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Weak Convergence Theorems for a System of Mixed Equilibrium Problems and Nonspreading Mappings in a Hilbert Space
Journal of Inequalities and Applications volume 2010, Article number: 246237 (2010)
Abstract
We introduce an iterative sequence and prove a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a quasi-nonexpansive mapping in Hilbert spaces. Moreover, we apply our result to obtain a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a nonspreading mapping. The result obtained in this paper improves and extends the recent ones announced by Moudafi (2009),Iemoto and Takahashi (2009), and many others. Using this result, we improve and unify several results in fixed point problems and equilibrium problems.
1. Introduction
Let be a nonempty closed convex subset of a real Hilbert space
. A mapping
of
into itself is said to be nonexpansive if
for all
, and a mapping
is said to be firmly nonexpansive if
for all
. Let
be a smooth,strictly convex and reflexive Banach space; let
be the duality mapping of
and
a nonempty closed convex subset of
. A mapping
is said to be nonspreading if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ1_HTML.gif)
for all , where
for all
; see, for instance, Kohsaka and Takahashi [1]. In the case when
is a Hilbert space, we know that
for all
Then a nonspreading mapping
in a Hilbert space
is defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ2_HTML.gif)
for all Let
be the set of fixed points of
, and
be nonempty; a mapping
is said to be quasi-nonexpansive if
for all
and
Remark 1.1.
In a Hilbert space, we know that every firmly nonexpansive mapping is nonspreading and if the set of fixed points of a nonspreading mapping is nonempty, the nonspreading mapping is quasi-nonexpansive; see [1].
Fixed point iterations process for nonexpansive mappings and asymptotically nonexpansive mappings in Banach spaces including Mann and Ishikawa iterations process have been studied extensively by many authors to solve the nonlinear operator equations. In 1953, Mann [2] introduced Mann iterative process defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ3_HTML.gif)
where and satisfies the assumptions
and
and proved that in case
is a Banach space, and
is closed, and
is continuous, then the convergence of
to a point
implies that
. Recently, Dotson [3] proved that a Mann iteration process was applied to the approximation of fixed points of quasi-nonexpansive mappings in Hilbert space and in uniformly convex and strictly convex Banach spaces.
On the other hand, Kohsaka and Takahashi [1] proved an existence theorem of fixed points for nonspreading mappings in a Banach space. Very recently, Iemoto and Takahashi [4] studied the approximation theorem of common fixed points for a nonexpansive mapping of
into itself and a nonspreading mapping
of
into itself in a Hilbert space. In particular, this result reduces to approximation fixed points of a nonspreading mapping
of
into itself in a Hilbert space by using iterative scheme
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ4_HTML.gif)
Let be a real-valued function and
an equilibrium bifunction, that is,
for each
. The mixed equilibrium problem is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ5_HTML.gif)
Denote the set of solutions of (1.5) by . The mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equiliubrium problems, and the equilibrium problems as special cases (see, e.g., Blum and Oettli [5]). In particular, if
, this problem reduces to the equilibrium problem, which is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ6_HTML.gif)
The set of solutions of (1.6) is denoted by Numerous problems in physics, optimization, and economics reduce to find a solution of (1.6). Let
be two monotone bifunctions and
is constant. In 2009, Moudafi [6] introduced an alternating algorithm for approximating a solution of the system of equilibrium problems: finding
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ7_HTML.gif)
Let be two monotone bifunctions and
are two constants. In this paper, we consider the following problem for finding
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ8_HTML.gif)
which is called a system of mixed equilibrium problems. In particular, if , then problem (1.8) reduces to finding
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ9_HTML.gif)
The system of nonlinear variational inequalities close to these introduced by Verma [7] is also a special case: by taking ,
, and
, where
are two nonlinear mappings. In this case, we can reformulate problem (1.7) to finding
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ10_HTML.gif)
which is called a general system of variational inequalities where and
are two constants. Moreover, if we add up the requirement that
, then problem (1.10) reduces to the classical variational inequality
.
In 2008,Ceng and Yao[8] considered a new iterative scheme for finding a common element of the set of solutions of MEP and the set of common fixed points of finitely many nonexpansive mappings.They also proved a strong convergence theorem for the iterative scheme. In the same year, Yao et al. [9] introduced a new hybrid iterative algorithm for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of the variational inequality of a monotone mapping, and the set of solutions of a mixed equilibrium problem. Very recently, Ceng et al. [10] introduced and studied a relaxed extragradient method for finding a common of the set of solution (1.10) for the and
-inverse strongly monotones and the set of fixed points of a nonexpansive mapping
of
into a real Hilbert space
. Let
, and
are given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ11_HTML.gif)
Then, they proved that the iterative sequence converges strongly to a common element of the set of fixed points of a nonexpansive mapping and a general system of variational inequalities with inverse-strongly monotone mappings under some parameters controlling conditions.
In this paper, we introduce an iterative sequence and prove a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a quasi-nonexpansive mapping in Hilbert spaces. Moreover, we apply our result to obtain a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a nonspreading mapping.
2. Preliminaries
Let be a real Hilbert space with inner product
and norm
, and let
be a closed convex subset of
. For every point
, there exists a unique nearest point in
, denoted by
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ12_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_IEq89_HTML.gif)
is called the metric projection of onto
. It is well known that
is a nonexpansive mapping of
onto
and satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ13_HTML.gif)
for all . Moreover,
is characterized by the following properties:
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ14_HTML.gif)
for all Further, for all
and
,
if and only if
, for all
.
A space is said to satisfy Opial's condition if for each sequence
in
which converges weakly to point
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ15_HTML.gif)
The following lemmas will be useful for proving the convergence result of this paper.
Lemma 2.1 (see [11]).
Let be an inner product space. Then for all
and
with
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ16_HTML.gif)
Lemma 2.2 (see [12]).
Let be a Hilbert space,
a closed convex subset of
, and
a nonexpansive mapping with
. If
is a sequence in
weakly converging to
and if
converges strongly to y, then
.
Lemma 2.3 (see [1]).
Let be a Hilbert space and
a nonempty closed convex subset of
. Let
be a nonspreading mapping of
into itself. Then the following are equivalent:
(1)there exists such that
is bounded;
(2) is nonempty.
Lemma 2.4 (see [1]).
Let be a Hilbert space and
a nonempty closed convex subset of
. Let
be a nonspreading mapping of
into itself. Then
is closed and convex.
Lemma 2.5 (see [4]).
Let be a Hilbert space,
a closed convex subset of
, and
a nonspreading mapping with
. Then
is demiclosed, that is,
and
imply
Lemma 2.6 (see [13]).
Let be a closed convex subset of a real Hilbert space
and let
be a sequence in
. Suppose that for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ17_HTML.gif)
for every . Then,
converges strongly to some
.
Lemma 2.7 (see [14]).
Assume is a sequence of nonnegative real numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ18_HTML.gif)
where is a sequence in
and
is a sequence in
such that
(1),
(2) or
Then
For solving the mixed equilibrium problems for an equilibrium bifunction , let us assume that
satisfies the following conditions:
(A1) for all
;
(A2) is monotone, that is,
for all
(A3)for each ,
is weakly upper semicontinuous;
(A4)for each ,
is convex, semicontinuous.
The following lemma appears implicitly in [5, 15].
Lemma 2.8 (see [5]).
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%2F246237/MediaObjects/13660_2010_Article_2093_Equ19_HTML.gif)
The following lemma was also given in [15].
Lemma 2.9 (see [15]).
Assume that satisfies (A1)–(A4). For
and
, define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ20_HTML.gif)
for all . Then, the following hold:
(1) is single-valued;
(2) is firmly nonexpansive, that is, for any
,
(3)
(4) is closed and convex.
We note that Lemma 2.9 is equivalent to the following lemma.
Lemma 2.10.
Let a nonempty closed convex subset of a real Hilbert space
. Let
be an equilibrium bifunction satisfying (A1)–(A4) and let
be a lower semicontinuous and convex functional. For
and
define a mapping
as follows.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ21_HTML.gif)
Then, the following results hold:
(i)for each ,
;
(ii) is single-valued;
(iii) is firmly nonexpansive, that is, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ22_HTML.gif)
(iv)
(v) is closed and convex.
Proof.
Define , for all
. Thus, the bifunction
satisfies, (A1)–(A4). Hence, by Lemmas 2.8 and 2.9, we have (i)–(v).
Lemma 2.11 (see [6]).
Let be a closed convex subset of a real Hilbert space
. Let
and
be two mappings from
satisfying (A1)–(A4) and let
and
be defined as in Lemma 2.10 associated to
and
, respectively. For given
is a solution of problem (1.8) if and only if
is a fixed point of the mapping
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ23_HTML.gif)
where .
Proof.
By a similar argument as in the proof of Proposition  2.1 in [6], we obtain the desired result.
We note from Lemma 2.11 that the mapping is nonexpansive. Moreover, if
is a closed bounded convex subset of
, then the solution of problem (1.8) always exists. Throughout this paper, we denote the set of solutions of (1.8) by
3. Main Result
In this section, we prove a weak convergence theorem for finding a common element of the set of fixed points of a quasi-nonexpansive mapping and the set of solutions of the system of mixed equilibrium problems.
Theorem 3.1.
Let be a closed convex subset of a real Hilbert space
. Let
and
be two bifunctions from
satisfying (A1)–(A4). Let
and let
and
be defined as in Lemma 2.10 associated to
and
, respectively. Let
be a quasi-nonexpansive mapping of
into itself such that
Suppose
and
,
are given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ24_HTML.gif)
for all , where
for some
and satisfy
. Then
converges weakly to
and
is a solution of problem (1.8), where
Proof.
Let . Then
and
Put
,
and
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ25_HTML.gif)
it follows by Lemma 2.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ26_HTML.gif)
Hence is a decreasing sequence and therefore
exists. This implies that
,
,
, and
are bounded. From (3.3), we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ27_HTML.gif)
Since and
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ28_HTML.gif)
This implies that . Since
and
are firmly nonexpansive, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ29_HTML.gif)
and so . By the convexity of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ30_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ31_HTML.gif)
Since and
, we obtain
Similarly, we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ32_HTML.gif)
and so . Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ33_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ34_HTML.gif)
It follows from and
that
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ35_HTML.gif)
and therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ36_HTML.gif)
Since is a bounded sequence, there exists a subsequence
such that
converges weakly to
. From Lemma 2.5, we have
. Let
be a mapping which is defined as in Lemma 2.11. Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ37_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ38_HTML.gif)
From and
, we get
According to Lemmas 2.2 and 2.11, we have
. Hence
Since
and
, we obtain
. Let
be another subsequence of
such that
converges weakly to
. We may show that
, suppose not. Since
exists for all
, it follows by the Opial's condition that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ39_HTML.gif)
This is a contradiction. Thus, we have . This implies that
converges weakly to
. Put
. Finally, we show that
. Now from (2.2) and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ40_HTML.gif)
Since is nonnegative and decreasing for any
it follows by Lemma 2.6 that
converges strongly to some
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ41_HTML.gif)
Therefore, .
Corollary 3.2.
Let be a closed convex subset of a real Hilbert space
. Let
and
be two bifunctions from
satisfying (A1)–(A4). Let
and let
and
be defined as in Lemma 2.10 associated to
and
, respectively. Let
be a nonspreading mapping of
into itself such that
Suppose
and
,
,
are given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ42_HTML.gif)
for all , where
for some
and satisfy
. Then
converges weakly to
and
is a solution of problem (1.8), where
Setting and
in Theorem 3.1, we have following result.
Corollary 3.3.
Let be a closed convex subset of a real Hilbert space
. Let
and
be two bifunctions from
satisfying (A1)–(A4). Let
and let
and
be defined as in Lemma 2.10 associated to
and
, respectively, such that
Suppose
and
,
,
are given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ43_HTML.gif)
for all , where
for some
and satisfy
. Then
converges weakly to
and
is a solution of problem (1.9), where
Setting in Theorem 3.1, we have the following result.
Corollary 3.4.
Let be a closed convex subset of a real Hilbert space
. Let
be a bifunction from
satisfying (A1)–(A4). Let
and let
be defined as in Lemma 2.9 associated to
. Let
be a quasi-nonexpansive mapping of
into itself such that
Suppose
and
and
are given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246237/MediaObjects/13660_2010_Article_2093_Equ44_HTML.gif)
for all , where
for some
and satisfy
. Then
converges weakly to
and
is a solution of problem (1.7), where
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Acknowledgments
The authors would like to thank the referee for the insightful comments and suggestions. The first author is thankful to the Thailand Research Fund for financial support under Grant BRG5280016. Moreover, the second author would like to thank the Office of the Higher Education Commission, Thailand for supporting by grant fund under Grant CHE-Ph.D-THA-SUP/86/2550, Thailand.
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Plubtieng, S., Sombut, K. Weak Convergence Theorems for a System of Mixed Equilibrium Problems and Nonspreading Mappings in a Hilbert Space. J Inequal Appl 2010, 246237 (2010). https://doi.org/10.1155/2010/246237
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DOI: https://doi.org/10.1155/2010/246237