- Research Article
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Strong Convergence Theorem for Equilibrium Problems and Fixed Points of a Nonspreading Mapping in Hilbert Spaces
Fixed Point Theory and Applications volume 2010, Article number: 296759 (2010)
Abstract
We introduce an iterative method for finding a common element of the set of solutions of equilibrium problems and the set of fixed points of a nonspreading mapping in a Hilbert space. Then, we prove a strong convergence theorem which is connected with the work of S. Takahashi and W. Takahashi (2007) and Iemoto and Takahashi (2009).
1. Introduction
Let be a real Hilbert space with inner product
and norm
, respectively, and let
be a closed convex subset of
. Let
be bifunction, where
is the set of real numbers. The equilibrium problem for
is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ1_HTML.gif)
The set of solution of (1.1) is denoted by . Given a mapping
, let
for all
. Then,
if and only if
for all
, that is,
is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.1); see, for example, [1–9] and the references therein.
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, and 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%2F296759/MediaObjects/13663_2010_Article_1253_Equ2_HTML.gif)
for all , where
for all
; see, for instance, Kohsaka and Takahashi [10]. 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%2F296759/MediaObjects/13663_2010_Article_1253_Equ3_HTML.gif)
for all . Let
be the set of fixed points of
, and
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 that if the set of fixed points of a nonspreading mapping is nonempty, the nonspreading mapping is quasi-nonexpansive; see [10, 11].
In 1953, Mann [12] introduced the iteration as follows: a sequence defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ4_HTML.gif)
where the initial guess element is arbitrary and
is a real sequence in
. Mann iteration has been extensively investigated for nonexpansive mappings. In an infinite-dimensional Hilbert space, Mann iteration can conclude only weak convergence (see [12, 13]). Fourteen years later, Halpern [14] introduced the following iterative scheme for approximating a fixed point of
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ5_HTML.gif)
for all , where
and
is a sequence of
. Strong convergence of this type iterative sequence has been widely studied: Wittmann [15] discussed such a sequence in a Hilbert space.
On the other hand, Kohsaka and Takahashi [10] proved an existence theorem of fixed point for nonspreading mappings in a Banach space. Recently, Lemoto and Takahashi [16] 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%2F296759/MediaObjects/13663_2010_Article_1253_Equ6_HTML.gif)
Some methods have been proposed to solve the equilibrium problem and fixed point problem of nonexpansive mapping: see, for instance, [1, 2, 6, 7, 17–20] and the references therein. In 1997, 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. Recently, S. Takahashi and W. Takahashi [8] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solution of equilibrium problems and the set of fixed points of a nonexpansive mapping in a Hilbert space. Let
be a nonexpansive mapping. In 2008, Plubtieng and Punpaeng [7] introduced a new iterative sequence for finding a common element of the set of solution of equilibrium problems and the set of fixed points of a nonexpansive mapping in a Hilbert space which is the optimality condition for the minimization problem. Very recently, S. Takahashi and W. Takahashi [9] introduced an iterative method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space and then obtain that the sequence converges strongly to a common element of two sets.
In this paper, motivated by S. Takahashi and W. Takahashi [8] and Lemoto and Takahashi [16], we introduce an iterative sequence and prove a strong convergence theorem for finding solution of equilibrium problems and the set of fixed points of a nonspreading mapping in Hilbert spaces.
2. Preliminaries
Let be a real Hilbert space. When
is a sequence in
,
implies that
converges weakly to
and
means the strong convergence. Let
be a nonempty closed convex subset of
. For every point
, there exists a unique nearest point in
; denote by
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ7_HTML.gif)
is called the metric projection of
onto
. We know that
is nonexpansive. Further, for
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ8_HTML.gif)
Moreover, is characterized by the following properties:
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ9_HTML.gif)
for all ,
. We also know that
satisfies Opial's condition [21], that is, for any sequence
with
, the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ10_HTML.gif)
holds for every with
; see [21, 22] for more details.
The following lemmas will be useful for proving the convergence result of this paper.
Lemma 2.1 (see [23]).
Let be an inner product space. Then for all
and
with
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ11_HTML.gif)
Lemma 2.2 (see [10]).
Let be a Hilbert space,
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 2.3 (see [10]).
Let be a Hilbert space,
a nonempty closed convex subset of
. Let
be a nonspreading mapping of
into itself. Then
is closed and convex.
Lemma 2.4.
Let be a real Hilbert space. Then for all
,
(1);
(2).
Lemma 2.5 (see [24]).
Let , and let
be sequences of real numbers such that
, for all
,
and
.
Then, .
Lemma 2.6 (see [16]).
Let be a Hilbert space,
a closed convex subset of
, and
a nonspreading mapping with
. Then
is demiclosed, that is,
and
imply
.
Lemma 2.7 (see [16]).
Let be a Hilbert space,
a nonempty closed convex subset of a real Hilbert space
, and let
be a nonspreading mapping of
into itself, and let
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ12_HTML.gif)
Lemma 2.8 (see [25]).
Assume is a sequence of nonnegative real numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ13_HTML.gif)
where is a sequence in
and
is a sequence in
such that
(1);
(2) or
.
Then .
For solving the equilibrium problems for a bifunction , let us assume that
satisfies the following conditions:
(A1);
(A2) is monotone, that is,
;
(A3)for each ,  
;
(A4)for each ,
is convex and lower semicontinuous.
The following lemma appears implicitly in [26].
Lemma 2.9 (see [26]).
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%2F296759/MediaObjects/13663_2010_Article_1253_Equ14_HTML.gif)
The following lemma was also given in [4].
Lemma 2.10 (see [4]).
Assume that satisfies (A1)–(A4). For
and
, define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ15_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.
Lemma 2.11 (see [27]).
Let be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence
of
which satisfies
for all
. Also consider the sequence of integers
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ16_HTML.gif)
Then is a nondecreasing sequence verifying
, and the following properties are satisfied for all
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ17_HTML.gif)
3. Main Result
In this section, we prove a strong convergence theorem for finding a common element of the set of fixed points of a nonspreading mapping and the set of solutions of the equilibrium problems.
Theorem 3.1.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a bifunctions from
satisfying (A1)–(A4), and let
be a nonspreading mapping of
into itself such that
. Let
, and let
and
be sequences generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ18_HTML.gif)
for all , where
and
satisfy
,
,
,
,
,
, and
.
Then converges strongly to
, where
.
Proof.
Let . From
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ19_HTML.gif)
for all . Put
. We divide the proof into several steps.
Step 1.
We claim that the sequences ,
,
, and
are bounded. First, we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ20_HTML.gif)
and so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ21_HTML.gif)
Putting , we note that
for all
. In fact, it is obvious that
. Assume that
for all
. Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ22_HTML.gif)
By induction, we obtain that for all
. So,
is bound. Hence,
,
, and
are also bounded.
Step 2.
Put . We claim that
as
. We note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ23_HTML.gif)
where . On the other hand, from
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ24_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ25_HTML.gif)
for all . Putting
in (3.7) and
in (3.8), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ26_HTML.gif)
So, from (A2), we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ27_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ28_HTML.gif)
Without loss of generality, let us assume that there exists a real number such that
for all
. Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ29_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ30_HTML.gif)
where . So, from (3.6), we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ31_HTML.gif)
By Lemma 2.5, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ32_HTML.gif)
for . We note from
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ33_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ34_HTML.gif)
Therefore, from the convexity of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ35_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ36_HTML.gif)
So, we have . Indeed, since
, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ37_HTML.gif)
Then, we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ38_HTML.gif)
Since, and
, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ39_HTML.gif)
Step 3.
Put . From
, it follows by Lemma 2.7 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ40_HTML.gif)
Since , we have
. Therefore, by (3.23), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ41_HTML.gif)
Step 4.
Putting , we claim that the sequence
converges strongly to
. Indeed, we discuss two possible cases.
Case 1.
Assume that there exists such that the sequence
is a nonincreasing sequence for all
. Then we have
(for
), and hence
exists. Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ42_HTML.gif)
By (3.22), (3.24), and (3.25), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ43_HTML.gif)
Let be a subsequence of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ44_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 and convex, we note that
is weakly closed. So, we have
. Since
, it follows by Lemma 2.6 that
. From (3.27) and the property of metric projection, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ45_HTML.gif)
Finally, we prove that . In fact, since
, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ46_HTML.gif)
By (3.28) and , we immediately deduce by Lemma 2.8 that
.
Case 2.
Assume that for all , there exits
such that
. Put
for all
. Thus, it follows that there exists a subsequence
of
such that
for all
. Let
be a mapping defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ47_HTML.gif)
where . By Lemma 2.11, we note that
is a nondecreasing sequence such that
as
and that the following properties are satisfied by all numbers
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ48_HTML.gif)
From (3.24), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ49_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ50_HTML.gif)
Take a subsequence of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ51_HTML.gif)
From the boundedness of , we can assume that
. Since
is closed and convex, it follows that
is weakly closed. So, we have
. Since
, it follows by Lemma 2.6 that
. From (3.34) and the property of metric projection, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ52_HTML.gif)
By the same argument as (3.29) in Case 1, we conclude immediately that, for all ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ53_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ54_HTML.gif)
By (3.35), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ55_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ56_HTML.gif)
Since for all
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ57_HTML.gif)
This completes the proof.
As direct consequences of Theorem 3.1, we obtain corollaries.
Corollary 3.2.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a bifunctions from
satisfying (A1)–(A4), and let
be a firmly nonexpansive mapping of
into itself such that
. Let
, and let
and
be sequences generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F296759/MediaObjects/13663_2010_Article_1253_Equ58_HTML.gif)
for all , where
and
satisfy
,
,
,
,
,
, and
.
Then converges strongly to
, where
.
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The authors would like to thank the referees for the insightful comments and suggestions. Moreover, the authors gratefully acknowledge the Thailand Research Fund Master Research Grants (TRF-MAG, MRG-WII515S029) for funding this paper.
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Plubtieng, S., Chornphrom, S. Strong Convergence Theorem for Equilibrium Problems and Fixed Points of a Nonspreading Mapping in Hilbert Spaces. Fixed Point Theory Appl 2010, 296759 (2010). https://doi.org/10.1155/2010/296759
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DOI: https://doi.org/10.1155/2010/296759