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A Hybrid Iterative Scheme for Variational Inequality Problems for Finite Families of Relatively Weak Quasi-Nonexpansive Mappings
Journal of Inequalities and Applications volume 2010, Article number: 724851 (2010)
Abstract
We consider a hybrid projection algorithm basing on the shrinking projection method for two families of relatively weak quasi-nonexpansive mappings. We establish strong convergence theorems for approximating the common fixed point of the set of the common fixed points of such two families and the set of solutions of the variational inequality for an inverse-strongly monotone operator in the framework of Banach spaces. At the end of the paper, we apply our results to consider the problem of finding a solution of the complementarity problem. Our results improve and extend the corresponding results announced by recent results.
1. Introduction
Let be a Banach space and let
be a nonempty, closed and convex subset of
. Let
be an operator. The classical variational inequality problem [1] for
is to find
such that

where denotes the dual space of
and
the generalized duality pairing between
and
. The set of all solutions of (1.1) is denoted by
. Such a problem is connected with the convex minimization problem, the complementarity, the problem of finding a point
satisfying
, and so on. First, we recall that a mapping
is said to be
(i)monotone if , for all
(ii)-inverse-strongly monotone if there exists a positive real number
such that

Let be the normalized duality mapping from
into
given by

It is well known that if is uniformly convex, then
is uniformly continuous on bounded subsets of
. Some properties of the duality mapping are given in [2–4].
Recall that a mappings is said to be nonexpansive if

If is a nonempty closed convex subset of a Hilbert space
and
is the metric projection of
onto
, then
is a nonexpansive mapping. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. In this connection, Alber [5] recently introduced a generalized projection operator
in a Banach space
which is an analogue of the metric projection in Hilbert spaces.
Consider the functional defined by

for all , where
is the normalized duality mapping from
to
. Observe that, in a Hilbert space
, (1.5) reduces to
=
for all
. The generalized projection
is a mapping that assigns to an arbitrary point
the minimum point of the functional
, that is,
=
, where
is the solution to the minimization problem:

The existence and uniqueness of the operator follows from the properties of the functional
and strict monotonicity of the mapping
(see, e.g., [3, 5–7]). In Hilbert spaces,
=
. It is obvious from the definition of the function
that
(1) for all
,
(2) for all
,
(3) for all
,
(4)If is a reflexive, strictly convex and smooth Banach space, then, for all
,

For more detail see [2, 3]. Let be a closed convex subset of
, and let
be a mapping from
into itself. We denote by
the set of fixed point of
. A point
in
is said to be an asymptotic fixed point of
[8] if
contains a sequence
which converges weakly to
such that
. The set of asymptotic fixed points of
will be denoted by
. A mapping
from
into itself is called relatively nonexpansive [7, 9, 10] if
=
and
for all
and
. The asymptotic behavior of relatively nonexpansive mappings were studied in [7, 9]. A point
in
is said to be a strong asymptotic fixed point of
if
contains a sequence
which converges strongly to
such that
. The set of strong asymptotic fixed points of
will be denoted by
. A mapping
from
into itself is called relatively weak nonexpansive if
and
for all
and
. If
is a smooth strictly convex and reflexive Banach space, and
is a continuous monotone mapping with
, then it is proved in [11] that
, for
is relatively weak nonexpansive.
is called relatively weak quasi-nonexpansive if
and
for all
and
.
Remark 1.1.
The class of relatively weak quasi-nonexpansive mappings is more general than the class of relatively weak nonexpansive mappings [7, 9, 12–14] which requires the strong restriction =
.
Remark 1.2.
If is relatively weak quasi-nonexpansive, then using the definition of
(i.e., the same argument as in the proof of [12, page 260]) one can show that
is closed and convex. It is obvious that relatively nonexpansive mapping is relatively weak nonexpansive mapping. In fact, for any mapping
we have
. Therefore, if
is a relatively nonexpansive mapping, then
=
=
.
Iiduka and Takahashi [15] introduced the following algorithm for finding a solution of the variational inequality for an -inverse-strongly monotone mapping
with
for all
and
in a
-uniformly convex and uniformly smooth Banach space
. For an initial point
, define a sequence
by

where is the duality mapping on
, and
is the generalized projection of
onto
. Assume that
for some
with
where
is the 2-uniformly convexity constant of
. They proved that if
is weakly sequentially continuous, then the sequence
converges weakly to some element
in
where
=
.
The problem of finding a common element of the set of the variational inequalities for monotone mappings in the framework of Hilbert spaces and Banach spaces has been intensively studied by many authors; see, for instance, [16–18] and the references cited therein.
On the other hand, in 2001, Xu and Ori [19] introduced the following implicit iterative process for a finite family of nonexpansive mappings , with
a real sequence in
, and an initial point
:

which can be rewritten in the following compact form:

where (here the
function takes values in
. They obtained the following result in a real Hilbert space.
Theorem XO
Let be a real Hilbert space,
a nonempty closed convex subset of
, and let
be a finite family of nonexpansive self-mappings on
such that
. Let
be a sequence defined by (1.10). If
is chosen so that
, as
, then
converges weakly to a common fixed point of the family of
.
On the other hand, Halpern [20] considered the following explicit iteration:

where is a nonexpansive mapping and
is a fixed point. He proved the strong convergence of
to a fixed point of
provided that
=
, where
.
Very recently, Qin et al. [21] proposed the following modification of the Halpern iteration for a single relatively quasi-nonexpansive mapping in a real Banach space. More precisely, they proved the following theorem.
Theorem 1 QCKZ.
Let be a nonempty and closed convex subset of a uniformly convex and uniformly smooth Banach space
and
a closed and quasi-
-nonexpansive mapping such that
. Let
be a sequence generated by the following manner:

Assume that satisfies the restriction:
, then
converges strongly to
.
Motivated and inspired by the above results, Cai and Hu [22] introduced the hybrid projection algorithm to modify the iterative processes (1.10), (1.11), and (1.12) to have strong convergence for a finite family of relatively weak quasi-nonexpansive mappings in Banach spaces. More precisely, they obtained the following theorem.
Theorem CH
Let be a nonempty and closed convex subset of a uniformly convex and uniformly smooth Banach space
and
be finite family of closed relatively weak quasi-nonexpansive mappings of
into itself with
. Assume that
is uniformly continuous for all
. Let
be a sequence generated by the following algorithm:

Assume that and
are the sequences in
satisfying
and
. Then
converges strongly to
, where
is the generalized projection from
onto
.
Motivated and inspired by Iiduka and Takahashi [15], Xu and Ori [19], Qin et al.[21], and Cai and Hu [22], we introduce a new hybrid projection algorithm basing on the shrinking projection method for two finite families of closed relatively weak quasi-nonexpansive mappings to have strong convergence theorems for approximating the common element of the set of common fixed points of two finite families of such mappings and the set of solutions of the variational inequality for an inverse-strongly monotone operator in the framework of Banach spaces. Our results improve and extend the corresponding results announced by recent results.
2. Preliminaries
A Banach space is said to be strictly convex if
for all
with
and
. It is also said to be uniformly convex if
for any two sequences
in
such that
and
=
. Let
be the unit sphere of
. Then the Banach space
is said to be smooth provided

exists for each . It is also said to be uniformly smooth if the limit is attained uniformly for
. It is well know that if
is smooth, then the duality mapping
is single valued. It is also known that if
is uniformly smooth, then
is uniformly norm-to-norm continuous on each bounded subset of
. Some properties of the duality mapping have been given in [3, 23–25]. A Banach space
is said to have Kadec-Klee property if a sequence
of
satisfying that
and
, then
. It is known that if
is uniformly convex, then
has the Kadec-Klee property; see [3, 23, 25] for more details.
We define the function which is called the modulus of convexity of
as following

Then is said to be
-uniformly convex if there exists a constant
such that constant
for all
. Constant
is called the
-uniformly convexity constant of
. A 2-uniformly convex Banach space is uniformly convex, see [26, 27] for more details. We know the following lemma of 2-uniformly convex Banach spaces.
Let be a
-uniformly convex Banach, then for all
from any bounded set of
and
,

where is the
-uniformly convexity constant of
.
Now we present some definitions and lemmas which will be applied in the proof of the main result in the next section.
Lemma 2.2 (Kamimura and Takahashi [30]).
Let be a uniformly convex and smooth Banach space and let
,
be two sequences of
such that either
or
is bounded. If
, then
.
Lemma 2.3 (Alber [5]).
Let be a nonempty closed convex subset of a smooth Banach space
and
. Then,
if and only if
for any
.
Lemma 2.4 (Alber [5]).
Let be a reflexive, strictly convex and smooth Banach space, let
be a nonempty closed convex subset of
and let
. Then

for all .
Let be a reflexive strictly convex, smooth and uniformly Banach space and the duality mapping
from
to
. Then
is also single-valued, one to one, surjective, and it is the duality mapping from
to
. We need the following mapping
which studied in Alber [5]:

for all and
. Obviously,
. We know the following lemma.
Lemma 2.5 (Kamimura and Takahashi [30]).
Let be a reflexive, strictly convex and smooth Banach space, and let
be as in (2.5). Then

for all and
.
Lemma 2.6 ([31, Lemma ]).
Let be a uniformly convex Banach space and
be a closed ball of
. Then there exists a continuous strictly increasing convex function
with
such that

for all and
with
.
An operator of
into
is said to be hemicontinuous if for all
, the mapping
of
into
defined by
is continuous with respect to the
topology of
. We denote by
the normal cone for
at a point
, that is

Lemma 2.7 (see [32]).
Let be a nonempty closed convex subset of a Banach space
and
a monotone, hemicontinuous operator of
into
. Let
be an operator defined as follows:

Then is maximal monotone and
=
.
3. Main Results
In this section, we prove strong convergence theorem which is our main result.
Theorem 3.1.
Let be a nonempty, closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space
, let
be an
-inverse-strongly monotone mapping of
into
with
for all
and
. Let
and
be two finite families of closed relatively weak quasi-nonexpansive mappings from
into itself with
, where
. Assume that
and
are uniformly continuous for all
. Let
be a sequence generated by the following algolithm:

where , and
is the normalized duality mapping on
. Assume that
,
and
are the sequences in
satisfying the restrictions:
(C1);
(C2) for some
with
, where
is the 2-uniformly convexity constant of
;
(C3) and if one of the following conditions is satisfied
(a) and
and
(b) and
.
Then converges strongly to
, where
is the generalized projection from
onto
.
Proof.
By the same method as in the proof of Cai and Hu [22], we can show that is closed and convex. Next, we show
for all
. In fact,
is obvious. Suppose
for some
. Then, for all
, we know from Lemma 2.5 that

Since and
is
-inverse-strongly monotone, we have

Therefore, from Lemma 2.1 and the assumption that for all
and
, we obtain that

Substituting (3.3) and (3.4) into (3.2) and using the condition that , we get

Using (3.5) and the convexity of , for each
, we obtain

It follows from (3.6) that

So, . Then by induction,
for all
and hence the sequence
generated by (3.1) is well defined. Next, we show that
is a convergent sequence in
. From
, we have

It follows from for all
that

From Lemma 2.4, we have

for each and for all
. Therefore, the sequence
is bounded. Furthermore, since
=
and
=
, we have

This implies that is nondecreasing and hence
exists. Similarly, by Lemma 2.4, we have, for any positive integer
, that

The existence of implies that
as
. From Lemma 2.2, we have

Hence, is a Cauchy sequence. Therefore, there exists a point
such that
as
Now, we will show that .
-
(I)
We first show that
. Indeed, taking
in (3.12), we have

It follows from Lemma 2.2 that

This implies that

The property of the function implies that

Since , we obtain

It follows from the condition (3.14) and (3.17) that

From Lemma 2.2, we have

Combining (3.15) and (3.20), we have

Since is uniformly norm-to-norm continuous on any bounded sets, we have

On the other hand, noticing

Since is uniformly norm-to-norm continuous on any bounded sets, we have

Using (3.15), (3.20), and (3.24) that

Taking the constant , we have, from Lemma 2.6, that there exists a continuous strictly increasing convex function
satisfying the inequality (2.7) and
.
Case 1.
Assume that (a) holds. Applying (2.7) and (3.5), we can calculate

This implies that

We observe that

It follows from (3.15), (3.22), (3.23) and (3.25) that

From and (3.27), we get

By the property of function , we obtain that

Since is uniformly norm-to-norm continuous on any bounded sets, we have

From (3.15) and (3.32), we have

Noticing that

for all . By the uniformly continuity of
, (3.16) and (3.33), we obtain

Thus

From the closeness of , we get
. Therefore
. In the same manner, we can apply the condition
to conclude that

Again, by (C2) and (3.26), we have

It follows from (3.29) and that

Since , we have

From Lemmas 2.4, 2.5, and (3.4), we have

It follows from (3.40) that

Lemma 2.2 implies that

Since is uniformly norm-to-norm continuous on any bounded sets, we have

Combining (3.37) and (3.43), we also obtain

Moreover

By (3.43), (3.15), we have

This implies that

Noticing that

for all . Since
is uniformly continuous, we can show that
. From the closeness of
, we get
. Therefore
. Hence
.
Case 2.
Assume that (b) holds. Using the inequalities (2.7) and (3.5), we obtain

This implies that

It follows from (3.21), (3.24) and the condition that

By the property of function , we obtain that

Since is uniformly norm-to-norm continuous on any bounded sets, we have

On the other hand, we can calculate

Observe that

It follows from (3.53) and (3.54) that

Applying and (3.57) and the fact that
is bounded to (3.55), we obtain

From Lemma 2.2, one obtains

We observe that

This together with (3.25) and (3.59), we obtain

Noticing that

for all . By the uniformly continuity of
, (3.16) and (3.61), we obtain

Thus

From the closeness of , we get
. Therefore
. By the same proof as in Case 1, we obtain that

Hence as
for each
and

Combining (3.54), (3.61), and (3.65), we also have

Moreover

By (3.43), (3.15), we have

This implies that

Noticing that

for all . Since
is uniformly continuous, we can show that
. From the closeness of
, we get
. Therefore
. Hence
.
-
(II)
We next show that
.
Let be an operator defined by:

By Lemma 2.7, is maximal monotone and
. Let
, since
, we have
. From
, we get

Since is
-inverse-strong monotone, we have

On other hand, from and Lemma 2.3, we have
, and hence

Because is
constricted, it holds from (3.74) and (3.75) that

for all . By taking the limit as
in (3.76) and from (3.43) and (3.44), we have
as
. By the maximality of
we obtain
and hence
. Hence we conclude that

Finally, we show that . Indeed, taking the limit as
in (3.9), we obtain

and hence by Lemma 2.3. This complete the proof.
Remark 3.2.
Theorem 3.1 improves and extends main results of Iiduka and Takahashi [15], Xu and Ori [19], Qin et al. [21], and Cai and Hu [22] because it can be applied to solving the problem of finding the common element of the set of common fixed points of two families of relatively weak quasi-nonexpansive mappings and the set of solutions of the variational inequality for an inverse-strongly monotone operator.
Strong convergence theorem for approximating a common fixed point of two finite families of closed relatively weak quasi-nonexpansive mappings in Banach spaces may not require that is 2-uniformly convex. In fact, we have the following theorem.
Corollary 3.3.
Let be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth Banach space
. Let
and
be two finite families of closed relatively weak quasi-nonexpansive mappings from
into itself with
, where
. Assume that
and
are uniformly continuous for all
. Let
be a sequence generated by the following algolithm:

where , and
is the normalized duality mapping on
. Assume that
,
and
are the sequences in
satisfying the following restrictions:
(C1) ;
(C2) and if one of the following conditions is satisfied
(a) and
and
(b) and
.
Then converges strongly to
, where
is the generalized projection from
onto
.
Proof.
Put in Theorem 3.1. Then, we get that
. Thus, the method of the proof of Theorem 3.1 gives the required assertion without the requirement that
is 2-uniformly convex.
Remark 3.4.
Corollary 3.3 improves Theorem 3.1 of Cai and Hu [22] from a finite family of of relatively weak quasi-nonexpansive mappings to two finite families of relatively weak quasi-nonexpansive mappings.
If , a Hilbert space, then
is
-uniformly convex (we can choose
) and uniformly smooth real Banach space and closed relatively weak quasi-nonexpansive map reduces to closed weak quasi-nonexpansive map. Furthermore,
, identity operator on
and
, projection mapping from
into
. Thus, the following corollaries hold.
Corollary 3.5.
Let be a nonempty, closed and convex subset of a Hilbert space
. Let
and
be two finite families of closed weak quasi-nonexpansive mappings from
into itself with
, where
with
for all
and
. Assume that
and
are uniformly continuous for all
. Let
be a sequence generated by the following algorithm:

where , and
is the normalized duality mapping on
. Assume that
,
,
,
, and
are the sequences in
satisfying the restrictions:
(C1) ;
(C2)  for some
with
, where
is the 2-uniformly convexity constant of
;
(C3)  and if one of the following conditions is satisfied
(a) and
and
(b) and
.
Then converges strongly to
, where
is the metric projection from
onto
.
Let be a nonempty closed convex cone in
, and let
be an operator from
into
. We define its polar in
to be the set

Then an element in
is called a solution of the complementarity problem if

The set of all solutions of the complementarity problem is denoted by . Several problem arising in different fields, such as mathematical programming, game theory, mechanics, and geometry, are to find solutions of the complementarity problems.
Theorem 3.6.
Let be a nonempty, closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space
, let
be an
-inverse-strongly monotone mapping of
into
with
for all
and
. Let
and
be two finite families of closed relatively weak quasi-nonexpansive mappings from
into itself with
, where
. Assume that
and
are uniformly continuous for all
. Let
be a sequence generated by the following algorithm:

where =
=
, and
is the normalized duality mapping on
. Assume that
,
and
are the sequences in
satisfying the restrictions:
(C1);
(C2) for some
with
, where
is the 2-uniformly convexity constant of
;
(C3) and if one of the following conditions is satisfied
(a) and
and
(b) and
.
Then converges strongly to
, where
is the generalized projection from
onto
.
Proof.
From [25, Lemma ], we have
. From Theorem 3.1, we can obtain the desired conclusion easily.
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Acknowledgments
The first author is supported by grant from under the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission, Thailand and the second author is supported by the Thailand Research Fund under Grant TRG5280011.
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Kamraksa, U., Wangkeeree, R. A Hybrid Iterative Scheme for Variational Inequality Problems for Finite Families of Relatively Weak Quasi-Nonexpansive Mappings. J Inequal Appl 2010, 724851 (2010). https://doi.org/10.1155/2010/724851
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DOI: https://doi.org/10.1155/2010/724851
Keywords
- Banach Space
- Variational Inequality
- Convex Subset
- Maximal Monotone
- Common Fixed Point