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Strong Convergence Theorems by Hybrid Methods for Strict Pseudocontractions and Equilibrium Problems
Fixed Point Theory and Applications volume 2010, Article number: 528307 (2010)
Abstract
Let be N strict pseudocontractions defined on a closed convex subset
of a real Hilbert space
. Consider the problem of finding a common element of the set of fixed point of these mappings and the set of solutions of an equilibrium problem with the parallel and cyclic algorithms. In this paper, we propose new iterative schemes for solving this problem and prove these schemes converge strongly by hybrid methods.
1. Introduction
Let be a real Hilbert space and let
be a nonempty closed convex subset of
. Let
be a bifunction from
to
, where
is the set of real numbers.
The equilibrium problem for is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ1_HTML.gif)
for all . The set of such solutions is denoted by
.
A mapping of
is said to be a
-strict pseudocontraction if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ2_HTML.gif)
for all ; see [1]. We denote the set of fixed points of
by
(i.e.,
).
Note that the class of strict pseudocontractions strictly includes the class of nonexpansive mappings which are mapping on
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ3_HTML.gif)
for all . That is,
is nonexpansive if and only if
is a
-strict pseudocontraction.
Numerous problems in physics, optimization, and economics reduce to finding a solution of the equilibrium problem. Some methods have been proposed to solve the equilibrium problem (1.1); see for instance [2–5]. In particular, Combettes and Hirstoaga [6] proposed several methods for solving the equilibrium problem. On the other hand, Mann [7], Nakajo and Takahashi [8] considered iterative schemes for finding a fixed point of a nonexpansive mapping.
Recently, Acedo and Xu [9] considered the problem of finding a common fixed point of a finite family of strict pseudocontractive mappings by the parallel and cyclic algorithms. Very recently, Liu [3] considered a general iterative method for equilibrium problems and strict pseudocontractions. In this paper, motivated by [3, 5, 9–12], applying parallel and cyclic algorithms, we obtain strong convergence theorems for finding a common element of the set of fixed points of a finite family of strict pseudocontractions and the set of solutions of the equilibrium problem (1.1) by the hybrid methods.
We will use the notation
(1) for weak convergence and
for strong convergence,
(2) denotes the weak
-limit set of
.
2. Preliminaries
We need some facts and tools in a real Hilbert space which are listed as below.
Lemma 2.1.
Let be a real Hilbert space. There hold the following identities.
(i), for all
(ii), for all
, for all
Lemma 2.2 (see [4]).
Let be a real Hilbert space. Given a nonempty closed convex subset
and points
and given also a real number
, the set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ4_HTML.gif)
is convex (and closed).
Recall that given a nonempty closed convex subset of a real Hilbert space
, for any
, there exists a unique nearest point in
, denoted by
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ5_HTML.gif)
for all . Such a
is called the metric (or the nearest point) projection of
onto
.
Lemma 2.3 (see [4]).
Let be a nonempty 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%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ6_HTML.gif)
Lemma 2.4 (see [13]).
Let be a nonempty closed convex subset of
. Let
is a sequence in
and
. Let
. Suppose
is such that
and satisfies the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ7_HTML.gif)
Then .
Lemma 2.5 (see [9]).
Let be a nonempty closed convex subset of
. Let
is a sequence in
and
. Assume
(i)the weak -limit set
,
(ii)for each exists.
Then is weakly convergent to a point in
.
Proposition 2.6 (see [9]).
Assume be a nonempty closed convex subset of a real Hilbert space
.
(i)If is a
-strict pseudocontraction, then
satisfies the Lipschitz condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ8_HTML.gif)
(ii)If is a
-strict pseudocontraction, then the mapping
is demiclosed (at 0). That is, if
is a sequence in
such that
and
, then
.
(iii)If is a
-strict pseudocontraction, then the fixed point set of
of
is closed and convex so that the projection
is well defined.
(iv)Given an integer , assume, for each
be a
-strict pseudocontraction for some
. Assume
is a positive sequence such that
. Then
is a
-strict pseudocontraction, with
(v)Let and
be given as in (iv) above. Suppose that
has a common fixed point. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ9_HTML.gif)
Lemma 2.7 (see [1]).
Let be a
-strict pseudocontraction. Define
by
for each
. Then, as
is a nonexpansive mapping such that
.
For solving the equilibrium problem, let us assume that the bifunction satisfies the following conditions:
(A1) for all
(A2) is monotone, that is,
for any
(A3)for each
(A4) is convex and lower semicontionuous for each
We recall some lemmas which will be needed in the rest of this paper.
Lemma 2.8 (see [14]).
Let be a nonempty closed convex subset of
, let
be bifunction from
to
satisfying (A1)–(A4) and let
and
. Then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ10_HTML.gif)
Lemma 2.9 (see [6]).
For , define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ11_HTML.gif)
for all . Then, the following statements hold:
(i) is single-valued;
(ii) is firmly nonexpansive, that is, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ12_HTML.gif)
(iii);
(iv) is closed and convex.
3. Parallel Algorithm
In this section, we apply the hybrid methods to the parallel algorithm for finding a common element of the set of fixed points of strict pseudocontractions and the set of solutions of the equilibrium problem (1.1) in Hilbert spaces.
Theorem 3.1.
Let be a nonempty closed convex subset of a real Hilbert space
and
a bifunction from
to
satisfying (A1)–(A4). Let
be an integer. Let, for each
be a
-strict pseudocontraction for some
. Let
Assume the set
. Assume also
is a finite sequence of positive numbers such that
for all
and
for all
. Let the mapping
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ13_HTML.gif)
Given , let
, and
be sequences generated by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ14_HTML.gif)
for every , where
for some
for some
, and
satisfies
. Then,
converge strongly to
.
Proof.
The proof is divided into several steps.
Step 1.
Show first that is well defined.
It is obvious that is closed and
is closed convex for every
. From Lemma 2.2, we also get
is convex.
Step 2.
Show for all
.
Indeed, take , from
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ15_HTML.gif)
for all . From Proposition 2.6, Lemma 2.7, and (3.3), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ16_HTML.gif)
So for all
. Next we show that
for all
by induction. For
, we have
. Assume that
for some
. Since
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ17_HTML.gif)
As by induction assumption, the inequality holds, in particular, for all
. This together with the definition of
implies that
. Hence
holds for all
.
Step 3.
Show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ18_HTML.gif)
Notice that the definition of actually
. This together with the fact
further implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ19_HTML.gif)
Then is bounded and (3.6) holds. From (3.3), (3.4), and Proposition 2.6(i), we also obtain
and
are bounded.
Step 4.
Show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ20_HTML.gif)
From and
, we get
. This together with Lemma 2.1(i) implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ21_HTML.gif)
Then , that is, the sequence
is nondecreasing. Since
is bounded,
exists. Then (3.8) holds.
Step 5.
Show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ22_HTML.gif)
From , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ23_HTML.gif)
By (3.8), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ24_HTML.gif)
For , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ25_HTML.gif)
hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ26_HTML.gif)
Therefore, by the convexity of , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ27_HTML.gif)
Since , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ28_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ29_HTML.gif)
from (3.12). Observe that we also have
. On the other hand, from
, we compute
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ30_HTML.gif)
From , (3.17), and
, we obtain
. It is easy to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ31_HTML.gif)
Combining the above results, we obtain From (3.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ32_HTML.gif)
It follows from that
Step 6.
Show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ33_HTML.gif)
We first show . To see this, we take
and assume that
as
for some subsequence
of
.
Without loss of generality, we may assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ34_HTML.gif)
It is easily seen that each and
. We also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ35_HTML.gif)
where Note that by Proposition 2.6,
is
-strict pseudocontraction and
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ36_HTML.gif)
we obtain by virtue of (3.10) and (3.22)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ37_HTML.gif)
So by the demiclosedness principle (Proposition 2.6(ii)), it follows that and hence
holds.
Next we show take
, and assume that
as
for some subsequence
of
. From (3.17), we obtain
. Since
and
is closed convex, we get
By we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ38_HTML.gif)
From the monotonicity of , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ39_HTML.gif)
hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ40_HTML.gif)
From (3.17) and condition (A4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ41_HTML.gif)
For with
and
, let
. Since
and
, we obtain
and hence
. So, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ42_HTML.gif)
Dividing by , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ43_HTML.gif)
Letting and from (A3), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ44_HTML.gif)
for all and
. Hence (3.21) holds.
Step 7.
From (3.6) and Lemma 2.4, we conclude that , where
.
A very similar result obtained in a way completely different is Theorem of [11].
Theorem 3.2.
Let be a nonempty closed convex subset of a real Hilbert space
and
a bifunction from
to
satisfying (A1)–(A4). Let
be an integer. Let, for each
be a
-strict pseudocontraction for some
. Let
Assume the set
. Assume also
is a finite sequence of positive numbers such that
for all
and
for all
. Let the mapping
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ45_HTML.gif)
Given , let
, and
be sequences generated by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ46_HTML.gif)
for every , where
for some
for some
, and
satisfies
. Then,
converge strongly to
.
Proof.
The proof of this theorem is similar to that of Theorem 3.1.
Step 1.
is well defined for all
We show is closed convex for all
by induction. For
, we have
is closed convex. Assume that
for some
is closed convex, from Lemma 2.2, we have
is also closed convex. The assumption holds.
Step 2.
.
Step 3.
for all
, where
Step 4.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_IEq301_HTML.gif)
Step 5.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_IEq302_HTML.gif)
Step 6.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_IEq303_HTML.gif)
Step 7.
.
The proof of Steps 2–7 is similar to that of Theorem 3.1.
A very similar result obtained in a way completely different is Theorem of [10].
4. Cyclic Algorithm
Let be a closed convex subset of a Hilbert space
and let
be
-strict pseudocontractions on
such that the common fixed point set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ47_HTML.gif)
Let and let
be a sequence in
. The cyclic algorithm generates a sequence
in the following way:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ48_HTML.gif)
In general, is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ49_HTML.gif)
where , with
.
Theorem 4.1.
Let be a nonempty closed convex subset of a real Hilbert space
and
a bifunction from
to
satisfying (A1)–(A4). Let
be an integer. Let, for each
be a
-strict pseudocontraction for some
. Let
Assume the set
. Given
, let
, and
be sequences generated by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ50_HTML.gif)
for every , where
for some
for some
, and
satisfies
. Then,
converge strongly to
.
Proof.
The proof of this theorem is similar to that of Theorem 3.1. The main points include the following.
Step 1.
is well defined for all
Step 2.
.
Step 3.
for all
, where
Step 4.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_IEq347_HTML.gif)
Step 5.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_IEq348_HTML.gif)
To prove the above steps, one simply replaces with
in the proof of Theorem 3.1.
Step 6.
Show that
Indeed, assume and
for some subsequence
of
. We may further assume
for all
. Since by
, we also have
for all
, we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ51_HTML.gif)
Then the demiclosedness principle (Proposition 2.6(ii)) implies that for all
. This ensures that
.
The proof of is similar to that of Theorem 3.1.
Step 7.
Show that
The strong convergence to of
is the consequence of Step 3, Step 5, and Lemma 2.4.
Theorem 4.2.
Let be a nonempty closed convex subset of a real Hilbert space
and
a bifunction from
to
satisfying (A1)–(A4). Let
be an integer. Let, for each
be a
-strict pseudocontraction for some
. Let
Assume the set
. Given
, let
, and
be sequences generated by the following algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F528307/MediaObjects/13663_2009_Article_1300_Equ52_HTML.gif)
for every , where
for some
for some
, and
satisfies
. Then,
converge strongly to
.
Proof.
The proof of this theorem can consult Step 1 of Theorem 3.2 and Steps 2–7 of Theorem 4.1.
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Acknowledgments
The authors would like to thank the referee for valuable suggestions to improve the manuscript and the Fundamental Research Funds for the Central Universities (Grant no. ZXH2009D021) and the science research foundation program in Civil Aviation University of China (04-CAUC-15S) for their financial support.
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Duan, P., Zhao, J. Strong Convergence Theorems by Hybrid Methods for Strict Pseudocontractions and Equilibrium Problems. Fixed Point Theory Appl 2010, 528307 (2010). https://doi.org/10.1155/2010/528307
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DOI: https://doi.org/10.1155/2010/528307