- Research Article
- Open access
- Published:
Shrinking Projection Method of Common Solutions for Generalized Equilibrium Quasi-
-Nonexpansive Mapping and Relatively Nonexpansive Mapping
Journal of Inequalities and Applications volume 2010, Article number: 101690 (2010)
Abstract
We prove a strong convergence theorem for finding a common element of the set of solutions for generalized equilibrium problems, the set of fixed points of a relatively nonexpansive mapping, and the set of fixed points of a quasi--nonexpansive mapping in a Banach space by using the shrinking Projection method. Our results improve the main results in S. Takahashi and W. Takahashi (2008) and Takahashi and Zembayashi (2008). Moreover, the method of proof adopted in the paper is different from that of S. Takahashi and W. Zembayashi (2008).
1. Introduction
Let be a Banach space and let
be a closed convex subsets of
. Let
be an equilibrium bifunction from
into
and let
be a nonlinear mapping. Then, we consider the following generalized equilibrium problem: find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ1_HTML.gif)
The set of solutions of (1.1) is denoted by , that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ2_HTML.gif)
In the case of ,
is denoted by
. In the case of
,
is denoted by
.
A mapping is said to be nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ3_HTML.gif)
We denote the set of fixed points of by
.
A mapping is said to be Quasi-
-nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ4_HTML.gif)
where is defined by (2.3).
Recently, in Hilbert space, Tada and Takahashi [1], and S. Takahashi and W. Takahashi [2] considered iterative methods for finding an element of . Very recently, S. Takahashi and W. Takahashi [3] introduced an iterative method for finding an element of
, where
is an inverse-strongly monotone mapping and then proved a strong convergence theorem. On the other hand, Takahashi and Zembayashi [4] prove 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 the shrinking Projection method which is different from S. Takahashi and W. Takahashi's hybrid method [3].
In this paper, motivated by Takahashi and Zembayashi [4], in Banach space, we prove a strong convergence theorem for finding an element of , where
is a continuous and monotone operator,
is a relatively nonexpansive mapping, and
is quasi-
-nonexpansive mapping. Moreover, the method of proof adopted in the paper is different from that of [3].
2. Preliminaries
Throughout this paper, all the Banach spaces are real. We denote by and
the sets of positive integers and real numbers, respectively. Let
be a Banach space and let
be the topological dual of
. For all
and
, we denote the value of
at
by
. Then, the duality mapping
on
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ5_HTML.gif)
for every . By the Hahn-Banach theorem,
is nonempty; see [5] for more details. We denote the weak convergence and the strong convergence of a sequence
to
in
by
and
, respectively. We also denote the
convergence of a sequence
to
in
by
. A Banach space
is said to be strictly convex if
for
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
. The space
is said to be smooth if the limit
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ6_HTML.gif)
exists for all . It is also said to be uniformly smooth if the limit exists uniformly in
. We know that if
is smooth, strictly convex, and reflexive, then the duality mapping
is single valued, one to one, and onto; see [6] for more details.
Let be a smooth, strictly convex, and reflexive Banach space and let
be a 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%2F101690/MediaObjects/13660_2009_Article_2049_Equ7_HTML.gif)
Following Alber [7], the generalized projection from
onto
is defined by
, where
is the solution to the following minimization problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ8_HTML.gif)
The generalized projection from
onto
is well defined, single valued and satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ9_HTML.gif)
If is a Hilbert space, then
and
is the metric projection of
onto
. It is well know that the following conclusions for generalized projections hold.
Lemma 2.1 (Alber [7] and Kamimura and Takahashi [8]).
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%2F101690/MediaObjects/13660_2009_Article_2049_Equ10_HTML.gif)
Lemma 2.2.
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%2F101690/MediaObjects/13660_2009_Article_2049_Equ11_HTML.gif)
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space
, and let
be a mapping from
into itself. We denoted by
the set of fixed points of
. A point
is said to be an asymptotic fixed point of
[9, 10] if there exists
in
which converges weakly to
and
. We denote the set of all asymptotic fixed point of
by
. Following Matsushita and Takahashi [11], a mapping
is said to be relatively nonexpansive if the following conditions are satisfied:
(1) is nonempty;
(2) for all
;
(3).
The following lemma is due to Matsushita and Takahashi [11].
Lemma 2.3 (Matsushita and Takahashi [11]).
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 lemmas.
Lemma 2.4 (see [8]).
Let be a smooth and uniformly convex Banach space and let
and
be sequences in
such that either
or
is bounded. If
, then
.
Lemma 2.5 (see [12]).
Let be a uniformly convex Banach space and
be a closed ball of
. Then there exists a continuous, stricting increasing and convex function
with
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ12_HTML.gif)
for all and
.
For solving the equilibrium problem for bifunction , let us assume that
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%2F101690/MediaObjects/13660_2009_Article_2049_Equ13_HTML.gif)
for each is a convex and lower semicontinuous.
If an equilibrium bifunction satisfies conditions (
)–(
), then we have the following two important results.
Lemma 2.6 (see [13]).
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space
, let
be an equilibrium bifunction
satisfying conditions (
)–(
), and let
for any given
. Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ14_HTML.gif)
Lemma 2.7 (see [4]).
Let be a nonempty closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space
; let
be an equilibrium bifunction
satisfying conditions (
)–(
). For
and
, define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ15_HTML.gif)
for all . Then, the following holds:
(1) is single-valued;
(2) is a firmly nonexpansive-type mapping
that is, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ16_HTML.gif)
(3);
(4) is a closed and convex set.
Lemma 2.8 (see [4]).
Let be a nonempty closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space
let
be an equilibrium bifunction
satisfying conditions (
)–(
). For
,
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ17_HTML.gif)
3. The Main Results
In this section, we prove a strong convergence theorem which is the main result in the paper.
Theorem 3.1.
Let be a uniformly smooth and uniformly convex Banach space, and let
be a nonempty closed convex subset of
. Let
be a continuous and monotone operator. Let
be a bifunction from
to
which satisfies (
)–(
), let
be a relatively nonexpansive mapping of
into itself such that
, and let
be a closed
nonexpansive mapping. Let
be the sequence generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ18_HTML.gif)
for every , where
is the duality mapping on
,
, and
for some
. If the following conditions are satisfied
,
,
then converges strongly to
where
is the generalized projection of
onto
.
Proof.
We define a bifunction by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ19_HTML.gif)
Next, we prove that the bifunction satisfies conditions (
)–(
).
for all
.
Since for all
.
is monotone, that is,
for all
.
Since is a continuous and monotone operator, hence from the definition of
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ20_HTML.gif)
For each ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ21_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ22_HTML.gif)
For each is a convex and lower semicontinuous.
For each and
, since
satisfies
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ23_HTML.gif)
So, is convex.
Similarly, we can prove that is lower semicontinuous.
Therefore, the generalized equilibrium problem (1.1) is equivalent to the following equilibrium problem: find such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ24_HTML.gif)
and (3.1) can be written as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ25_HTML.gif)
Since the bifunction satisfies conditions (
)–(
), from Lemma 2.7, for a given
and
, we can define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ26_HTML.gif)
Moreover, satisfies the conclusions in Lemma 2.7.
Putting for all
, we have from Lemmas 2.7 and 2.8 that
are relatively nonexpansive.
We divide the proof of Theorem 3.1 into six steps.
Step 1.
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%2F101690/MediaObjects/13660_2009_Article_2049_Equ27_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_IEq260_HTML.gif)
is convex. So, is a closed convex subset of
for all
.
Step 2.
Next we show by induction that for all
. From
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ28_HTML.gif)
Suppose that for some
. For any
since
and
are relatively nonexpansive,
is
nonexpansive, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ29_HTML.gif)
Hence, we have This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ30_HTML.gif)
So, is well defined.
Step 3.
Next we prove that the sequences are bounded. From the definition of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ31_HTML.gif)
for all . Then
is bounded. Therefore,
,
, and
are bounded.
Step 4.
Next we prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ32_HTML.gif)
From and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ33_HTML.gif)
Thus, is nondecreasing. So, the limit of
exists. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ34_HTML.gif)
for all , we have
. From
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ35_HTML.gif)
Therefore, we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ36_HTML.gif)
Since and
is uniformly convex and smooth, we have from Lemma 2.4 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ37_HTML.gif)
So, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ38_HTML.gif)
Since is uniformly norm-to-norm continuous on bounded sets and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ39_HTML.gif)
For any , from Lemma 2.5 and (3.8), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ40_HTML.gif)
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ41_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ42_HTML.gif)
from (3.21) and (3.22), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ43_HTML.gif)
Since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ44_HTML.gif)
Therefore, from the property of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ45_HTML.gif)
Since is uniformly norm-to-norm continuous on bounded sets, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ46_HTML.gif)
Similarly, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ47_HTML.gif)
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ48_HTML.gif)
From (3.26) and , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ49_HTML.gif)
Therefore, from the property of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ50_HTML.gif)
Since is uniformly norm-to-norm continuous on bounded sets, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ51_HTML.gif)
Step 5.
Next we prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ52_HTML.gif)
where .
(a)We prove that
In fact, for any given , there exists a subsequence
of
such that
. Since
and
is relatively nonexpansive, we have
, that is,
.
(b)We prove that .
In fact, from , (3.12) and Lemma 2.8, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ53_HTML.gif)
Hence it follows from (3.26) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ54_HTML.gif)
Since is uniformly convex and smooth and
is bounded, we have from Lemma 2.4 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ55_HTML.gif)
For any given , there exists a subsequence
such that
. Since
, we have
.
Since is uniformly norm-to-norm continuous on bounded sets, from (3.38), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ56_HTML.gif)
From , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ57_HTML.gif)
By , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ58_HTML.gif)
Replacing by
, we have from
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ59_HTML.gif)
Since is convex and lower semicontinuous, it is also weakly lower semicontinuous. So, letting
, we have from (3.42) and
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ60_HTML.gif)
For any with
and
, let
. Since
and hence
, from conditions
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ61_HTML.gif)
This implies that . Hence from condition
, we have
for all
, and hence
.
(c)Now we prove that .
In fact, for any given , there exists a subsequence
such that
. Since
, we have
Since
is a closed convex subset of
. we have
, that is,
. From (3.14) and (3.16), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ62_HTML.gif)
Since the norm is weakly lower semicontinuous, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ63_HTML.gif)
that is, , then,
. Since
is uniformly convex Banach space,
has a Kadec-Klee property, we have
. From (3.34) and
being closed, we have
, that is,
.
Step 6.
Finally we prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ64_HTML.gif)
where . From
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ65_HTML.gif)
From the definition of , we have
. Hence,
. Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ66_HTML.gif)
Since has the Kadec-Klee property, we have that
. Therefore,
converges strongly to
.
This completes the proof of Theorem 3.1.
Theorem 3.2.
Let be a uniformly smooth and uniformly convex Banach space, and let
be a nonempty closed convex subset of
. and
be a continuous and monotone operator. Let
be a bifunction from
which satisfies (
)–(
) and let
be a relatively nonexpansive mapping of
into itself such that
. Let
be the sequence generated by
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F101690/MediaObjects/13660_2009_Article_2049_Equ67_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.
In Theorem 3.1, take , we get
Therefore, the conclusion of Theorem 3.2 can be obtained from Theorem 3.1.
Remark 3.3.
Theorem 3.1 in [3] and Theorem 3.1 in [4] are special cases of Theorem 3.2.
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Liu, M., Chang, SS. & Zuo, P. Shrinking Projection Method of Common Solutions for Generalized Equilibrium Quasi--Nonexpansive Mapping and Relatively Nonexpansive Mapping.
J Inequal Appl 2010, 101690 (2010). https://doi.org/10.1155/2010/101690
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DOI: https://doi.org/10.1155/2010/101690