- Research Article
- Open access
- Published:
An Iterative Algorithm Combining Viscosity Method with Parallel Method for a Generalized Equilibrium Problem and Strict Pseudocontractions
Fixed Point Theory and Applications volume 2009, Article number: 794178 (2009)
Abstract
We introduce a new approximation scheme combining the viscosity method with parallel method for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a family of finitely strict pseudocontractions. We obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces. Based on this result, we also get some new and interesting results. The results in this paper extend and improve some well-known results in the literature.
1. Introduction
Let be a real Hilbert space with inner product
and induced norm
and let
be a nonempty-closed convex subset of
. Let
be a function and let
be a bifunction from
to
such that
, where
is the set of real numbers and
. Flores-Bazán [1] introduced the following generalized equilibrium problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ1_HTML.gif)
The set of solutions of (1.1) is denoted by . Flores-Bazán [1] provided some characterizations of the nonemptiness of the solution set for problem (1.1) in reflexive Banach spaces in the quasiconvex case. Bigi et al. [2] studied a dual problem associated with the problem (1.1) with
.
Let . Here
denotes the indicator function of the set
; that is,
if
and
otherwise. Then the problem (1.1) becomes the following equilibrium problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ2_HTML.gif)
The set of solutions of (1.2) is denoted by . The problem (1.2) includes, as special cases, the optimization problem, the variational inequality problem, the fixed point problem, the nonlinear complementarity problem, the Nash equilibrium problem in noncooperative games, and the vector optimization problem. For more detail, please see [3–5] and the references therein.
If for all
, where
is a function, then the problem (1.1) becomes a problem of finding
which is a solution of the following minimization problem:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ3_HTML.gif)
The set of solutions of (1.3) is denoted by .
If is replaced by a real-valued function
, the problem (1.1) reduces to the following mixed equilibrium problem introduced by Ceng and Yao [6]:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ4_HTML.gif)
Recall that a mapping is said to be a
-strict pseudocontraction [7] if there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ5_HTML.gif)
where denotes the identity operator on
. When
,
is said to be nonexpansive. Note that the class of strict pseudocontraction mappings strictly includes the class of nonexpansive mappings. We denote the set of fixed points of
by
.
Ceng and Yao [6], Yao et al. [8], and Peng and Yao [9, 10] introduced some iterative schemes for finding a common element of the set of solutions of the mixed equilibrium problem (1.4) and the set of common fixed points of a family of finitely (infinitely) nonexpansive mappings (strict pseudocontractions) in a Hilbert space and obtained some strong convergence theorems(weak convergence theorems). Some methods have been proposed to solve the problem (1.2); see, for instance, [3–5, 11–18] and the references therein. Recently, S. Takahashi and W. Takahashi [12] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping in a Hilbert space and proved a strong convergence theorem. Su et al. [13] introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for an -inverse strongly monotone mapping in a Hilbert space. Tada and Takahashi [14] introduced two iterative schemes for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a nonexpansive mapping in a Hilbert space and obtained both strong convergence theorem and weak convergence theorem. Ceng et al. [15] introduced an iterative algorithm for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a strict pseudocontraction mapping. Chang et al. [16] introduced some iterative processes based on the extragradient method for finding the common element of the set of fixed points of a family of infinitely nonexpansive mappings, the set of problem (1.2), and the set of solutions of a variational inequality problem for an
-inverse strongly monotone mapping. Colao et al. [17] introduced an iterative method for finding a common element of the set of solutions of problem (1.2) and the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space and proved the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem. To the best of our knowledge, there is not any algorithms for solving problem (1.1).
On the other hand, Marino and Xu [19] and Zhou [20] introduced and researched some iterative scheme for finding a fixed point of a strict pseudocontraction mapping. Acedo and Xu [21] introduced some parallel and cyclic algorithms for finding a common fixed point of a family of finite strict pseudocontraction mappings and obtained both weak and strong convergence theorems for the sequences generated by the iterative schemes.
In the present paper, we introduce a new approximation scheme combining the viscosity method with parallel method for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of a family of finitely strict pseudocontractions. We obtain a strong convergence theorem for the sequences generated by these processes. Based on this result, we also get some new and interesting results. The results in this paper extend and improve some well-known results in the literature.
2. Preliminaries
Let be a real Hilbert space with inner product
and norm
. Let
be a nonempty-closed convex subset of
. Let symbols
and
denote strong and weak convergences, respectively. In a real Hilbert space
, it is well known that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ6_HTML.gif)
for all and
.
For any , there exists a unique nearest point in
, denoted by
, such that
for all
. The mapping
is called the metric projection of
onto
. We know that
is a nonexpansive mapping from
onto
. It is also known that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ7_HTML.gif)
for all and
.
For each , we denote by
the convex hull of
. A multivalued mapping
is said to be a KKM map if, for every finite subset
,
We will use the following results in the sequel.
Lemma 2.1 (see [22]).
Let be a nonempty subset of a Hausdorff topological vector space
and let
be a KKM map. If
is closed for all
and is compact for at least one
, then
For solving the generalized equilibrium problem, let us give the following assumptions for the bifunction ,
and the set
:
(A1) for all
;
(A2) is monotone, that is,
for any
;
(A3) for each ,
is weakly upper semicontinuous;
(A4)for each is convex;
(A5)for each is lower semicontinuous;
(B1)For each and
, there exist a bounded subset
and
such that for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ8_HTML.gif)
(B2) is a bounded set.
Lemma 2.2.
Let be a nonempty-closed convex subset of
. Let
be a bifunction from
to
satisfying (A1)–(A4) and let
be a proper lower semicontinuous and convex function such that
. For
and
define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ9_HTML.gif)
for all . Assume that either (B1) or (B2) holds. Then, the following conclusions hold:
(1)for each ,
;
(2) is single-valued;
(3) is firmly nonexpansive, that is, for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ10_HTML.gif)
(4)
(5) is closed and convex.
Proof.
Let be any given point in
. For each
, we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ11_HTML.gif)
Note that for each ,
is nonempty since
and for each
,
. We will prove that
is a KKM map on
. Suppose that there exists a finite subset
of
and
for all
with
such that
for each
. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ12_HTML.gif)
for each . By (A4) and the convexity of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ13_HTML.gif)
which is a contradiction. Hence, is a KKM map on
. Note that
(the weak closure of
) is a weakly closed subset of
for each
. Moreover, if (B2) holds, then
is also weakly compact for each
. If (B1) holds, then for
, there exists a bounded subset
and
such that for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ14_HTML.gif)
This shows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ15_HTML.gif)
Hence, is weakly compact. Thus, in both cases, we can use Lemma 2.1 and have
.
Next, we will prove that for each
; that is,
is weakly closed. Let
and let
be a sequence in
such that
. Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ16_HTML.gif)
Since is weakly lower semicontinuous, we can show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ17_HTML.gif)
It follows from (A3) and the weak lower semicontinuity of that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ18_HTML.gif)
This implies that . Hence,
is weakly closed. Hence,
. Hence, from the arbitrariness of
, we conclude that
,
.
We observe that . So by similar argument with that in the proof of Lemma 2.3 in [9], we can easily show that
is single-valued and
is a firmly nonexpansive-type map. Next, we claim that
. Indeed, we have the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ19_HTML.gif)
At last, we claim that is a closed convex. Indeed, Since
is firmly nonexpansive,
is also nonexpansive. By [23, Proposition 5.3], we know that
is closed and convex.
Remark 2.3.
It is easy to see that Lemma 2.2 is a generalization of [9, Lemma 2.3].
Assume that is a sequence of nonnegative real numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ20_HTML.gif)
where is a sequence in
and
is a sequence such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ21_HTML.gif)
Then,
Lemma 2.5.
In a real Hilbert space , there holds the following inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ22_HTML.gif)
for all
3. Strong Convergence Theorems
In this section, we show a strong convergence of an iterative algorithm based on both viscosity approximation method and parallel method which solves the problem of finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a family of finitely strict pseudocontractions in a Hilbert space.
We need the following assumptions for the parameters and
:
(C1) and
;
(C2);
(C3) for some
and
;
(C4) and
;
(C5) for all
.
Theorem 3.1.
Let be a nonempty-closed convex subset of a real Hilbert space
. Let
be a bifunction from
to
satisfying (A1)–(A5), and let
be a proper lower semicontinuous and convex function such that
. Let
be an integer. For each
, let
be an
-strict pseudocontraction for some
such that
. Assume for each
,
is a finite sequence of positive numbers such that
for all
and
for all
. Let
. Assume that either (B1) or (B2) holds. Let
be a contraction of
into itself and let
,
and
be sequences generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ23_HTML.gif)
for every , where
and
are sequences of numbers satisfying the conditions (C1)–(C5). Then,
,
and
converge strongly to
.
Proof.
We show that is a contraction of
into itself. In fact, there exists
such that
for all
. So, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ24_HTML.gif)
for all Since
is complete, there exists a unique element
such that
.
Let and let
be a sequence of mappings defined as in Lemma 2.2. From
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ25_HTML.gif)
We define a mapping by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ26_HTML.gif)
By [21, Proposition 2.6], we know that is an
-strict pseudocontraction and
. It follows from (3.3),
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ27_HTML.gif)
Put It is obvious that
Suppose
From (3.3), (3.5), and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ28_HTML.gif)
for every Therefore,
is bounded. From (3.3) and (3.5), we also obtain that
and
are bounded.
Following [26], define by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ29_HTML.gif)
As shown in [26], each is a nonexpansive mapping on
. Set
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ30_HTML.gif)
On the other hand, from and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ31_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ32_HTML.gif)
Putting in (3.9) and
in (3.10), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ33_HTML.gif)
So, from the monotonicity of , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ34_HTML.gif)
hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ35_HTML.gif)
Without loss of generality, let us assume that there exists a real number such that
for all
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ36_HTML.gif)
hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ37_HTML.gif)
where
It follows from (3.8) and (3.15) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ38_HTML.gif)
Define a sequence such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ39_HTML.gif)
Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ40_HTML.gif)
From (3.18) and (3.16), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ41_HTML.gif)
It follows from (C1)–(C5) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ42_HTML.gif)
Hence, by [27, Lemma 2.2], we have . Consequently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ43_HTML.gif)
Since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ44_HTML.gif)
thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ45_HTML.gif)
It follows from (C1) and (C2) that .
Since for
, it follows from (3.5) and (3.3) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ46_HTML.gif)
from which it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ47_HTML.gif)
It follows from (C1)–(C3) and that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ48_HTML.gif)
For , we have from Lemma 2.2,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ49_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ50_HTML.gif)
By (3.24) and (3.28), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ51_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ52_HTML.gif)
It follows from (C1), (C2), and that
.
Next, we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ53_HTML.gif)
where . To show this inequality, we can choose a subsequence
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ54_HTML.gif)
Since is bounded, there exists a subsequence
of
which converges weakly to
. Without loss of generality, we can assume that
From
, we obtain that
. From
, we also obtain that
. Since
and
is closed and convex, we obtain
.
We first show that To see this, we observe that we may assume (by passing to a further subsequence if necessary)
(as
) for
It is easy to see that
and
. We also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ55_HTML.gif)
where . Note that by [21, Proposition 2.6],
is an
-strict pseudocontraction and
. Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ56_HTML.gif)
it follows from (3.26) and that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ57_HTML.gif)
So by the demiclosedness principle [21, Proposition 2.6(ii)], it follows that .
We now show By
we know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ58_HTML.gif)
It follows from (A2) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ59_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ60_HTML.gif)
It follows from (A4), (A5) and the weakly lower semicontinuity of ,
and
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ61_HTML.gif)
For with
and
, let
Since
and
, we obtain
and hence
. So by (A4) and the convexity of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ62_HTML.gif)
Dividing by , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ63_HTML.gif)
Letting , it follows from (A3) and the weakly lower semicontinuity of
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ64_HTML.gif)
for all . Observe that if
, then
holds. Moreover, hence
. This implies
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ65_HTML.gif)
Finally, we show that , where
.
From Lemma 2.5, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ66_HTML.gif)
thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ67_HTML.gif)
It follows from (C1), (3.43), (3.45), and Lemma 2.4 that . From
and
, we have
and
. The proof is now complete.
Theorem 3.2.
Let be a nonempty-closed convex subset of a real Hilbert space
. Let
be a bifunction from
to
satisfying (A1)–(A5), and let
be a proper lower semicontinuous and convex function such that
. Let
be an integer. For each
, let
be an
-strict pseudocontraction for some
such that
. Assume for each
,
is a finite sequence of positive numbers such that
for all
and
for all
. Let
. Assume that either (B1) or (B2) holds. Let
be an arbitrary point in
and let
,
and
be sequences generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ68_HTML.gif)
for every where
and
are sequences of numbers satisfying the conditions (C1)–(C5). Then,
,
and
converge strongly to
.
Proof.
Let for all
, by Theorem 3.1, we obtain the desired result.
4. Applications
By Theorems 3.1 and 3.2, we can obtain many new and interesting strong convergence theorems. Now, give some examples as follows: for , let
, by Theorems 3.1 and 3.2, respectively, we have the following results.
Theorem 4.1.
Let be a nonempty-closed convex subset of a real Hilbert space
. Let
be a bifunction from
to
satisfying (A1)–(A5), and let
be a proper lower semicontinuous and convex function such that
. Let
be an
-strict pseudocontraction for some
such that
. Assume that either (B1) or (B2) holds. Let
be a contraction of
into itself and let
,
and
be sequences generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ69_HTML.gif)
for every where
,
,
and
are sequences of numbers satisfying the conditions (C1)–(C4). Then,
,
and
converge strongly to
.
Theorem 4.2.
Let be a nonempty-closed convex subset of a real Hilbert space
. Let
be a bifunction from
to
satisfying (A1)–(A5), and let
be a proper lower semicontinuous and convex function such that
. Let
be an
-strict pseudocontraction for some
such that
. Assume that either (B1) or (B2) holds. Let
be an arbitrary point in
and let
,
and
be sequences generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ70_HTML.gif)
for every where
,
,
and
are sequences of numbers satisfying the conditions (C1)–(C4). Then,
,
and
converge strongly to
.
We need the following two assumptions.
(B3)For each and
, there exist a bounded subset
and
such that for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ71_HTML.gif)
(B4)For each and
, there exist a bounded subset
and
such that for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ72_HTML.gif)
Let , by Theorems 3.1 and 3.2, respectively, we obtain the following results.
Theorem 4.3.
Let be a nonempty-closed convex subset of a real Hilbert space
. Let
be a bifunction from
to
satisfying (A1)–(A5). Let
be an integer. For each
, let
be an
-strict pseudocontraction for some
such that
. Assume for each
,
is a finite sequence of positive numbers such that
for all
and
for all
. Let
. Assume that either (B3) or (B2) holds. Let
be a contraction of
into itself, and let
,
and
be sequences generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ73_HTML.gif)
for every where
and
are sequences of numbers satisfying the conditions (C1)–(C5). Then,
,
and
converge strongly to
.
Theorem 4.4.
Let be a nonempty-closed convex subset of a real Hilbert space
. Let
be a bifunction from
to
satisfying (A1)–(A5). Let
be an integer. For each
, let
be an
-strict pseudocontraction for some
such that
. Assume for each
,
is a finite sequence of positive numbers such that
for all
and
for all
. Let
. Assume that either (B3) or (B2) holds. Let
be an arbitrary point in
and let
,
and
be sequences generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ74_HTML.gif)
for every where
and
are sequences of numbers satisfying the conditions (C1)–(C5). Then,
,
and
converge strongly to
.
Let for all
, by Theorems 3.1 and 3.2, respectively, we obtain the following results.
Theorem 4.5.
Let be a nonempty-closed convex subset of a real Hilbert space
. Let
be a lower semicontinuous and convex function, and let
be a proper lower semicontinuous and convex function such that
. Let
be an integer. For each
, let
be an
-strict pseudocontraction for some
such that
. Assume for each
,
is a finite sequence of positive numbers such that
for all
and
for all
. Let
. Assume that either (B4) or (B2) holds. Let
be a contraction of
into itself, and let
,
and
be sequences generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ75_HTML.gif)
for every , where
and
are sequences of numbers satisfying the conditions (C1)–(C5). Then,
,
and
converge strongly to
.
Theorem 4.6.
Let be a nonempty-closed convex subset of a real Hilbert space
. Let
be a lower semicontinuous and convex function, and let
be a proper lower semicontinuous and convex function such that
. Let
be an integer. For each
, let
be an
-strict pseudocontraction for some
such that
. Assume for each
,
is a finite sequence of positive numbers such that
for all
and
for all
. Let
. Assume that either (B4) or (B2) holds. Let
be an arbitrary point in
and let
,
and
be sequences generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ76_HTML.gif)
for every where
and
are sequences of numbers satisfying the conditions (C1)–(C5). Then,
,
and
converge strongly to
.
Let , and let
for all
. Then
. By Theorems 3.1 and 3.2, we obtain the following results.
Theorem 4.7.
Let be a nonempty-closed convex subset of a real Hilbert space
. Let
be an integer. For each
, let
be an
-strict pseudocontraction for some
such that
. Assume for each
,
is a finite sequence of positive numbers such that
for all
and
for all
. Let
. Let
be a contraction of
into itself, and let
and
be sequences generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ77_HTML.gif)
for every , where
and
are sequences of numbers satisfying the conditions (C1)–(C3) and (C5). Then,
and
converge strongly to
.
Theorem 4.8.
Let be a nonempty-closed convex subset of a real Hilbert space
. Let
be an integer. For each
, let
be an
-strict pseudocontraction for some
such that
. Assume for each
,
is a finite sequence of positive numbers such that
for all
and
for all
. Let
. Let
be an arbitrary point in
and let
and
be sequences generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F794178/MediaObjects/13663_2008_Article_1178_Equ78_HTML.gif)
for every , where
and
are sequences of numbers satisfying the conditions (C1)–(C3) and (C5). Then,
and
converge strongly to
.
Remark 4.9.
-
(1)
Since the nonexpansive mappings have been replaced by the strict pseudocontractions, Theorems 3.1, 3.2, 4.1 and 4.2 extend and improve [6, Theorem 3.1], [8, Theorem 3.5], [9, Theorems 4.1 and 4.2], [18, Theorem 4.1], and the main results in [9–11, 13–16].
-
(2)
Since the weak convergence has been replaced by strong convergence, Theorems 3.1, 3.2, 4.1−4.4 extend and improve [12, Theorem 3.1], [10, Corollary 4.1].
-
(3)
Theorems 4.7 and 4.8 are strong convergence theorems for strict pseudocontractions without
constraints and hence they improve the corresponding results in [19, 21]. Theorems 3.1 and 3.2 also improve [10, Corollary 3.1].
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Acknowledgments
The first author was supported by the National Natural Science Foundation of China (Grants No. 10771228 and No. 10831009), the Science and Technology Research Project of Chinese Ministry of Education (Grant no. 206123), the Education Committee project Research Foundation of Chongqing Normal University (Grant no. KJ070816); the second and third authors were partially supported by the Grants NSC97-2221-E-230-017 and NSC96-2628-E-110-014-MY3, respectively.
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Peng, JW., Liou, YC. & Yao, JC. An Iterative Algorithm Combining Viscosity Method with Parallel Method for a Generalized Equilibrium Problem and Strict Pseudocontractions. Fixed Point Theory Appl 2009, 794178 (2009). https://doi.org/10.1155/2009/794178
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DOI: https://doi.org/10.1155/2009/794178