- Research Article
- Open access
- Published:
A General Iterative Method for Solving the Variational Inequality Problem and Fixed Point Problem of an Infinite Family of Nonexpansive Mappings in Hilbert Spaces
Fixed Point Theory and Applications volume 2009, Article number: 369215 (2009)
Abstract
We introduce an iterative scheme for finding a common element of the set of common fixed points of a family of infinitely nonexpansive mappings, and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in a Hilbert space. Under suitable conditions, some strong convergence theorems for approximating a common element of the above two sets are obtained. As applications, at the end of the paper we utilize our results to study the problem of finding a common element of the set of fixed points of a family of infinitely nonexpansive mappings and the set of fixed points of a finite family of -strictly pseudocontractive mappings. The results presented in the paper improve some recent results of Qin and Cho (2008).
1. Introduction
Throughout this paper, we always assume that is a real Hilbert space with inner product
and norm
, respectively,
is a nonempty closed convex subset of
, and
is the metric projection of
onto
. In the following, we denote by
strong convergence and by
weak convergence. Recall that a mapping
is called nonexpansive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ1_HTML.gif)
We denote by the set of fixed points of
. Recall that a mapping
is said to be
(i)monotone if , for all
;
(ii)-Lipschitz if there exists a constant
such that
, for all
;
(iii)-inverse-strongly monotone [1, 2] if there exists a positive real number
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ2_HTML.gif)
Remark 1.1.
It is obvious that any -inverse-strongly monotone mapping
is monotone and
-Lipschitz continuous.
Let be a mapping. The classical variational inequality problem is to find a
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ3_HTML.gif)
The set of solutions of variational inequality (1.3) is denoted by . The variational inequality has been extensively studied in the literature; see, for example, [3, 4] and the references therein.
A self-mapping is a contraction if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ4_HTML.gif)
Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [5–8] and the references therein. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences. A typical problem is to minimize a quadratic function over the set of the fixed points a nonexpansive mapping on a real Hilbert space:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ5_HTML.gif)
where is a linear bounded operator,
is the fixed point set of a nonexpansive mapping
, and
is a given point in
. Let
be a real Hilbert space. Recall that a linear bounded operator
is strongly positive if there is a constant
with property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ6_HTML.gif)
Recently, Marino and Xu [9] introduced the following general iterative scheme based on the viscosity approximation method introduced by Moudafi [10]:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ7_HTML.gif)
where is a strongly positive bounded linear operator on
. They proved that if the sequence
of parameters satisfies appropriate conditions, then the sequence
generated by (1.7) converges strongly to the unique solution of the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ8_HTML.gif)
which is the optimality condition for the minimization problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ9_HTML.gif)
where is a potential function for
(i.e.,
for
).
On the other hand, two classical iteration processes are often used to approximate a fixed point of a nonexpansive mapping. The first one is introduced by Mann [11] and is defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ10_HTML.gif)
where the sequence is in the interval
.
The second iteration process is referred to as Ishikawa's iteration process [12] which is defined recursively by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ11_HTML.gif)
where and
are sequences in the interval
. However, both (1.16) and (1.11) have only weak convergence in general (see [13], e.g.). Very recently, Qin and Cho [14] introduced a composite iterative algorithm
defined as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ12_HTML.gif)
where is a contraction,
is a nonexpansive mapping, and
is a strongly positive linear bounded self-adjoint operator, proved that, under certain appropriate assumptions on the parameters,
defined by (1.12) converges strongly to a fixed point of
, which solves some variational inequality and is also the optimality condition for the minimization problem (1.9).
On the other hand, for finding an element of , under the assumption that a set
is nonempty, closed, and convex, a mapping
is nonexpansive and a mapping
is
-inverse-strongly monotone, Takahashi and Toyoda [15] introduced the following iterative scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ13_HTML.gif)
where is a sequence in
, and
is a sequence in
. They proved that if
, then the sequence
generated by (1.13) converges weakly to some
. Recently, Iiduka and Takahashi [16] proposed another iterative scheme as follows
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ14_HTML.gif)
where is an
-inverse strongly monotone mapping,
and
satisfy some parameters controlling conditions. They showed that if
is nonempty, then the sequence
generated by (1.14) converges strongly to some
.
The existence of common fixed points for a finite family of nonexpansive mappings has been considered by many authors (see [17–20] and the references therein). The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappings (see [21, 22]). The problem of finding an optimal point that minimizes a given cost function over the common set of fixed points of a family of nonexpansive mappings is of wide interdisciplinary interest and practical importance (see [18, 23]). A simple algorithmic solution to the problem of minimizing a quadratic function over the common set of fixed points of a family of nonexpansive mappings is of extreme value in many applications including set theoretic signal estimation (see [23, 24]).
In this paper, we study the mapping defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ15_HTML.gif)
where is a nonnegative real sequence with
, for all
,
, form a family of infinitely nonexpansive mappings of
into itself. Nonexpansivity of each
ensures the nonexpansivity of
. Such a
is nonexpansive from
to
and it is called a
-mapping generated by
and
.
In this paper, motivated and inspired by Su et al. [25], Marino and Xu [9], Takahashi and Toyoda [15], and Iiduka and Takahashi [16], we will introduce a new iterative scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ16_HTML.gif)
where is a mapping defined by (1.15),
is a contraction,
is strongly positive linear bounded self-adjoint operator,
is a
-inverse strongly monotone, and we prove that under certain appropriate assumptions on the sequences
,
,
, and
, the sequences
defined by (1.16) converge strongly to a common element of the set of common fixed points of a family of
and the set of solutions of the variational inequality for an inverse-strongly monotone mapping, which solves some variational inequality and is also the optimality condition for the minimization problem (1.9).
2. Preliminaries
Let be a real Hilbert space. It is well known that for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ17_HTML.gif)
Let be a nonempty closed convex subset of
. For every point
, there exists a unique nearest point in
, denoted by
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ18_HTML.gif)
is called the metric projection of
onto
. It is well known that
is a nonexpansive mapping of
onto
and satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ19_HTML.gif)
for every Moreover,
is characterized by the following properties:
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ20_HTML.gif)
for all . It is easy to see that the following is true:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ21_HTML.gif)
A Banach space is said to satisfy the Opial's 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%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ22_HTML.gif)
It is well known that each Hilbert space satisfies the Opial's condition.
A set-valued mapping is called monotone if for all
,
and
imply
. A monotone mapping
is maximal if the graph of
of
is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping
is maximal if and only if for
,
for every
implies
. Let
be a monotone map of
into
and let
be the normal cone to
at
, that is,
and define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ23_HTML.gif)
Then is the maximal monotone and
if and only if
; see [26].
Now we collect some useful lemmas for proving the convergence result of this paper.
Lemma 2.1.
In a Hilbert space . Then the following inequality holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ24_HTML.gif)
Lemma 2.2 (see [27]).
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 [28]).
Assume is a sequence of nonnegative real numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ25_HTML.gif)
where is a sequence in
and
is a sequence in
such that
(1)
(2) or
Then
Lemma 2.4 (see [9]).
Assume that is a strongly positive linear bounded self-adjoint operator on a Hilbert space
with coefficient
and
. Then
.
Throughout this paper, we will assume that , for all
. Concerning
defined by (1.15), we have the following lemmas which are important to prove our main result.
Lemma 2.5 (see [29]).
Let be a nonempty closed convex subset of a Hilbert space
, let
be a family of infinitely nonexpansive mapping with
, and let
be a real sequence such that
, for all
. Then
(1) is nonexpansive and
for each
;
(2)for each and for each positive integer
, the limit
exists;
(3)the mapping define by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ26_HTML.gif)
is a nonexpansive mapping satisfying and it is called the
-mapping generated by
and
Lemma 2.6 (see [30]).
Let be a nonempty closed convex subset of a Hilbert space
, let
be a family of infinitely nonexpansive mappings with
, and let
be a real sequence such that
, for all
. If
is any bounded subset of
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ27_HTML.gif)
3. Main Results
Now we are in a position to state and prove the main result in this paper.
Theorem 3.1.
Let be a closed convex subset of a real Hilbert space
, let
be a contraction of
into itself, let
be an
-inverse strongly monotone mapping of
into
, and let
be a family of infinitely nonexpansive mappings with
. Let
be a strongly positive linear bounded self-adjoint operator with the coefficient
such that
. Assume that
. Let
,
,
, and
be sequences in
satisfying the following conditions:
(C1)
(C2)
(C3)
(C4)
(C5).
Then the sequence defined by (1.16) converges strongly to
, where
which solves the following variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ28_HTML.gif)
Proof.
Since as
by the condition (C1), we may assume, without loss of generality that
for all
. First, we will show that
is nonexpansive. Indeed, for all
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ29_HTML.gif)
which implies that is nonexpansive. Noticing that
is a linear bounded self-adjoint operator, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ30_HTML.gif)
Observing that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equa_HTML.gif)
we obtain is positive. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equb_HTML.gif)
Next, we observe that is bounded. Indeed, pick
and notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ31_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ32_HTML.gif)
By simple induction, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ33_HTML.gif)
which gives that the sequence is bounded, and so are
and
.
Next, we claim that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ34_HTML.gif)
Since and
are nonexpansive, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ35_HTML.gif)
where is a constant such that
. Similarly, there exists
such that
.
Observing that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ36_HTML.gif)
we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ37_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ38_HTML.gif)
Noticing that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ39_HTML.gif)
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ40_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ41_HTML.gif)
Substituting (3.11) into (3.14), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ42_HTML.gif)
where is an appropriate constant such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ43_HTML.gif)
Putting , we get,
.
Now, we compute . Observing that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ44_HTML.gif)
It follows from (3.15) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ45_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ46_HTML.gif)
Observing the conditions (C1) and (C4) and taking the superior limit as , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ47_HTML.gif)
We can obtain easily by Lemma 2.2 since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ48_HTML.gif)
one obtains that (3.7) holds. Setting , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ49_HTML.gif)
Observing that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ50_HTML.gif)
we arrive at
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ51_HTML.gif)
This implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ52_HTML.gif)
From (3.7) and (C1) we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ53_HTML.gif)
Next, we will show that as
for any
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ54_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ55_HTML.gif)
This impies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ56_HTML.gif)
Since and from (3.7), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ57_HTML.gif)
From (2.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ58_HTML.gif)
so, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ59_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ60_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ61_HTML.gif)
Applying (3.7), (3.30), and to the last inequality, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ62_HTML.gif)
It follows from (3.26) and (3.35) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ63_HTML.gif)
On the other hand, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ64_HTML.gif)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ65_HTML.gif)
From the conditions (C3), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ66_HTML.gif)
Applying Lemma 2.6 and (3.39), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ67_HTML.gif)
It follows from (3.26) and (3.40) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ68_HTML.gif)
We observe that is a contraction. Indeed, for all
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ69_HTML.gif)
Banach's Contraction Mapping Principle guarantees that has a unique fixed point, say
. That is,
.
Next, we claim that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ70_HTML.gif)
Indeed, we choose a subsequence of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ71_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
Therefore, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ72_HTML.gif)
Next we prove that .
First, we prove that .
Suppose the contrary, , that is,
. Since
, by the Opial's condition and (3.41), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ73_HTML.gif)
This is a contradiction, which shows that .
Next, we prove . For this purpose, let
be the maximal monotone mapping defined by (2.7):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ74_HTML.gif)
For any given , hence
. Since
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ75_HTML.gif)
On the other hand, from , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ76_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ77_HTML.gif)
Therefore, we obtian
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ78_HTML.gif)
Noting that as
and
is Lipschitz continuous, hence from (3.18), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ79_HTML.gif)
Since is maximal monotone, we have
, and hence
.
The conclusion is proved.
Hence by (3.45), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ80_HTML.gif)
Since , it follows from (3.39), (3.41), and (3.53) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ81_HTML.gif)
Hence (3.43) holds. Using (3.26) and (3.54), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ82_HTML.gif)
Now, from Lemma 2.1, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ83_HTML.gif)
Since ,
, and
are bounded, we can take a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ84_HTML.gif)
for all . It then follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ85_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ86_HTML.gif)
Using (C1), (3.54), and (3.55), we get . Now applying Lemma 2.3 to (3.58), we conclude that
. This completes the proof.
Remark 3.2.
Theorem 3.1 mainly improve the results of Qin and Cho [14] from a single nonexpansive mapping to an infinite family of nonexpansive mappings.
4. Applications
In this section, we obtain two results by using a special case of the proposed method.
Theorem 4.1.
Let be a real Hilbert space, let
be an
-inverse strongly monotone mapping on
, let
be a family of infinitely nonexpansive mappings with
. Let
a contraction with coefficient
, and let
be a strongly positive bounded linear operator on
with coefficient
and
. Suppose the sequences
,
, and
be generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ87_HTML.gif)
where ,
,
, and
are sequences in
satisfying the following conditions:
(C1)
(C2)
(C3)
(C4)
(C5).
Then ,
, and
converge strongly to
which solves the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ88_HTML.gif)
Proof.
We have and
. Applying Theorem 3.1, we obtain the desired result.
Next, we will apply the main results to the problem for finding a common element of the set of fixed points of a family of infinitely nonexpansive mappings and the set of fixed points of a finite family of -strictly pseudocontractive mappings.
Definition 4.2.
A mappings is said to be a
-strictly pseudocontractive mapping if there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ89_HTML.gif)
The following lemmas can be obtained from [31, Proposition 2.6] by Acedo and Xu, easily.
Lemma 4.3.
Let be a Hilbert space, let
be a closed convex subset of
. For any integer
, assume that, for each
is a
-strictly pseudocontractive mapping for some
. Assume that
is a positive sequence such that
. Then
is a
-strictly pseudocontractive mapping with
.
Lemma 4.4.
Let and
be as in Lemma 4.3. Suppose that
has a common fixed point in
. Then
.
Let be a
-strictly pseudocontractive mapping for some
. We define a mapping
, where
is a positive sequence such that
, then
is a
-inverse-strongly monotone mapping with
. In fact, from Lemma 4.3, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ90_HTML.gif)
That is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ91_HTML.gif)
On the other hand
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ92_HTML.gif)
Hence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ93_HTML.gif)
This shows that is
-inverse-strongly monotone.
Theorem 4.5.
Let be a closed convex subset of a real Hilbert space
. For any integer
, assume that, for each
is a
-strictly pseudocontractive mapping for some
. Let
be a family of infinitely nonexpansive mappings with
. Let
a contraction with coefficient
and let
be a strongly positive bounded linear operator with coefficient
and
. Let the sequences
,
, and
be generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ94_HTML.gif)
where ,
,
, and
are the sequences in
satisfying the following conditions:
(C1)
(C2)
(C3)
(C4)
(C5)
Then ,
, and
converge strongly to
which solves the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ95_HTML.gif)
Proof.
Taking , we know that
is
-inverse strongly monotone with
. Hence,
is a monotone
-Lipschitz continuous mapping with
. From Lemma 4.4, we know that
is a
-strictly pseudocontractive mapping with
and then
by Chang [30, Proposition 1.3.5]. Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F369215/MediaObjects/13663_2008_Article_1136_Equ96_HTML.gif)
The conclusion of Theorem 4.5 can be obtained from Theorem 3.1.
Remark 4.6.
Theorem 4.5 is a generalization and improvement of the theorems by Qin and Cho [14], Iiduka and Takahashi [16, Thorem 3.1], and Takahashi and Toyoda [15].
References
Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space. Journal of Mathematical Analysis and Applications 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6
Liu F, Nashed MZ: Regularization of nonlinear ill-posed variational inequalities and convergence rates. Set-Valued Analysis 1998,6(4):313–344. 10.1023/A:1008643727926
Yao J-C, Chadli O: Pseudomonotone complementarity problems and variational inequalities. In Handbook of Generalized Convexity and Generalized Monotonicity, Nonconvex Optimization and Its Applications. Volume 76. Springer, New York, NY, USA; 2005:501–558.
Zeng LC, Schaible S, Yao JC: Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities. Journal of Optimization Theory and Applications 2005,124(3):725–738. 10.1007/s10957-004-1182-z
Deutsch F, Yamada I: Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numerical Functional Analysis and Optimization 1998,19(1–2):33–56.
Xu H-K: Iterative algorithms for nonlinear operators. Journal of the London Mathematical Society. Second Series 2002,66(1):240–256. 10.1112/S0024610702003332
Xu HK: An iterative approach to quadratic optimization. Journal of Optimization Theory and Applications 2003,116(3):659–678. 10.1023/A:1023073621589
Yamada I: The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In Inherently Parallel Algorithm for Feasibility and Optimization and Their Applications, Studies in Computational Mathematics. Volume 8. Edited by: Butnariu D, Censor Y, Reich S. North-Holland, Amsterdam, The Netherlands; 2001:473–504.
Marino G, Xu H-K: A general iterative method for nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2006,318(1):43–52. 10.1016/j.jmaa.2005.05.028
Moudafi A: Viscosity approximation methods for fixed-points problems. Journal of Mathematical Analysis and Applications 2000,241(1):46–55. 10.1006/jmaa.1999.6615
Mann WR: Mean value methods in iteration. Proceedings of the American Mathematical Society 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3
Ishikawa S: Fixed points by a new iteration method. Proceedings of the American Mathematical Society 1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5
Genel A, Lindenstrauss J: An example concerning fixed points. Israel Journal of Mathematics 1975,22(1):81–86. 10.1007/BF02757276
Qin X, Cho Y: Convergence of a general iterative method for nonexpansive mappings in Hilbert spaces. Journal of Computational and Applied Mathematics. In press
Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications 2003,118(2):417–428. 10.1023/A:1025407607560
Iiduka H, Takahashi W: Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings. Nonlinear Analysis: Theory, Methods & Applications 2005,61(3):341–350. 10.1016/j.na.2003.07.023
Aoyama K, Kimura Y, Takahashi W, Toyoda M: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Analysis: Theory, Methods & Applications 2007,67(8):2350–2360. 10.1016/j.na.2006.08.032
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
Shang M, Su Y, Qin X: Strong convergence theorems for a finite family of nonexpansive mappings. Fixed Point Theory and Applications 2007, Article ID 76971 2007:-9.
Shimoji K, Takahashi W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese Journal of Mathematics 2001,5(2):387–404.
Bauschke HH, Borwein JM: On projection algorithms for solving convex feasibility problems. SIAM Review 1996,38(3):367–426. 10.1137/S0036144593251710
Combettes PL: The foundations of set theoretic estimation. Proceedings of the IEEE 1993,81(2):182–208.
Youla DC: Mathematical theory of image restoration by the method of convex projections. In Image Recovery: Theory and Application. Edited by: Star H. Academic Press, Orlando, Fla, USA; 1987:29–77.
Iusem AN, De Pierro AR: On the convergence of Han's method for convex programming with quadratic objective. Mathematical Programming Series B 1991,52(1–3):265–284.
Su Y, Shang M, Qin X: A general iterative scheme for nonexpansive mappings and inverse-strongly monotone mappings. Journal of Applied Mathematics and Computing 2008,28(1–2):283–294. 10.1007/s12190-008-0103-y
Rockafellar RT: On the maximality of sums of nonlinear monotone operators. Transactions of the American Mathematical Society 1970,149(1):75–88. 10.1090/S0002-9947-1970-0282272-5
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
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
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
Chang SS: Variational Inequalities and Related Problems. Chongqing Publishing House, Chongqing, China; 2007.
Acedo GL, Xu H-K: Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Analysis: Theory, Methods & Applications 2007,67(7):2258–2271. 10.1016/j.na.2006.08.036
Acknowledgments
The authors would like to thank the referee for the comments which improve the manuscript. R. Wangkeeree was supported for CHE-PhD-THA-SUP/2551 from the Commission on Higher Education and the Thailand Research Fund under Grant TRG5280011.
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
Wangkeeree, R., Kamraksa, U. A General Iterative Method for Solving the Variational Inequality Problem and Fixed Point Problem of an Infinite Family of Nonexpansive Mappings in Hilbert Spaces. Fixed Point Theory Appl 2009, 369215 (2009). https://doi.org/10.1155/2009/369215
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/369215