- Research Article
- Open access
- Published:
Strong Convergence for Mixed Equilibrium Problems of Infinitely Nonexpansive Mappings
Fixed Point Theory and Applications volume 2010, Article number: 756492 (2010)
Abstract
We introduce a new iterative scheme for finding a common element of infinitely nonexpansive mappings, the set of solutions of a mixed equilibrium problems, and the set of solutions of the variational inequality for an -inverse-strongly monotone mapping in a Hilbert Space. Then, the strong converge theorem is proved under some parameter controlling conditions. The results of this paper extend and improve the results of Jing Zhao and Songnian He(2009)and many others. Using this theorem, we obtain some interesting corollaries.
1. Introduction
Let be a real Hilbert space with norm
and inner product
. And let
be a nonempty closed convex subset of
. Let
be a real-valued function and let
be an equilibrium bifunction, that is,
for each
. Ceng and Yao [1] considered the following mixed equilibrium problem.
Find such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ1_HTML.gif)
The set of solutions of (1.1) is denoted by It is easy to see that
is the solution of problem (1.1) and
. In particular, if
, the mixed equilibrium problem (1.1) reduced to the equilibrium problem.
Find such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ2_HTML.gif)
The set of solutions of (1.2) is denoted by If
and
for all
, where
is a mapping from
to
, then the mixed equilibrium problem (1.1) becomes the following variational inequality.
Find such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ3_HTML.gif)
The set of solutions of (1.3) is denoted by .
The variational inequality and the mixed equilibrium problems which include fixed point problems, optimization problems, variational inequality problems have been extensively studied in literature. See, for example, [2–8].
In 1997, Combettes and Hirstoaga [9] introduced an iterative method for finding the best approximation to the initial data and proved a strong convergence theorem. Subsequently, Takahashi and Takahashi [7] introduced another iterative scheme for finding a common element of and the set of fixed points of nonexpansive mappings. Furthermore,Yao et al.[8, 10] introduced an iterative scheme for finding a common element of
and the set of fixed points of finitely (infinitely) nonexpansive mappings.
Very recently, Ceng and Yao [1] considered a new iterative scheme for finding a common element of and the set of common fixed points of finitely many nonexpansive mappings in a Hilbert space and obtained a strong convergence theorem.
Now, we recall that a mapping is said to be
(i)monotone if
(ii)-Lipschitz if there exists a constant
such that
(iii)-inverse strongly monotone if there exists a positive real number
such that
It is obvious that any -inverse strongly monotone mapping
is monotone and Lipscitz. A mapping
is called nonexpansive if
We denote by
the set of fixed point of
.
In 2006, Yao and Yao [11] introduced the following iterative scheme.
Let be a closed convex subset of a real Hilbert space. Let
be an
-inverse strongly monotone mapping of
into
and let
be a nonexpansive mapping of
into itself such that
. Suppose that
and
and
are given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ4_HTML.gif)
where ,
, and
are sequence in
and
is a sequence in [0,2
]. They proved that the sequence
defined by (1.4) converges strongly to a common element of
under some parameter controlling conditions.
Moreover, Plubtieng and Punpaeng [12] introduced an iterative scheme (1.5) for finding a common element of the set of fixed point of nonexpansive mappings, the set of solutions of an equilibrium problems, and the set of solutions of the variational of inequality problem for an -inverse strongly monotone mapping in a real Hilbert space. Suppose that
and
,
, and
are given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ5_HTML.gif)
where ,
, and
are sequence in
,
is a sequence in [0,2
], and
. Under some parameter controlling conditions, they proved that the sequence
defined by (1.5) converges strongly to
.
On the other hand, Yao et al. [8] introduced an iterative scheme (1.7) for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed point of infinitely many nonexpansive mappings in . Let
be a sequence of nonexpansive mappings of
into itself and let
be a sequence of real number in
. For each
, define a mapping
of
into itself as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ6_HTML.gif)
Such a mapping is called the
-mapping generated by
and
. In [8], given
arbitrarily, the sequences
and
are generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ7_HTML.gif)
They proved that under some parameter controlling conditions, generated by (1.7) converges strongly to
, where
.
Subsequently, Ceng and Yao [13] introduced an iterative scheme by the viscosity approximation method:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ8_HTML.gif)
where ,
and
are sequence in (0,1) such that
. Under some parameter controlling conditions, they proved that the sequence
defined by (1.8) converges strongly to
, where
.
Recently, Zhao and He [14] introduced the following iterative process.
Suppose that =
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ9_HTML.gif)
where ,
,
, and
such that
. Under some parameter controlling conditions, they proved that the sequence
defined by (1.9) converges strongly to
, where
Motivated by the ongoing research in this field, in this paper we suggest and analyze an iterative scheme for finding a common element of the set of fixed point of infinitely nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the variational of inequality problem for an -inverse strongly monotone mapping in a real Hilbert space. Under some appropriate conditions imposed on the parameters, we prove another strong convergence theorem and show that the approximate solution converges to a unique solution of some variational inequality which is the optimality condition for the minimization problem. The results of this paper extend and improve the results of Zhao and He [14] and many others. For some related works, we refer the readers to [15–22] and the references therein.
2. Preliminaries
Let be a real Hilbert space and let
be a closed convex subset of
. Then, 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%2F756492/MediaObjects/13663_2010_Article_1338_Equ10_HTML.gif)
is called the metric projection of
onto
. It is well known that
is nonexpansive mapping and satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ11_HTML.gif)
Moreover, is characterized by the following properties:
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ12_HTML.gif)
It is clear that
A space is said to satisfy Opials condition if for each sequence
in
which converges weakly to a point
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ13_HTML.gif)
The following lemmas will be useful for proving the convergence result of this paper.
Lemma 2.1 (see [23]).
Let and
be bounded sequences in a Banach space
and let
be a sequence in
with
Suppose that
for all integer
and
Then
Lemma 2.2 (see [24]).
Let be a real Hilbert space, let
be a closed convex subset of
, and let
be a nonexpansive mapping with
If
is a sequence in
weakly converging to
and if
converge strongly to
, then
.
Lemma 2.3 (see [25]).
Assume that is a sequence of nonnegative real numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ14_HTML.gif)
where is a sequence in
and
is a sequence in
such that
and
or
.
Then
In this paper, for solving the mixed equilibrium problem, let us give the following assumptions for a bifunction and the set
:
(A1) for all
;
(A2) is monotone, that is,
for any
;
(A3) is upper-hemicontinuous, that is, for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ15_HTML.gif)
(A4) is convex and lower semicontinuous for each
;
(B1) for each and
, there exists a bounded subset
and
such that for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ16_HTML.gif)
(B2) is a bounded set.
By a similar argument as in the proof of Lemma 2.3 in [26], we have the following result.
Lemma 2.4.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a bifunction from
that satisfies (A1)–(A4) and let
be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For
and
, define a mapping
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ17_HTML.gif)
for all Then, the following conditions hold:
(1)for each ;
(2) is single-valued;
(3) is firmly nonexpansive, that is, for any
,
;
(4);
(5) is closed and convex.
Let be a sequence of nonexpansive mappings of
into itself, where
is a nonempty closed convex subset of a real Hilbert space
. Given a sequence
in
, we define a sequence
of self-mappings on
by (1.6). Then We have the following result.
Lemma 2.5 (see [27]).
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a sequence of nonexpansive self-mappings on
such that
and let
be a sequence in
for some
. Then, for every
and
,
exists.
Remark 2.6 (see [8]).
It can be shown from Lemma 2.5 that if is a nonempty bounded subset of
, then for
, there exists
such that for all
,
, where
.
Remark 2.7 (see [8]).
Using Lemma 2.5, we define a mapping as follows:
, for all
.
is called the
-mapping generated by
and
Since is nonexpansive,
is also nonexpansive.
Indeed, for all ,
.
If is a bounded sequence in
, then we put
. Hence it is clear from Remark 2.6 that for any arbitrary
, there exists
such that for all
,
This implies that
Lemma 2.8 (see [27]).
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a sequence of nonexpansive self-mappings on
such that
and let
be a sequence in
for some
. Then
3. Main Results
Theorem 3.1.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a lower semicontinuous and convex function. Let
be a bifunction from
satisfying (A1)–(A4), let
be an
-inverse-strongly monotone mapping of
into
, and let
be a sequence of nonexpansive self-mapping on
such that
. Suppose that
,
,
, and
are sequences in
,
is a sequence in
such that
for some
with
, and
is a real sequence. Suppose that the following conditions are satisfied:
(i),
(ii) and
,
(iii),
(iv) and
,
(v),
(vi) and
.
Let be a contraction of
into itself with coefficient
. Assume that either (B1) or (B2) holds. Let the sequences
,
, and
be generated by,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ18_HTML.gif)
for all , where
is defined by (1.6) and
is a sequence in
, for some
. Then the sequence
converges strongly to a point
, where
.
Proof.
For any and
, we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ19_HTML.gif)
which implies that is nonexpansive.
Let be a sequence of mappping defined as in Lemma 2.4 and let
. Then
and
. Put
. From (3.2) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ20_HTML.gif)
Hence, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ21_HTML.gif)
Therefore is bounded. Consequently,
, and
are also bounded.
Next, we claim that .
Indeed, setting it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ22_HTML.gif)
Now, we estimate and
.
From the definition of , (1.6), and since
,
are nonexpansive, we deduce that, for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ23_HTML.gif)
for some constant such that sup
And we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ24_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ25_HTML.gif)
where = sup
.
Combining (3.7) and (3.8), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ26_HTML.gif)
On the other hand, from and
, we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ27_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ28_HTML.gif)
Putting in (3.10) and
in (3.11), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ29_HTML.gif)
So, from (A2) we get .
Hence
Without loss of generality, we may assume that there exists a real number such that
, for all
. Then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ30_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ31_HTML.gif)
where = sup
. Hence from (3.9) and (3.14), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ32_HTML.gif)
Combining (3.5), (3.6), and (3.15), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ33_HTML.gif)
It follows from (3.16) and conditions (i)–(vi) and that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ34_HTML.gif)
By Lemma 2.1, we have Consequently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ35_HTML.gif)
From conditions (iv)–(vi), (3.7), (3.8), (3.14), and (3.18), we also get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ36_HTML.gif)
Since and from the definitionof
, we have
. Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ37_HTML.gif)
For , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ38_HTML.gif)
and hence .
From (3.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ39_HTML.gif)
That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ40_HTML.gif)
From (ii) and (3.18), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ41_HTML.gif)
From (3.2)-(3.3), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ42_HTML.gif)
Then we get,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ43_HTML.gif)
Since and
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ44_HTML.gif)
We note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ45_HTML.gif)
Then we derive
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ46_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ47_HTML.gif)
which imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ48_HTML.gif)
From condition (ii), (3.18), and (3.27), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ49_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ50_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ51_HTML.gif)
and then we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ52_HTML.gif)
So we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ53_HTML.gif)
From condition (iv) and (3.20), (3.24), and (3.32), we have Moreover, from Remark 2.7 we get
Next, we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ54_HTML.gif)
where . Indeed, we choose a subsequence
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ55_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
Next, we show that .
First, we show that . In fact by
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ56_HTML.gif)
From (A2), we also have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ57_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ58_HTML.gif)
From ,
and
, we get
. It follows from (A4) that
and from the lower semicontinuity of
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ59_HTML.gif)
For with
and
, let
Since
and
, we have
and hence
. So, from (A1) and (A4), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ60_HTML.gif)
Dividing by , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ61_HTML.gif)
Letting , it follows from the weakly semicontinuity of
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ62_HTML.gif)
Hence .
Second, we show that . Assume
. Since
and
, by Opial's condition, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ63_HTML.gif)
which derives a contradiction. Thus we have .
Finally, by the same argument in the proof of [28, Theorem ], we can show that
Hence .
Since and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ64_HTML.gif)
Therefore, (3.37) holds.
Finally, we show that . From definition of
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ65_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ66_HTML.gif)
By (3.47) and Lemma 2.3, we get that converges strongly to
.
This completes the proof.
Setting and
in Theorem 3.1., we have the following result.
Corollary 3.2 (see [14, Theorem ]).
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a bifunction from
satisfying (A1)–(A4), let
be an
-inverse-strongly monotone mapping of
into
, and let
be a sequence of nonexpansive self-mapping on
such that
. Suppose that
,
,
,
, and
are sequences in
,
is a sequence in
such that
for some
with
and
is a real sequence. Suppose that the following conditions are satisfied:
(i),
(ii) and
,
(iii),
(iv) and
,
(v),
(vi) and
.
Let the sequence be generated by,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ67_HTML.gif)
for all , where
is defined by (1.6) and
is a sequence in
, for some
. Then the sequence
converges strongly to a point
, where
.
Setting in Theorem 3.1, we have the following result.
Corollary 3.3.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a bifunction from
satisfying (A1)–(A4), let
be an
-inverse-strongly monotone mapping of
into
, and let
be a sequence of nonexpansive self-mapping on
such that
. Suppose that
,
,
, and
are sequences in
,
is a sequence in
such that
for some
with
, and
is a real sequence. Suppose that the following conditions are satisfied:
(i),
(ii) and
,
(iii),
(iv) and
,
(v),
(vi) and
.
Let be a contraction of
into itself with coefficient
and let the sequence
be generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ68_HTML.gif)
for all , where
is defined by (1.6) and
is a sequence in
, for some
. Then the sequence
converges strongly to a point
, where
.
By Theorem 3.1, we obtain some interesting strong convergence theorems.
Setting then we have
in Theorem 3.1, and we have the following result.
Corollary 3.4.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be a lower semicontinuous and convex function. Let
be a bifunction from
satisfying (A1)–(A4), and let
be an
-inverse-strongly monotone mapping of
into
such that
. Suppose that
,
,
, and
are sequences in
,
is a sequence in
such that
for some
with
and
is a real sequence. Suppose that the following conditions are satisfied:
(i),
(ii) and
,
(iii),
and
,
(v),
(vi) and
.
Let be a contraction of
into itself with coefficient
. Assume that either (B1) or (B2) holds. Then the sequences
,
, and
generated by,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ69_HTML.gif)
converge strongly to a point , where
.
Setting and
then we have
in Theorem 3.1, and we have the following result.
Corollary 3.5.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be an
-inverse-strongly monotone mapping of
into
and let
be a sequence of nonexpansive self-mapping on
such that
. Suppose that
,
,
, and
are sequences in
,
is a sequence in
such that
for some
with
. Suppose that the following conditions are satisfied:
(i),
(ii) and
,
(iii),
(iv) and
,
(v).
Let be a contraction of
into itself with coefficient
. Let the sequences
and
be generated by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F756492/MediaObjects/13663_2010_Article_1338_Equ70_HTML.gif)
for all , where
defined by (1.6) and
is a sequence in
, for some
. Then the sequences
and
converge strongly to a point
, where
.
References
Ceng L-C, Yao J-C: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. Journal of Computational and Applied Mathematics 2008,214(1):186–201. 10.1016/j.cam.2007.02.022
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.
Chadli O, Schaible S, Yao JC: Regularized equilibrium problems with application to noncoercive hemivariational inequalities. Journal of Optimization Theory and Applications 2004,121(3):571–596.
Chadli O, Wong NC, Yao JC: Equilibrium problems with applications to eigenvalue problems. Journal of Optimization Theory and Applications 2003,117(2):245–266. 10.1023/A:1023627606067
Konnov IV, Schaible S, Yao JC: Combined relaxation method for mixed equilibrium problems. Journal of Optimization Theory and Applications 2005,126(2):309–322. 10.1007/s10957-005-4716-0
Kumam P: A new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping. Journal of Applied Mathematics and Computing 2009,29(1–2):263–280. 10.1007/s12190-008-0129-1
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
Yao Y, Liou YC, Yao JC: Convergence theorem for equilibrium problems and fixed point problems of infinte family of nonexpansive. Fixed Point Theory and Applications 2007, 2007:-12.
Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. Journal of Nonlinear and Convex Analysis 1997, 78: 29–41.
Yao Y, Liou Y-C, Yao J-C: An extragradient method for fixed point problems and variational inequality problems. Journal of Inequalities and Applications 2007, 2007:-12.
Yao Y, Yao J-C: On modified iterative method for nonexpansive mappings and monotone mappings. Applied Mathematics and Computation 2007,186(2):1551–1558. 10.1016/j.amc.2006.08.062
Plubtieng S, Punpaeng R: A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings. Applied Mathematics and Computation 2008,197(2):548–558. 10.1016/j.amc.2007.07.075
Ceng L-C, Yao J-C: Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Applied Mathematics and Computation 2008,198(2):729–741. 10.1016/j.amc.2007.09.011
Zhao J, He S: A new iterative method for equilibrium problems and fixed point problems of infinitely nonexpansive mappings and monotone mappings. Applied Mathematics and Computation 2009,215(2):670–680. 10.1016/j.amc.2009.05.041
Ceng L-C, Ansari QH, Siegfried S, Yao JC: Iterative methods for generalized equilibrium problems, systems of more generalized equilibrium problems and fixed point problems for nonexpansive mappings in Hilbert spaces. Fixed Point Theory 2010.,11(2):
Ceng L-C, Ansari QH, Yao J-C: Viscosity approximation methods for generalized equilibrium problems and fixed point problems. Journal of Global Optimization 2009,43(4):487–502. 10.1007/s10898-008-9342-6
Ceng L-C, Petruşel A, Yao JC: Iterative approaches to solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Journal of Optimization Theory and Applications 2009,143(1):37–58. 10.1007/s10957-009-9549-9
Ceng L-C, Yao J-C: A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem. Nonlinear Analysis: Theory, Methods & Applications 2010,72(3–4):1922–1937. 10.1016/j.na.2009.09.033
Chadli O, Liu Z, Yao JC: Applications of equilibrium problems to a class of noncoercive variational inequalities. Journal of Optimization Theory and Applications 2007,132(1):89–110. 10.1007/s10957-006-9072-1
Petruşel A, Yao J-C: Viscosity approximation to common fixed points of families of nonexpansive mappings with generalized contractions mappings. Nonlinear Analysis: Theory, Methods & Applications 2008,69(4):1100–1111. 10.1016/j.na.2007.06.016
Zeng L-C, Ansari QH, Shyu DS, Yao J-C: Strong and weak convergence theorems for common solutions of generalized equilibrium problems and zeros of maximal monotone operators. Fixed Point Theory and Applications 2010, 2010:-33.
Zeng LC, Lin YC, Yao JC: Iterative schemes for generalized equilibrium problem and two maximal monotone operators. Journal of Inequalities and Applications 2009, 2009:-34.
Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals. Journal of Mathematical Analysis and Applications 2005,305(1):227–239. 10.1016/j.jmaa.2004.11.017
Bauschke HH: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. Journal of Mathematical Analysis and Applications 1996,202(1):150–159. 10.1006/jmaa.1996.0308
Xu H-K: Viscosity approximation methods for nonexpansive mappings. Journal of Mathematical Analysis and Applications 2004,298(1):279–291. 10.1016/j.jmaa.2004.04.059
Peng J-W, Yao J-C: A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems. Taiwanese Journal of Mathematics 2008,12(6):1401–1432.
O'Hara JG, Pillay P, Xu H-K: Iterative approaches to convex feasibility problems in Banach spaces. Nonlinear Analysis, Theory, Methods and Applications 2006,64(9):2022–2042. 10.1016/j.na.2005.07.036
Nadezhkina N, Takahashi W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications 2006,128(1):191–201. 10.1007/s10957-005-7564-z
Acknowledgment
The author would like to thank the referees for their helpful comments and suggestions, which improved the presentation of this paper.
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
Joomwong, J. Strong Convergence for Mixed Equilibrium Problems of Infinitely Nonexpansive Mappings. Fixed Point Theory Appl 2010, 756492 (2010). https://doi.org/10.1155/2010/756492
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/756492