- Research Article
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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
. 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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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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Keywords
- Banach Space
- Variational Inequality
- Convex Subset
- Maximal Monotone
- Common Fixed Point
