- Research Article
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A New Hybrid Iterative Algorithm for Fixed-Point Problems, Variational Inequality Problems, and Mixed Equilibrium Problems
Fixed Point Theory and Applications volume 2008, Article number: 417089 (2008)
Abstract
We introduce a new hybrid iterative algorithm for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of the variational inequality of a monotone mapping, and the set of solutions of a mixed equilibrium problem. This study, proves a strong convergence theorem by the proposed hybrid iterative algorithm which solves fixed-point problems, variational inequality problems, and mixed equilibrium problems.
1. Introduction
Let be a nonempty closed convex subset of a real Hilbert space
. Recall that a mapping
is called contractive if there exists a constant
such that
for all
. A mapping
is said to be nonexpansive if
for all
. Denote the set of fixed points of
by
.
Let be a real-valued function and
be an equilibrium bifunction, that is,
for each
. The mixed equilibrium problem (for short, MEP) is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ1_HTML.gif)
In particular, if , this problem reduces to the equilibrium problem (for short, EP), which is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ2_HTML.gif)
Denote the set of solutions of MEP by . The mixed equilibrium problems include fixed-point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and the equilibrium problems as special cases (see, e.g., [1–5]). Some methods have been proposed to solve the MEP and EP (see, e.g., [5–14]). In 1997, Combettes and Hirstoaga [13] introduced an iterative method of finding the best approximation to the initial data and proved a strong convergence theorem. Subsequently, S. Takahashi and W. Takahashi [8] introduced another iterative scheme for finding a common element of the set of solutions of EP and the set of fixed-point points of a nonexpansive mapping. Yao et al. [12] considered an iterative scheme for finding a common element of the set of solutions of EP and the set of common fixed points of an infinite nonexpansive mappings. Very recently, Zeng and Yao [14] considered a new iterative scheme for finding a common element of the set of solutions of MEP and the set of common fixed points of finitely many nonexpansive mappings. Their results extend and improve many results in the literature.
Let of
into
be a nonlinear mapping. It is well known that the variational inequality problem is to find
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ3_HTML.gif)
The set of solutions of the variational inequality problem is denoted by . A mapping
is called
-inverse-strongly monotone if there exists a positive real number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ4_HTML.gif)
Recently, some authors have proposed new iterative algorithms to approximate a common element of the set of fixed points of a nonxpansive mapping and the set of solutions of the variational inequality. For the details, see [15, 16] and the references therein.
Motivated by the recent works, in this paper we introduce a new hybrid iterative algorithm for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of the variational inequality of a monotone mapping, and the set of solutions of a mixed equilibrium problem. We prove a strong convergence theorem by the proposed hybrid iterative algorithm which solves fixed-point problems, variational inequality problems, and mixed equilibrium problems.
2. Preliminaries
Let be a real Hilbert space with inner product
and norm
. Let
be a nonempty 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%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ5_HTML.gif)
Such a is called the metric projection of
onto
. It is well known that
is a nonexpansive mapping and satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ6_HTML.gif)
Moreover, is characterized by the following properties:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ7_HTML.gif)
It is clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ8_HTML.gif)
In this paper, for solving the mixed equilibrium problems for an equilibrium bifunction , we assume that
satisfies the following conditions:
(H1) is monotone, that is,
for all
;
(H2) for each fixed ,
is concave and upper semicontinuous;
(H3) for each ,
is convex.
A mapping is called Lipschitz continuous if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ9_HTML.gif)
A differentiable function on a convex set
is called:
-
(i)
-convex if
(2.6)
where is the Fréchet derivative of
at
;
-
(ii)
-strongly convex if there exists a constant
such that
(2.7)
Let be a nonempty closed convex subset of a real Hilbert space
,
be a real-valued function, and
be an equilibrium bifunction. Let
be a positive number. For a given point
, the auxiliary problem for MEP consists of finding
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ12_HTML.gif)
Let be the mapping such that for each
,
is the solution set of the auxiliary problem MEP, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ13_HTML.gif)
We first need the following important and interesting result.
Lemma 2.1 (see [14]).
Let be a nonempty closed convex subset of a real Hilbert space
and let
be a lower semicontinuous and convex functional. Let
be an equilibrium bifunction satisfying conditions (H1)–(H3). Assume that
-
(i)
is Lipschitz continuous with constant
such that
-
(a)
for all
,
-
(b)
is affine in the first variable,
-
(c)
for each fixed
,
is sequentially continuous from the weak topology to the weak topology;
-
(ii)
is
-strongly convex with constant
and its derivative
is sequentially continuous from the weak topology to the strong topology;
-
(iii)
for each
, there exist a bounded subset
and
such that for any
,
(2.10)
Then there hold the following:
-
(i)
is single-valued;
-
(ii)
is nonexpansive if
is Lipschitz continuous with constant
such that
and
(2.11)
where for
;
-
(iii)
;
-
(vi)
is closed and convex.
We also need the following lemmas for proving our main results.
Lemma 2.2 (see [17]).
Let and
be bounded sequences in a Banach space
and let
be a sequence in
with
. Suppose
for all integers
and
. Then
.
Lemma 2.3 (see [18]).
Assume is a sequence of nonnegative real numbers such that
, where
is a sequence in
and
is a sequence such that
-
(1)
;
-
(2)
or
.
Then .
3. Iterative Algorithm and Strong Convergence Theorems
In this section, we first introduce a new iterative algorithm. Consequently, we will establish a strong convergence theorem for this iteration algorithm. To be more specific, let be infinite mappings of
into itself and let
be real numbers such that
for every
. For any
, define a mapping
of
into itself as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ16_HTML.gif)
Such is called the
-mapping generated by
and
. For the iterative algorithm for a finite family of nonexpansive mappings, we refer the reader to [19].
We have the following crucial Lemmas 3.1 and 3.2 concerning which can be found in [20]. Now we only need the following similar version in Hilbert spaces.
Lemma 3.1.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be nonexpansive mappings of
into itself such that
is nonempty, and let
be real numbers such that
for any
. Then for every
and
, the limit
exists.
Lemma 3.2.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be nonexpansive mappings of
into itself such that
is nonempty, and let
be real numbers such that
for any
. Then
.
The following remark [12] is important to prove our main results.
Remark 3.3.
Using Lemma 3.1, one can define a mapping of
into itself as
for every
. If
is a bounded sequence in
, then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ17_HTML.gif)
Throughout this paper, we will assume that for every
.
Now we introduce the following iteration algorithm.
Algorithm 3.4.
Let be a constant. Let
be a lower semicontinuous and convex functional and let
be an equilibrium bifunction. Let
be a
-inverse-strongly monotone mapping and
be the
-mapping defined by (3.1). Let
be a contraction of
into itself with coefficient
and given
arbitrarily. Suppose that the sequences
and
are generated iteratively by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ18_HTML.gif)
where ,
, and
are three sequences in
, and
is a sequence in
.
Now we study the strong convergence of the hybrid iterative algorithm (3.3).
Theorem 3.5.
Let be a nonempty closed convex subset of a real Hilbert space
and let
be a lower semicontinuous and convex functional. Let
be an equilibrium bifunction satisfying conditions (H1)–(H3) and let
be an infinite family of nonexpansive mappings of
into itself. Let
be a
-inverse-strongly monotone mapping such that
. Suppose
,
, and
are three sequences in
with
. Assume that
-
(i)
is Lipschitz continuous with constant
such that
-
(a)
for all
,
-
(b)
is affine in the first variable,
-
(c)
for each fixed
,
is sequentially continuous from the weak topology to the weak topology;
-
(ii)
is
- strongly convex with constant
and its derivative
is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant
such that
;
-
(iii)
for each
; there exist a bounded subset
and
such that for any
,
(3.4)
-
(iv)
,
,
,
, and
.
Let be a contraction of
into itself and given
arbitrarily. Then the sequence
generated by (3.3) converges strongly to
, where
provided that
is firmly nonexpansive.
Proof.
We first note that is a contraction with coefficient
. Then
for all
. Therefore
is a contraction of
into itself which implies that there exists a unique element
such that
.
Next we divide the following proofs into several steps.
Step 1 ( ,
, and
are bounded).
Let . From the definition of
, we know that
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ20_HTML.gif)
For all and
, we note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ21_HTML.gif)
which implies that is nonexpansive.
Set for all
. From (2.4), we have that
. It follows from (3.6) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ22_HTML.gif)
Hence we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ23_HTML.gif)
Therefore is bounded, so are
and
.
Step 2 ().
Setting for all
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ24_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ25_HTML.gif)
Now we estimate and
.
From (3.1), since and
are nonexpansive, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ26_HTML.gif)
where is a constant such that
.
At the same time, we observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ27_HTML.gif)
Since and
, from the nonexpansivity of
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ28_HTML.gif)
Substituting (3.11)–(3.13) into (3.10), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ29_HTML.gif)
Since , and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ30_HTML.gif)
Hence by Lemma 2.2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ31_HTML.gif)
Consequently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ32_HTML.gif)
Step 3 ().
Note that . Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ33_HTML.gif)
For , noting that
is firmly nonexpansive, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ34_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ35_HTML.gif)
So, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ36_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ37_HTML.gif)
From (3.6), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ38_HTML.gif)
Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ39_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ40_HTML.gif)
We note that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ41_HTML.gif)
Then we derive
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ42_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ43_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ44_HTML.gif)
Since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ45_HTML.gif)
Combining the above inequality, (3.18)–(3.29), and Remark 3.3, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ46_HTML.gif)
Step 4 (, where
).
To show the above inequality, we can choose a subsequence of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ47_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
.
First, we show . Assume that
. Since
and
, by Opial's condition, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ48_HTML.gif)
which is a contradiction. Hence we get . By the same argument as that in the proof of [21, Theorem 3.1], we can prove that
; and by the same argument as that in the proof of [14, Theorem 4.1], we also can prove that
. Hence
.
Since and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ49_HTML.gif)
Step 5 (, where
).
From (3.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ50_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ51_HTML.gif)
It is easy to see that and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ52_HTML.gif)
Applying Lemma 2.3 and (3.34) to (3.36), we conclude that as
. This completes the proof.
Concerning , we give the following remark.
Remark 3.6.
For each , we denote
and
. Then for all
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ53_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ54_HTML.gif)
Taking in (3.38) and
in (3.39), and adding up these two inequalities, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ55_HTML.gif)
Note that and
. Hence from (3.40), we deduce
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ56_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ57_HTML.gif)
Since is
-strongly monotone with constant
, then from (3.42), we conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ58_HTML.gif)
Take ,
, and
. Then from (3.43), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ59_HTML.gif)
This indicates that is firmly nonexpansive.
Corollary 3.7.
Let be a nonempty closed convex subset of a real Hilbert space
and let
be a lower semicontinuous and convex functional. Let
be an equilibrium bifunction satisfying conditions (H1)–(H3). Let
be a
-inverse-strongly monotone mapping such that
. Suppose
,
, and
are three sequences in
with
. Assume that
-
(i)
is Lipschitz continuous with constant
such that
-
(a)
for all
,
-
(b)
is affine in the first variable,
-
(c)
for each fixed
,
is sequentially continuous from the weak topology to the weak topology;
-
(ii)
is
-strongly convex with constant
and its derivative
is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with constant
such that
;
-
(iii)
for each
, there exist a bounded subset
and
such that, for any
,
(3.45)
-
(iv)
,
,
,
, and
.
Let be a contraction of
into itself and given
arbitrarily. Let the sequences
,
, and
be generated iteratively by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ61_HTML.gif)
Then the sequence generated by (3.46) converges strongly to
, where
provided that
is firmly nonexpansive.
Proof.
Take for all
, and for all
in (3.1). Then
for all
. The conclusion follows immediately from Theorem 3.5. This completes the proof.
Corollary 3.8.
Let be a nonempty closed convex subset of a real Hilbert space
. Let
be an infinite family of nonexpansive mappings of
into itself. Let
be a
-inverse-strongly monotone mapping such that
. Suppose
,
, and
are three sequences in
with
,
. Assume that
-
(i)
and
;
-
(ii)
;
-
(iii)
and
Let be a contraction of
into itself and given
arbitrarily. Then the sequence
, generated iteratively by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F417089/MediaObjects/13663_2007_Article_1082_Equ62_HTML.gif)
converges strongly to , where
.
Proof.
Set and
for all
and put
. Take
and
for all
. Then we have
. Hence the conclusion follows. This completes the proof.
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Acknowledgments
The authors are extremely grateful to the anonymous referees and Professor T. Suzuki for their useful comments and suggestions. The first author was partially supposed by National Natural Science Foundation of China Grant no. 10771050. The second author was partially supposed by the Grant no. NSC 96-2221-E-230-003.
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Yao, Y., Liou, YC. & Yao, JC. A New Hybrid Iterative Algorithm for Fixed-Point Problems, Variational Inequality Problems, and Mixed Equilibrium Problems. Fixed Point Theory Appl 2008, 417089 (2008). https://doi.org/10.1155/2008/417089
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DOI: https://doi.org/10.1155/2008/417089