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Strong Convergence Theorems for Countable Lipschitzian Mappings and Its Applications in Equilibrium and Optimization Problems
Fixed Point Theory and Applications volume 2009, Article number: 462489 (2009)
Abstract
The purpose of this paper is to propose a modified hybrid method in mathematical programming and to obtain some strong convergence theorems for common fixed points of a countable family of Lipschitzian mappings. Further, we apply our results to solve the equilibrium and optimization problems. The results of this paper improved and extended the results of W. Nilsrakoo and S. Saejung (2008) and some others in some respects.
1. Introduction and Preliminaries
Let be a real Hilbert space with inner product
and norm
and let
be a nonempty subset of
. A mapping
is said to be Lipschitzian if there exists a positive constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ1_HTML.gif)
In this case, is also said to be
-Lipschitzian. Throughout the paper, we assume that every Lipschitzian mapping is
-Lipschitzian with
. If
, then
is known as a nonexpansive mapping. We denote by
the set of fixed points of
. There are many methods for approximating the fixed points of a nonexpansive mapping. In 1953, Mann [1] introduced the following iteration method:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ2_HTML.gif)
where the initial guess element is arbitrary, and
is a real sequence in [0,1]. Mann iteration has been extensively investigated for nonexpansive mappings. One of the fundamental convergence results is proved by Qin and Su[2]. In an infinite-dimensional Hilbert space, Mann iteration could conclude only weak convergence [3]. Attempts to modify the Mann iteration method (1.2) so that strong convergence is guaranteed have recently been made. Nakajo and Takahashi [4] proposed the following modification of Mann iteration method(1.2):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ3_HTML.gif)
where denotes the metric projection from
onto a closed convex subset
of
. The iterative method (1.3) is said to be hybrid method or
method. In recent years, the hybrid method (1.3) has been modified by many authors for other nonlinear operators [2, 5–10].
In 2008, Nilsrakoo and Saejung [11] used the hybrid method to obtain a strong convergence theorem for countable Lipschitzian mappings as follows.
Theorem 1.
Let be a nonempty bounded closed convex subset of a real Hilbert space
. Let
be a sequence of
-Lipschitzian mappings from
into itself with
and let
be nonempty. Assume that
is a sequence in
with
. Let
be a sequence in
defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ4_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ5_HTML.gif)
as . Let
for any bounded subset
of
, and let
be a mapping of
into itself defined by
, for all
and suppose that
, then
converges strongly to
.
Nilsrakoo and Saejung also apply the aforementioned result to obtain an applied result for equilibrium problems.
The purpose of this paper is to propose a modified hybrid method in mathematical programming and to obtain some strong convergence theorems for common fixed points of a countable family of Lipschitzian mappings. Finally, we apply our results to solve the equilibrium and optimization problems. The results of this paper improved and extended the Nilsrakoo and Saejung results in some respects.
Recall that given a closed convex subset of a real Hilbert space
, the nearest point projection
from
onto
assigns to each
its nearest point denoted by
in
from
to
; that is,
is the unique point in
with the property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ6_HTML.gif)
The following lemma is well known.
Lemma 1.1.
Let be a closed convex subset of a real Hilbert space
. Given
and
Then
if and only if there holds the relation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ7_HTML.gif)
Definition 1.2.
Let be a nonempty closed convex subset of a real Hilbert space
and let
be a sequence of
-Lipschitzian mappings from
into itself with
.
is said to satisfy the (SU) condition, if the following conditions hold:
(1)for any strong convergence sequence , the sequence
is also strong convergent;
(2)the common fixed points set is nonempty;
(3), where
is defined by
, for all
,
denotes the fixed points set of
.
In Section 3 of this paper, we will give an important example of sequence of -Lipschitzian mappings which satisfies the (SU) condition.
Lemma 1.3.
Let
be a nonempty closed convex subset of a real Hilbert space
and let
be a sequence of
-Lipschitzian mappings from
into itself with
. If
satisfies the (SU) condition. Then
(1) is bounded implies
is uniformly continuous on the
;
(2) implies
is
-Lipschitzian;
(3) implies
is nonexpansive.
Proof.
Observe that, for all , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ8_HTML.gif)
The results (1)–(3) are easy to prove.
2. Main Results
Theorem 2.1.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a sequence of
-Lipschitzian mappings from
into itself with
, as
. Assume
satisfies the (SU) condition and
is a sequence in
with
. Let
be a sequence in
defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ9_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ10_HTML.gif)
Then converges strongly to
, where
is the metric projection from
onto closed convex subset
.
Proof.
We first prove that and
are closed and convex for each
. From the definition of
and
, it is obvious that
is closed and
is closed and convex for each
. We prove that
is convex. Since
is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ11_HTML.gif)
it follows that is convex. Next, we show that
, for all
. For any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ12_HTML.gif)
so that , therefore, we have
, for all
. Next, we show that
, for all
. We prove this by induction. For
, we have
. Suppose that
, then
and there exists a unique element
such that
. By using Lemma 1.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ13_HTML.gif)
for all . In particular,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ14_HTML.gif)
for all . It follows from the definition of
that
. By using the principle of induction, we claim that
, for all
. Therefore, we have
, for all
. Now the sequence
is well defined.
It follows from the definition of that
. Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ15_HTML.gif)
for all . This implies that the
is bounded. On the other hand, from
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ16_HTML.gif)
for all . Therefore,
is nondecreasing and bounded. So that
exists.
Note again that , hence for any positive integer
, we have
which implies that
. Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ17_HTML.gif)
From this inequality, we know that is a Cauchy sequence in
, so that there exists a point
such that
.
Since , then
this together with
, as
, implies that
. From
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ18_HTML.gif)
as . Since
is nonexpansive, therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ19_HTML.gif)
as . Then
.
Finally, we claim that . If not, we have
. There must exists a positive integer
, if
, then
, which leads to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ20_HTML.gif)
It follows that implies that
This is a contradiction, hence
. This completes the proof.
The following theorem directly follows from Theorem 2.1.
Theorem 2.2.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a sequence of nonexpansive mappings from
into itself. Assume
satisfies the (SU) condition and
is a sequence in
with
. Let
be a sequence in
defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ21_HTML.gif)
Then converges strongly to
. Where
is the metric projection from
onto closed convex subset
.
3. Application for Equilibrium and Optimization
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a bifunction of
into
, 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%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ22_HTML.gif)
The set of solutions of (3.1) 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 (3.1). Some methods have been proposed to solve the equilibrium problem; see, for instance, [12–16].
For solving the equilibrium problem for a bifunction , let us assume that
satisfies the following conditions:
(A1), for all
;
(A2) is monotone, that is,
, for all
;
(A3)for each ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ23_HTML.gif)
(A4)for each ,
is convex and lower semicontinuous.
We need the following lemmas for the proof of our main results.
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%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ24_HTML.gif)
Assume that satisfies (A1)–(A4). For
and
, define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ25_HTML.gif)
for all . Then, the following hold:
(1) is single-valued;
(2) is firmly nonexpansive, that is, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ26_HTML.gif)
(3);
(4) is closed and convex.
Remark 3.3.
is also nonexpansive, for all
.
Now, we prove the following lemma which is very important for the main results of this section.
Lemma 3.4.
Let be a nonempty closed convex subset of
and let
be a bifunction of
into
satisfying (A1)–(A4). Let
be a positive real sequence such that
. Then the sequence
satisfies the (SU) condition.
Proof.
-
(1)
Let
be a convergent sequence in
. Let
, for all
, then
(3.6)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ28_HTML.gif)
Putting in (3.6) and
in (3.7), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ29_HTML.gif)
So, from (A2) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ30_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ31_HTML.gif)
Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ32_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ33_HTML.gif)
Therefore, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ34_HTML.gif)
On the other hand, for any , from
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ35_HTML.gif)
so that is bounded. Since
, this together with (3.13) implies that the
is a Cauchy sequence. Hence
is convergent.
-
(2)
By using Lemma 3.2, we know that
(3.15)
-
(3)
From (1) we know that,
exists, for all
. So, we can define a mapping
from
into itself by
(3.16)
It is obvious that the is nonexpansive. It is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ38_HTML.gif)
On the other hand, let , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ39_HTML.gif)
By (A2) we know
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ40_HTML.gif)
Since and from (A4), we have
, for all
. Then, for
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ41_HTML.gif)
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ42_HTML.gif)
Letting and using (A3), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ43_HTML.gif)
and hence . From the aforementioned two respects, we know that
. This completes the proof.
Theorem 3.5.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a bifunction from
into
satisfying (A1)–(A4) and
. Let
and
be sequences generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ44_HTML.gif)
Assume and
. Then
converges strongly to
, where
is the metric projection from
onto
.
Proof.
By using Lemma 3.4, we know that satisfies the (SU) condition. Then
. By using Theorem 2.2, we obtain the result of Theorem 3.5
Now, we study a kind of optimization problem by using the aforementioned results of this paper. That is, we will give an iterative algorithm of solution for the following optimization problem with nonempty set of solutions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ45_HTML.gif)
where is a convex and lower semicontinuous functional defined on a closed convex subset
of a Hilbert space
. We denoted by
the set of solutions of (3.24). Let
be a bifunction from
to
defined by
. We consider the following equilibrium problem, that is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ46_HTML.gif)
It is obvious that , where
denotes the set of solutions of equilibrium Problem (3.25). In addition, it is easy to see that
satisfies the conditions (A1)–(A4) in Section 2. Therefore, from Theorem 3.5, we can obtain the following theorem.
Theorem 3.6.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a convex and lower semicontinuous functional defined on
. Let
and
be sequences generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equa_HTML.gif)
Assume and
. Then
converges strongly to
.
Remark 3.7 ..
It is easy to see that this paper hassome new methods and results than the results of Nilsrakoo and Saejung [11]:
(1)proposed a modified hybrid iterative scheme, so that the new simple method of proof has been used;
(2)removed the bounded restriction for closed convex set ;
(3)relax the conditions of sequence ;
(4)give an application for optimization problem;
(5)the sequence of sets satisfy the following relation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F462489/MediaObjects/13663_2008_Article_1144_Equ47_HTML.gif)
so that to raise the convergence rate of is possible.
References
Mann WR: Mean value methods in iteration. Proceedings of the American Mathematical Society 1953,4(3):506–510. 10.1090/S0002-9939-1953-0054846-3
Qin X, Su Y: Strong convergence theorems for relatively nonexpansive mappings in a Banach space. Nonlinear Analysis: Theory, Methods & Applications 2007,67(6):1958–1965. 10.1016/j.na.2006.08.021
Genel A, Lindenstrauss J: An example concerning fixed points. Israel Journal of Mathematics 1975,22(1):81–86. 10.1007/BF02757276
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
Kim T-H, Xu H-K: Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Analysis: Theory, Methods & Applications 2006,64(5):1140–1152. 10.1016/j.na.2005.05.059
Nakajo K, Shimoji K, Takahashi W: Strong convergence theorems by the hybrid method for families of nonexpansive mappings in Hilbert spaces. Taiwanese Journal of Mathematics 2006,10(2):339–360.
Su Y, Qin X: Monotone CQ iteration processes for nonexpansive semigroups and maximal monotone operators. Nonlinear Analysis: Theory, Methods & Applications 2008,68(12):3657–3664. 10.1016/j.na.2007.04.008
Su Y, Wang D, Shang M: Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings. Fixed Point Theory and Applications 2008, Article ID 284613, 2008:-8.
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, Article ID 528476, 2008:-11.
Cheng Y, Tian M: Strong convergence theorem by monotone hybrid algorithm for equilibrium problems, hemirelatively nonexpansive mappings, and maximal monotone operators. Fixed Point Theory and Applications 2008, Article ID 617248, 2008:-12.
Nilsrakoo W, Saejung S: Weak and strong convergence theorems for countable Lipschitzian mappings and its applications. Nonlinear Analysis: Theory, Methods & Applications 2008,69(8):2695–2708. 10.1016/j.na.2007.08.044
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.
Flåm SD, Antipin AS: Equilibrium programming using proximal-like algorithms. Mathematical Programming 1997,78(1):29–41.
Moudafi A, Théra M: Proximal and dynamical approaches to equilibrium problems. In Ill-Posed Variational Problems and Regularization Techniques (Trier, 1998), Lecture Notes in Economics and Mathematical Systems. Volume 477. Springer, Berlin, Germany; 1999:187–201.
Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. Journal of Mathematical Analysis and Applications 2007,331(1):506–515. 10.1016/j.jmaa.2006.08.036
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This project is supported by the National Natural Science Foundation of China under grant(10771050).
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Yang, L., Su, Y. Strong Convergence Theorems for Countable Lipschitzian Mappings and Its Applications in Equilibrium and Optimization Problems. Fixed Point Theory Appl 2009, 462489 (2009). https://doi.org/10.1155/2009/462489
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DOI: https://doi.org/10.1155/2009/462489