- Research Article
- Open access
- Published:
Hybrid Methods for Equilibrium Problems and Fixed Points Problems of a Countable Family of Relatively Nonexpansive Mappings in Banach Spaces
Fixed Point Theory and Applications volume 2010, Article number: 962628 (2012)
Abstract
The purpose of this paper is to introduce hybrid projection algorithms for finding a common element of the set of common fixed points of a countable family of relatively nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces. Moreover, we apply our result to the problem of finding a common element of an equilibrium problem and the problem of finding a zero of a maximal monotone operator. Our result improve and extend the corresponding results announced by Takahashi and Zembayashi (2008 and 2009), and many others.
1. Introduction
Let be a real Banach space and
the dual space of
. Let
be a nonempty closed convex subset of
and
a bifunction from
to
, where
denotes the set of real numbers. The equilibrium problem is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ1_HTML.gif)
The set of solutions 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 reduced to find a solution of (1.1). Some methods have been proposed to solve the equilibrium problem; see, for instance, Blum and Oettli [1], Combettes and Hirstoaga [2], and Moudafi [3].
Recall that a mapping is said to be nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ2_HTML.gif)
We denote by the set of fixed points of
. If a Banach space
is uniformly convex,
is bounded, closed and convex, and
is a nonexpansive mapping of
into itself, then
is nonempty; see [4] for more details. Recently, many authors studied the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in the framework of Hilbert spaces and Banach spaces, respectively; see, for instance, [5–13] and the references therein.
A popular method is the hybrid projection method developed by Nakajo and Takahashi [14], Kamimura and Takahashi [15], and Martinez-Yanes and Xu [16]; see also [5, 17–20] and references therein. Recently Takahashi et al. [21] introduced an alterative projection method, which is called the shrinking projection method, and they showed several strong convergence theorems for a family of nonexpansive mappings. In 2008, Takahashi and Zembayashi [12] introduced two iterative sequences for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solution of an equilibrium problem in a Banach space. Then they prove strong and weak convergence of the sequences. Very recently, Takahashi and Zembayashi [13] proved a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a relatively nonexpansive mapping in a Banach space by using a new hybrid method.
On the other hand, motivated by Nakajo and Takahashi [14], Matsushita and Takahashi [17] reformulated the definition of the notion and obtained weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping. Very recently, Aoyama et al. [22] introduce a Halpern type iterative sequence for finding a common fixed point of a countable family of nonexpansive mappings. Let and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ3_HTML.gif)
for all where
is a nonempty closed convex subset of a Banach space;
is a sequence in
and
is a sequence of nonexpansive mappings with some condition. They proved that
defined by (1.3) converges strongly to a common fixed point of
Motivated and inspired by the research going on in this direction, we prove a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a countable family of relatively nonexpansive mappings in a Banach space by using the shrinking projection method. Further, we apply our result to the problem of finding a common element of an equilibrium problem and the problem of finding a zero of a maximal monotone operator. The result obtained in this paper improves and extends the corresponding result of [13] and many others.
2. Preliminaries
Let be a real Banach space with norm
and let
be the dual of
. For all
and
, we denote the value of
at
by
. The normalized duality mapping
from
to
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ4_HTML.gif)
for . By Hahn-Banach theorem,
is nonempty; see [4] for more details. We denote the strong convergence and the weak convergence of a sequence
to
in
by
and
, respectively. We also denote the weak
convergence of a sequence
to
in
by
A Banach space
is said to be strictly convex if
for all
with
and
. It is also said to be uniformly convex if for each
, there exists
such that
for
with
and
. A uniformly convex Banach space has the Kadec-Klee property, that is,
and
imply
. Let
be the unit sphere of
. Then the Banach space
is said to be smooth provided that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ5_HTML.gif)
exists for each . It is also said to be uniformly smooth if the limit is attained uniformly for
. It is well know that if
is smooth, strictly convex and reflexive, then the duality mapping
is single valued, one-to-one and onto.
Let be a smooth, strictly convex and reflexive Banach space, and let
be a nonempty closed convex subset of
. Throughout this paper, we denote by
the function defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ6_HTML.gif)
It is obvious from the definition of the function that
for all
. Following Alber [23], the generalized projection
from
onto
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ7_HTML.gif)
If is a Hilbert space, then
and
is the metric projection of
onto
. We know the following lemmas for generalized projections.
Lemma 2.1 (Alber [23], Kamimura and Takahashi [15]).
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ8_HTML.gif)
Lemma 2.2 (Alber [23], Kamimura and Takahashi [15]).
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space
let
and let
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ9_HTML.gif)
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space
; let
be a mapping from
into itself. We denote by
the set of fixed point of
. A point
is said to be an asymptotic fixed point of
if there exists
in
which converges weakly to
and
. The set of asymptotic fixed points of
will be denoted by
. Following Matsushita and Takahashi [17], a mapping
from
into itself is said to be relatively nonexpansive if
is nonempty,
and
.
The following lemma is according to Matsushita and Takahashi [17].
Lemma 2.3 (Matsushita and Takahashi [17]).
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space
, and let
be a relatively nonexpansive mapping from
into itself. Then
is closed and convex.
We also know the following three lemmas.
Lemma 2.4 (Kamimura and Takahashi [15]).
Let be a uniformly convex and smooth Banach space, and let
,
be sequences in
such that either
or
is bounded. If
, then
.
Lemma 2.5 (Xu [24], Zlinescu [25, 26]).
Let be a uniformly convex Banach space, and let
. Then there exists a strictly increasing, continuous, and convex function
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ10_HTML.gif)
for all and
, where
Lemma 2.6 (Kamimura and Takahashi [15]).
Let be a smooth and uniformly convex Banach space and let
Then there exists a strictly increasing, continuous, and convex function
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ11_HTML.gif)
for all
For solving the equilibrium problem, let us assume that a bifunction satisfies the following conditions:
for all
;
is monotone, that is,
for all
;
for each ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ12_HTML.gif)
for each is convex and lower semicontinuous.
The following result is in Blum and Oettlli [1].
Lemma 2.7 (Blum and Oettlli [1]).
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space
, and let
be a bifunction from
satisfying
, and let
and
. Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ13_HTML.gif)
We also know the following lemmas.
Lemma 2.8 (Takahashi and Zembayashi [12]).
Let be a nonempty closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space
, and let
be a bifunction from
satisfying
. For
and
, define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ14_HTML.gif)
for all , Then, the following holds:
(1) is single-valued;
(2) is a firmly nonexpansive-type mapping, that is, for all
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ15_HTML.gif)
(3);
(4) is closed and convex.
Lemma 2.9 (Takahashi and Zembayashi [12]).
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space
, and let
be a bifunction from
satisfying
. Then for
,
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ16_HTML.gif)
3. Main Results
Let be a nonempty closed convex subset of a Banach space
let
be a family of mappings of
into itself with
and
denotes the set of all weak subsequential limits of a bounded sequence
in
.
is said to satisfy the NST
-condition [27] if for every bounded sequence
in
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ17_HTML.gif)
In this section, by using the NST-condition, we prove two strong convergence theorems for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a countable family of relatively nonexpansive mappings in a Banach space.
Theorem 3.1.
Let be a uniformly convex and uniformly smooth Banach space, and let
be a nonempty closed convex subset of
. Let
be a bifunction from
to
satisfying
and let
be a family of relatively nonexpansive mappings from
into itself such that
. Let
be a sequence generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ18_HTML.gif)
for every , where
is the duality mapping on
,
satisfies
and
for some
Suppose that
satisfy the NST
-condition. Then
converges strongly to
, where
is the generalized projection of
onto
.
Proof.
Putting for all
, we have from Lemma 2.9 that
are relatively nonexpansive.
We first show that is closed and convex. It is obvious that
is closed. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ19_HTML.gif)
is convex. So,
is a closed convex subset of
for all
.
Next, we show by induction that for all
. From
, we have
. Suppose that
for some
. Let
Since
and
are relatively nonexpansive, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ20_HTML.gif)
Hence . This implies that
for all
. So,
is well defined.
Next, we show that is bounded. From the definition of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ21_HTML.gif)
for all Thus
is bounded and therefore
and
are also bounded.
From and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ22_HTML.gif)
This implies that is nondecreasing and so
exists. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ23_HTML.gif)
for all , it follows that
From
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ24_HTML.gif)
Therefore, we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ25_HTML.gif)
Since and
is uniformly convex and smooth, it follows from Lemma 2.4 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ26_HTML.gif)
Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ27_HTML.gif)
Since is uniformly norm-to-norm continuous on bounded sets, it follows by (3.11) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ28_HTML.gif)
Let . Since
is a uniformly smooth Banach space, we note that
is a uniformly convex Banach space. Therefore, by Lemma 2.5, there exists a continuous, strictly increasing, and convex function
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ29_HTML.gif)
for and
So, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ30_HTML.gif)
for all Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ31_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ32_HTML.gif)
it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ33_HTML.gif)
From we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ34_HTML.gif)
Therefore, we note from the property of that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ35_HTML.gif)
Since is uniformly norm-to-norm continuous on bounded sets, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ36_HTML.gif)
Since satisfy the NST
-condition, we have
. So, we assume that a subsequence
of
converges weakly to
We shall show that
. From
and Lemma 2.9, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ37_HTML.gif)
So, we note from (3.17) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ38_HTML.gif)
Since is uniformly convex and smooth and
is bounded, it follows from Lemma 2.4 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ39_HTML.gif)
From ,
and
, we have
Since
is uniformly norm-to-norm continuous on bounded sets and (3.23), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ40_HTML.gif)
From we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ41_HTML.gif)
By the definition of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ42_HTML.gif)
Replacing by
, we have from (A2) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ43_HTML.gif)
Since is convex and lower semicontinuous, it is also weakly lower semicontinuous. Letting
, we note from (3.25) and (
) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ44_HTML.gif)
For with
and
, let
Since
and
, we have
and hence
So, from (
), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ45_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ46_HTML.gif)
Letting from (A3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ47_HTML.gif)
Therefore, we obtain Finally, we will show that
. Let
. From
and
we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ48_HTML.gif)
Since the norm is weakly lower semicontinuous, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ49_HTML.gif)
From the definition of , we have
. Hence
Therefore, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ50_HTML.gif)
Since has the Kadec-Klee property, it follows that
Therefore,
converges strongly to
As direct consequences of Theorem 3.1, we can obtain the following corollaries.
Corollary 3.2 (Takahashi and Zembayashi [13]).
Let be a uniformly convex and uniformly smooth Banach space, and let
be a nonempty closed convex subset of
. Let
be a bifunction from
to
satisfying
, and let
be a relatively nonexpansive mapping from
into itself such that
. Let
be a sequence generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ51_HTML.gif)
for every , where
is the duality mapping on
,
satisfies
and
for some
Then
converges strongly to
, where
is the generalized projection of
onto
.
Proof.
Put . Let
be a bounded sequence in
with
and let
. Then there exists subsequence
of
such that
. It follows directly from the definition of
that
. Hence
satisfies NST
-condition, by Theorem 3.1;
converges strongly to
.
Corollary 3.3 (Takahashi and Zembayashi [12]).
Let be a uniformly convex and uniformly smooth Banach space; let
be a nonempty closed convex subset of
. Let
be a bifunction from
to
satisfying
. Let
be a sequence generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ52_HTML.gif)
for every , where
is the duality mapping on
and
for some
Then,
converges strongly to
Proof.
Putting in Theorem 3.1, we obtain Corollary 3.3.
Corollary 3.4.
Let be a uniformly convex and uniformly smooth Banach space, and let
be a nonempty closed convex subset of
. Let
be a family of relatively nonexpansive mappings from
into itself such that
. Let
be a sequence generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ53_HTML.gif)
for every , where
is the duality mapping on
,
satisfies
. Suppose that
satisfy the NST
-condition. Then
converges strongly to
, where
is the generalized projection of
onto
.
Proof.
Putting for all
and
in Theorem 3.1, we obtain Corollary 3.4.
Similarly as in the proof of Theorem 3.1, we can prove the following theorem.
Theorem 3.5.
Let be a uniformly convex and uniformly smooth Banach space, and let
be a nonempty closed convex subset of
. Let
be a bifunction from
to
satisfying
and let
be a family of relatively nonexpansive mappings from
into itself such that
. Let
be a sequence generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ54_HTML.gif)
for every , where
is the duality mapping on
,
satisfies
and
for some
Suppose that
satisfy the NST
-condition. Then
converges strongly to
, where
is the generalized projection of
onto
.
Proof.
We first show that is closed and convex. It is obvious that
is closed and
is closed and convex. Since
it follows that
is convex. Hence
is a closed and convex subset of
for all
Similarly as in proof of Theorem 3.1, we note that
for all
Next, we show by induction that
for all
. From
, we note that
Suppose that
for some
Then there exists
such that
From the definition of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ55_HTML.gif)
for all . Since
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ56_HTML.gif)
and hence . So, we have
and therefore,
Hence
for all
. This implies that
is well defined. By the same argument as in proof of Theorem 3.1, we can prove that the sequence
converges strongly to
.
Setting in Theorem 3.5, we have the following result.
Corollary 3.6 (Takahashi and Zembayashi [12, Theorem ]).
Let be a uniformly convex and uniformly smooth Banach space; let
be a nonempty closed convex subset of
. Let
be a bifunction from
to
satisfying
, and let
be a relatively nonexpansive mapping from
into itself such that
. Let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ57_HTML.gif)
for every , where
is the duality mapping on
,
satisfies
and
for some
Then,
converges strongly to
where
is the generalized projection of
onto
.
4. Applications
In this section, we apply our result to the problem of finding a common element of an equilibrium problem and the problem of finding a zero of a maximal monotone operator in a Banach space by using the shrinking projection method.
Let be a real Banach space. An operator
is said to be monotone if
whenever
. We denote the set
by
A monotone
is said to be maximal if its graph
is not properly contained in the graph of any other monotone operator. If
is maximal monotone, then the solution set
is closed and convex.
Let be a smooth, strictly convex and reflexive Banach space, and let
be a maximal monotone operator. Then for each
and
, there corresponds a unique element
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ58_HTML.gif)
see Barbu [28] or Takahashi [4]. We define the resolvent of by
. In other words,
for all
. We know that
is relatively nonexpansive and
for all
(see [4, 17]), where
denotes the set of all fixed points of
. We can also define, for each
, the Yosida approximation of
by
We know that
for all
We now consider the strong convergence theorem for finding a common element of the solution set of an equilibrium problem and the problem of finding a zero of a maximal monotone operator.
Theorem 4.1.
Let be a uniformly convex and uniformly smooth Banach space. Let
be a maximal monotone operator, and let
for all
. Let
be a nonempty closed convex subset of
such that
Let
be a bifunction from
to
satisfying
with
. Let
be a sequence generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ59_HTML.gif)
for every , where
is the duality mapping on
,
,
satisfy
, and
for some
Then
converges strongly to
, where
is the generalized projection of
onto
.
Proof.
Let be a bounded sequence in
such that
and let
. Then, there exists a subsequence
of
such that
. By the uniform smoothness of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ60_HTML.gif)
Since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ61_HTML.gif)
Let . Then it holds from the monotonicity of
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ62_HTML.gif)
for all . Letting
, we get
. Then, the maximality of
implies
. Hence by Theorem 3.1,
converges strongly to
In case . Putting
for all
and
in Theorem 4.1, we obtain the following corollary.
Corollary 4.2.
Let be a uniformly convex and uniformly smooth Banach space. Let
be a maximal monotone operator, and let
for all
, with
. Let
be a sequence generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ63_HTML.gif)
for every , where
is the duality mapping on
,
,
satisfy
. Then
converges strongly to
, where
is the generalized projection of
onto
.
Similarly as in the proof of Theorem 4.1, we can prove the following theorem.
Theorem 4.3.
Let be a uniformly convex and uniformly smooth Banach space. Let
be a maximal monotone operator and let
for all
. Let
be a nonempty closed convex subset of
such that
. Let
be a bifunction from
to
satisfying
with
. Let
be a sequence generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ64_HTML.gif)
for every , where
is the duality mapping on
,
,
satisfy
, and
for some
Then
converges strongly to
, where
is the generalized projection of
onto
.
Corollary 4.4.
Let be a uniformly convex and uniformly smooth Banach space. Let
be a maximal monotone operator and let
for all
, with
. Let
be a sequence generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F962628/MediaObjects/13663_2009_Article_1372_Equ65_HTML.gif)
for every , where
is the duality mapping on
,
,
satisfy
. Then
converges strongly to
, where
is the generalized projection of
onto
.
References
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.
Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis 2005,6(1):117–136.
Moudafi A: Second-order differential proximal methods for equilibrium problems. Journal of Inequalities in Pure and Applied Mathematics 2003,4(1, article 18):7.
Takahashi W: Convex Analysis and Approximation Fixed Points, Mathematical Analysis Series. Volume 2. Yokohama Publishers, Yokohama, Japan; 2000:iv+280.
Ceng L-C, Yao J-C: Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Applied Mathematics and Computation 2008,198(2):729–741. 10.1016/j.amc.2007.09.011
Ceng L-C, Yao J-C: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. Journal of Computational and Applied Mathematics 2008,214(1):186–201. 10.1016/j.cam.2007.02.022
Ceng L-C, Schaible S, Yao J-C: Implicit iteration scheme with perturbed mapping for equilibrium problems and fixed point problems of finitely many nonexpansive mappings. Journal of Optimization Theory and Applications 2008,139(2):403–418. 10.1007/s10957-008-9361-y
Ceng L-C, Petruşel A, Yao J-C: Iterative approaches to solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Journal of Optimization Theory and Applications 2009,143(1):37–58. 10.1007/s10957-009-9549-9
Peng J-W, Yao J-C: Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems. Mathematical and Computer Modelling 2009,49(9–10):1816–1828. 10.1016/j.mcm.2008.11.014
Plubtieng S, Punpaeng R: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007,336(1):455–469. 10.1016/j.jmaa.2007.02.044
Plubtieng S, Punpaeng R: A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings. Applied Mathematics and Computation 2008,197(2):548–558. 10.1016/j.amc.2007.07.075
Takahashi W, Zembayashi K: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,70(1):45–57. 10.1016/j.na.2007.11.031
Takahashi W, Zembayashi K: Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings. Fixed Point Theory and Applications 2008, 2008:-11.
Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. Journal of Mathematical Analysis and Applications 2003,279(2):372–379. 10.1016/S0022-247X(02)00458-4
Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM Journal on Optimization 2002,13(3):938–945. 10.1137/S105262340139611X
Martinez-Yanes C, Xu H-K: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Analysis: Theory, Methods & Applications 2006,64(11):2400–2411. 10.1016/j.na.2005.08.018
Matsushita S, Takahashi W: A strong convergence theorem for relatively nonexpansive mappings in a Banach space. Journal of Approximation Theory 2005,134(2):257–266. 10.1016/j.jat.2005.02.007
Peng J-W, Yao J-C: A modified CQ method for equilibrium problems, fixed points and variational inequality. Fixed Point Theory 2008,9(2):515–531.
Peng J-W, Yao J-C: A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems. Taiwanese Journal of Mathematics 2008,12(6):1401–1432.
Plubtieng S, Ungchittrakool K: Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space. Journal of Approximation Theory 2007,149(2):103–115. 10.1016/j.jat.2007.04.014
Takahashi W, Takeuchi Y, Kubota R: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2008,341(1):276–286. 10.1016/j.jmaa.2007.09.062
Aoyama K, Kimura Y, Takahashi W, Toyoda M: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Analysis: Theory, Methods & Applications 2007,67(8):2350–2360. 10.1016/j.na.2006.08.032
Alber YaI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Applied Mathematics. Volume 178. Edited by: Kartsatos AG. Marcel Dekker, New York, NY, USA; 1996:15–50.
Xu HK: Inequalities in Banach spaces with applications. Nonlinear Analysis: Theory, Methods & Applications 1991,16(12):1127–1138. 10.1016/0362-546X(91)90200-K
Zălinescu C: On uniformly convex functions. Journal of Mathematical Analysis and Applications 1983,95(2):344–374. 10.1016/0022-247X(83)90112-9
Zălinescu C: Convex Analysis in General Vector Spaces. World Scientific, River Edge, NJ, USA; 2002:xx+367.
Nakajo K, Shimoji K, Takahashi W: On strong convergence by the hybrid method for families of mappings in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2009,71(1–2):112–119. 10.1016/j.na.2008.10.034
Barbu V: Nonlinear Semigroups and Differential Equations in Banach Spaces. Editura Academiei Republicii Socialiste România, Bucharest, Romania; 1976:352.
Acknowledgments
The first author thanks the National Research Council of Thailand to Naresuan University, 2009 for the financial support. Moreover, the second author would like to thank the National Centre of Excellence in Mathematics, PERDO, under the Commission on Higher Education, Ministry of Education, Thailand. This work is dedicated to Professor Wataru Takahashi on his retirement.
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Plubtieng, S., Sriprad, W. Hybrid Methods for Equilibrium Problems and Fixed Points Problems of a Countable Family of Relatively Nonexpansive Mappings in Banach Spaces. Fixed Point Theory Appl 2010, 962628 (2012). https://doi.org/10.1155/2010/962628
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DOI: https://doi.org/10.1155/2010/962628