), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ65_HTML.gif)
By nonexpansiveness of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ66_HTML.gif)
Hence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ67_HTML.gif)
By using the same method as (3.45), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ68_HTML.gif)
Substituting (3.45) and (3.46) into (3.35), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ69_HTML.gif)
By (3.47), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ70_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ71_HTML.gif)
From (3.17), (3.26), (3.27), and conditions (i)–(iii), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ72_HTML.gif)
By using the same method as (3.50), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ73_HTML.gif)
By (3.42) and (3.50), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ74_HTML.gif)
By (3.43) and (3.51), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ75_HTML.gif)
Since and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ76_HTML.gif)
By (3.52) and (3.53), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ77_HTML.gif)
Note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ78_HTML.gif)
From (3.20) and (3.55), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ79_HTML.gif)
Step 5.
We will show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ80_HTML.gif)
where . To show this inequality, take subsequence
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ81_HTML.gif)
Since is bounded, there exists a subsequence
of
which converges weakly to
. Without loss of generality, we can assume that
. Since
is closed convex,
is weakly closed. So, we have
. Let us show that
. We first show that
. From (3.42), we have
. Since
, for any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ82_HTML.gif)
From (A2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ83_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ84_HTML.gif)
Put for all
and
. Then, we have
. So, from (3.62), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ85_HTML.gif)
Since , we have
. Further, from monotonicity of
, we have
. So, from (A4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ86_HTML.gif)
From (A1), (A4), and (3.64), we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ87_HTML.gif)
Thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ88_HTML.gif)
Letting , we have, for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ89_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ90_HTML.gif)
From (3.43), we have . Since
, for any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ91_HTML.gif)
From (A2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ92_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ93_HTML.gif)
Put for all
and
. Then, we have
. So, from (3.71) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ94_HTML.gif)
Since , we have
. Further, from monotonicity of
, we have
. So, from (A4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ95_HTML.gif)
From (A1), (A4), and (3.64), we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ96_HTML.gif)
hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ97_HTML.gif)
Letting , we have, for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ98_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ99_HTML.gif)
Define a mapping by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ100_HTML.gif)
where . From Lemma 2.3, we have that
is nonexpansive with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ101_HTML.gif)
Next, we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ102_HTML.gif)
By nonexpansiveness of and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ103_HTML.gif)
where . From (3.17), (3.42), (3.43), (3.55), and condition (iii), we have
. Since
, it follows from (3.80) that,
. By Lemma 2.4, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ104_HTML.gif)
Assume that . Using Opial
property, (3.57) and Lemma 2.10 we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ105_HTML.gif)
This is a contradiction, so we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ106_HTML.gif)
From (3.68), (3.77) (3.82), and (3.84), we have . Since
is contraction with the coefficient
,
has a unique fixed point. Let
be a fixed point of
, that is
. Since
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ107_HTML.gif)
Step 6.
Finally, we will show that as
. By nonexpansiveness of
, we can show that
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ108_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ109_HTML.gif)
By Step 5, (3.87), and Lemma 2.2, we have , where
. It easy to see that sequences
,
, and
converge strongly to
.
4. Application
Using our main theorem (Theorem 3.1), we obtain the following strong convergence theorems involving finite family of -strict pseudocontractions.
To prove strong convergence theorem in this section, we need definition and lemma as follows.
Definition 4.1.
A mapping is said to be a
-strongly pseudo contraction mapping, if there exist
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ110_HTML.gif)
Lemma 4.2 (see [20]).
Let be a nonempty closed convex subset of a real Hilbert space
and
a
-strict pseudo contraction. Define
by
for each
. Then, as
  
is nonexpansive such that
.
Theorem 4.3.
Let be a nonempty closed convex subset of a Hilbert space
. Let
and
be two bifunctions from
into
satisfying (A1)–(A4), respectively. Let
is a
-inverse strongly monotone mapping and
be a
-inverse strongly monotone mapping. Let
be a finite family of
-psuedo contractions with
, where
are defined by
,
for all
. Define a mapping
by
, for all
. Let
be a contraction with the coefficient
. Let
be the S-mappings generated by
and
, where
,
and
for all
for all
and
for all
for all
. Let
be sequences generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ111_HTML.gif)
where ,
,
such that
,
,
,
,
. Assume that
(i),
(ii),
(iii),
(iv),
,
,
,
,
,
(v)and
, for all
.
Then the sequence ,
,
,
converges strongly to
, and
is solution of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ112_HTML.gif)
Proof.
For every , by Lemma 4.2, we have
is nonexpansive mappings. From Theorem 3.1, we can concluded the desired conclusion.
Theorem 4.4.
Let be a nonempty closed convex subset of a Hilbert space
. Let
and
be two bifunctions from
into
satisfying (A1)–(A4), respectively. Let
be a
-inverse strongly monotone mapping. Let
be a finite family of
-strict pseudo contractions with
, where
defined by
. Define a mapping
by
,
. Let
a contraction with the coefficient
. Let
be the S-mappings generated by
and
, where
,
and
for all
for all
and
for all
for all
. Let
be sequences generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ113_HTML.gif)
where ,
,
such that
,
,
. Assume that
(i)and
,
(ii),
(iii),
(iv)and
, for all
.
Then the sequence ,
,
converges strongly to
, and
is solution of
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F836714/MediaObjects/13663_2010_Article_1352_Equ114_HTML.gif)
Proof.
For every , by Lemma 4.2, we have that
is nonexpansive mappings, putting
,
,
,
, and
. From Theorem 3.1, we can conclude the desired conclusion.
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Acknowledgment
The authors would like to thank Professor Dr. Suthep Suantai for his suggestion in doing and improving this paper.
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Kangtunyakarn, A. Iterative Methods for Finding Common Solution of Generalized Equilibrium Problems and Variational Inequality Problems and Fixed Point Problems of a Finite Family of Nonexpansive Mappings. Fixed Point Theory Appl 2010, 836714 (2010). https://doi.org/10.1155/2010/836714
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DOI: https://doi.org/10.1155/2010/836714