- Research Article
- Open access
- Published:
Strong Convergence Theorem by Monotone Hybrid Algorithm for Equilibrium Problems, Hemirelatively Nonexpansive Mappings, and Maximal Monotone Operators
Fixed Point Theory and Applications volume 2008, Article number: 617248 (2008)
Abstract
We introduce a new hybrid iterative algorithm for finding a common element of the set of fixed points of hemirelatively nonexpansive mappings and the set of solutions of an equilibrium problem and for finding a common element of the set of zero points of maximal monotone operators and the set of solutions of an equilibrium problem in a Banach space. Using this theorem, we obtain three new results for finding a solution of an equilibrium problem, a fixed point of a hemirelatively nonexpnasive mapping, and a zero point of maximal monotone operators in a Banach space.
1. Introduction
Let be a Banach space, let
be a closed convex subset of
, and let
be a bifunction from
to
, where
is the set of real numbers. The equilibrium problem is to find
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ1_HTML.gif)
The set of such solutions is denoted by
.
In 2006, Martinez-Yanes and Xu [1] obtained strong convergence theorems for finding a fixed point of a nonexpansive mapping by a new hybrid method in a Hilbert space. In particular, Takahashi and Zembayashi [2] established 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 nonexpansive mapping in a uniformly convex and uniformly smooth Banach space. Very recently, Su et al. [3] proved the following theorem by a monotone hybrid method.
Theorem 1.1 (see Su et al. [3]).
Let be a uniformly convex and uniformly smooth real Banach space, let
be a nonempty closed convex subset of
, and let
be a closed hemirelatively nonexpansive mapping such that
. Assume that
is a sequence in
such that
. Define a sequence
in
by the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ2_HTML.gif)
where is the duality mapping on
. Then,
converges strongly to
, where
is the generalized projection from
onto
.
In this paper, motivated by Su et al. [3], we 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 hemirelatively nonexpansive mapping and for finding a common element of the set of zero points of maximal monotone operators and the set of solutions of an equilibrium problem in a Banach space by using the monotone hybrid method. Using this theorem, we obtain three new strong convergence results for finding a solution of an equilibrium problem, a fixed point of a hemirelatively nonexpnasive mapping, and a zero point of maximal monotone operators in a Banach space.
2. Preliminaries
Let be a real Banach space with dual
. We denote by
the normalized duality mapping from
to
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ3_HTML.gif)
where denotes the generalized duality pairing. It is well known that if
is uniformly convex, then
is uniformly continuous on bounded subsets of
. In this case,
is single valued and also one to one.
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%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ4_HTML.gif)
Following Alber [4], the generalized projection from
onto
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ5_HTML.gif)
The generalized projection from
onto
is well defined and single valued, and it satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ6_HTML.gif)
If is a Hilbert space, then
and
is the metric projection of
onto
.
If is a reflexive strict convex and smooth Banach space, then for
if and only if
. It is sufficient to show that if
, then
. From (2.4), we have
. This implies
. From the definition of
, we have
, that is,
.
Let be a closed convex subset of
and let
be a mapping from
into itself. We denote by
the set of fixed points of
.
is called hemirelatively nonexpansive if
for all
and
.
A point in
is said to be an asymptotic fixed point of
[5] if
contains a sequence
which converges weakly to
such that the strong
. The set of asymptotic fixed points of
will be denoted by
. A hemirelatively nonexpansive mapping
from
into itself is called relatively nonexpansive [1, 5, 6] if
.
We need the following lemmas for the proof of our main results.
Lemma 2.1 (see Alber [4]).
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%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ7_HTML.gif)
Lemma 2.2 (see Alber [4]).
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%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ8_HTML.gif)
Lemma 2.3 (see Kamimura and Takahashi [7]).
Let be a smooth and uniformly convex Banach space and let
and
be sequences in
such that either
or
is bounded. If
. Then
.
Lemma 2.4 (see Xu [8]).
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%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ9_HTML.gif)
where .
Lemma 2.5 (see Kamimura and Takahashi [7]).
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%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ10_HTML.gif)
For solving the equilibrium problem, let us assume that a bifunction satisfies the following conditions:
-
(A1)
for all
-
(A2)
is monotone, that is,
for all
-
(A3) for all
;
-
(A4) for all
is convex.
Lemma 2.6 (see Blum and Oettli [9]).
Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space
, let
be a bifunction from
to
satisfying (A1)–(A4), let
, and let
. Then, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ11_HTML.gif)
Lemma 2.7 (see Takahashi and Zembayashi [10]).
Let be a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach space
, let
be a bifunction from
to
satisfying (A1)–(A4), and let
, for
. Define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ12_HTML.gif)
Then, the following holds:
-
(1)
is single valued;
-
(2)
is a firmly nonexpansive-type mapping [11], that is, for all
,
(211) -
(3)
;
-
(4)
is closed and convex.
Lemma 2.8 (see Takahashi and Zembayashi [10]).
Let be a closed convex subset of a smooth, strictly convex, and reflexive Banach space
and let
be a bifunction from
to
satisfying (A1)–(A4). Then, for
and
, and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ14_HTML.gif)
Lemma 2.9 (see Su et al. [3]).
Let be a strictly convex and smooth real Banach space, let
be a closed convex subset of
, and let
be a hemirelatively nonexpansive mapping from
into itself. Then,
is closed and convex.
Recall that an operator in a Banach space is called closed, if
, then
.
3. Strong Convergence Theorem
Theorem 3.1.
Let be a uniformly convex and uniformly smooth real Banach space, let
be a nonempty closed convex subset of
, let
be a bifunction from
to
satisfying (A1)–(A4), and let
be a closed hemirelatively nonexpansive mapping such that
. Define a sequence
in
by the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ15_HTML.gif)
for every , where
is the duality mapping on
are sequences in
such that
and
for some
. Then,
converges strongly to
, where
is the generalized projection of
onto
.
Proof.
First, we can easily show that and
are closed and convex for each
.
Next, we show that for all
. Let
. Putting
for all
, from Lemma 2.8, we have
relatively nonexpansive. Since
are relatively nonexpansive and
is hemirelatively nonexpansive, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ16_HTML.gif)
Hence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ17_HTML.gif)
Next, we show that for all
. We prove this by induction. For
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ18_HTML.gif)
Suppose that , by Lemma 2.2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ19_HTML.gif)
As , by the induction assumptions, the last inequality holds, in particular, for all
. This, together with the definition of
, implies that
. So,
is well defined.
Since and
for all
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ20_HTML.gif)
Therefore, is nondecreasing. In addition, from the definition of
and Lemma 2.2,
. Therefore, for each
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ21_HTML.gif)
Therefore, and
are bounded. This, together with (3.6), implies that the limit of
exists. From Lemma 2.1, we have, for any positive integer
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ22_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ23_HTML.gif)
From (3.9), we can prove that is a Cauchy sequence. Therefore, there exists a point
such that
converges strongly to
.
Since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ24_HTML.gif)
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ25_HTML.gif)
From Lemma 2.3, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ26_HTML.gif)
So, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ27_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%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ28_HTML.gif)
Let Since
is a uniformly smooth Banach space, we know that
is a uniformly convex Banach space. Therefore, from Lemma 2.4, there exists a continuous, strictly increasing, and convex function
with
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ29_HTML.gif)
for , and
. So, we have that for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ30_HTML.gif)
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ31_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ32_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ33_HTML.gif)
From , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ34_HTML.gif)
Therefore, from the property of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ35_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%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ36_HTML.gif)
Since is a closed operator and
, then
is a fixed point of
.
On the other hand,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ37_HTML.gif)
So, we have from (3.19) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ38_HTML.gif)
From Lemma 2.3, we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ39_HTML.gif)
From and
, we have
.
From (3.25), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ40_HTML.gif)
From , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ41_HTML.gif)
By , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ42_HTML.gif)
From (A2), we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ43_HTML.gif)
From (3.27) and (A4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ44_HTML.gif)
For with
and
, let
. We have
. So, from (A1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ45_HTML.gif)
Dividing by , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ46_HTML.gif)
Letting , from (A3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ47_HTML.gif)
Therefore, . Finally, we prove that
. From Lemma 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ48_HTML.gif)
Since and
, for all
, we get from Lemma 2.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ49_HTML.gif)
By the definition of , it follows that
and
, whence
. Therefore, it follows from the uniqueness of
that
. This completes the proof.
Corollary 3.2.
Let be a uniformly convex and uniformly smooth real Banach space, let
be a nonempty closed convex subset of
, and let
be a bifunction from
to
satisfying (A1)–(A4). Define a sequence
in
by the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ50_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 3.2.
Corollary 3.3.
Let be a uniformly convex and uniformly smooth real Banach space, let
be a nonempty closed convex subset of
, and let
be a closed hemirelatively nonexpansive mapping. Define a sequence
in
by the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ51_HTML.gif)
for every , where
is the duality mapping on
are sequences in
such that
. Then,
converges strongly to
.
Proof.
Putting for all
and
for all
in Theorem 3.1, we obtain Corollary 3.3.
Corollary 3.4.
Let be a uniformly convex and uniformly smooth real Banach space, let
be a nonempty closed convex subset of
, let
be a bifunction from
to
satisfying (A1)–(A4), and let
be a closed relatively nonexpansive mapping such that
. Define a sequence
in
by the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ52_HTML.gif)
for every , where
is the duality mapping on
are sequences in
such that
and
for some
. Then,
converges strongly to
.
Proof.
Since every relatively nonexpansive mapping is a hemirelatively one, Corollary 3.4 is implied by Theorem 3.1.
Remark 3.5 (see Rockafellar [12]).
Let be a reflexive, strictly convex, and smooth Banach space and let
be a monotone operator from
to
. Then,
is maximal if and only if
for all
.
Let be a reflexive, strictly convex, and smooth Banach space and let
be a maximal monotone operator from
to
. Using Remark 3.5 and strict convexity of
, we obtain that for every
and
, there exists a unique
such that
If
, then we can define a single-valued mapping
by
, and such a
is called the resolvent of
. We know that
for all
and
is relatively nonexpansive mapping (see [2] for more details). Using Theorem 3.1, we can consider the problem of strong convergence concerning maximal monotone operators in a Banach space.
Theorem 3.6.
Let be a uniformly convex and uniformly smooth real Banach space, let
be a nonempty closed convex subset of
, let
be a bifunction from
to
satisfying (A1)–(A4), and let
be a resolvent of
and a closed mapping such that
, where
. Define a sequence
in
by the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F617248/MediaObjects/13663_2008_Article_1095_Equ53_HTML.gif)
for every , where
is the duality mapping on
,
is a sequences in
such that
and
for some
, Then,
converges strongly to
.
Proof.
Since is a closed relatively nonexpansive mapping and
, from Corollary 3.4, we obtain Theorem 3.6.
References
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
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.
Su Y, Wang D, Shang M: Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings. Fixed Point Theory and Applications 2008, 2008:-8.
Alber YI: 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.
Butnariu D, Reich S, Zaslavski AJ: Asymptotic behavior of relatively nonexpansive operators in Banach spaces. Journal of Applied Analysis 2001, 7(2):151-174. 10.1515/JAA.2001.151
Matsushita S-Y, 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
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
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
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994, 63(1–4):123-145.
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
Kohsaka F, Takahashi W: Existence and approximation of fixed points of firmly nonexpansive type mappings in Banach spaces. to appear in SIAM Journal on Optimization
Rockafellar RT: On the maximality of sums of nonlinear monotone operators. Transactions of the American Mathematical Society 1970, 149: 75-88. 10.1090/S0002-9947-1970-0282272-5
Acknowledgment
This work is supported by Tianjin Natural Science Foundation in China Grant no. 06YFJMJC12500.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Cheng, Y., Tian, M. Strong Convergence Theorem by Monotone Hybrid Algorithm for Equilibrium Problems, Hemirelatively Nonexpansive Mappings, and Maximal Monotone Operators. Fixed Point Theory Appl 2008, 617248 (2008). https://doi.org/10.1155/2008/617248
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2008/617248