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Strong Convergence Theorem for a New General System of Variational Inequalities in Banach Spaces
Fixed Point Theory and Applications volume 2010, Article number: 246808 (2011)
Abstract
We introduce a new system of general variational inequalities in Banach spaces. The equivalence between this system of variational inequalities and fixed point problems concerning the nonexpansive mapping is established. By using this equivalent formulation, we introduce an iterative scheme for finding a solution of the system of variational inequalities in Banach spaces. Our main result extends a recent result acheived by Yao, Noor, Noor, Liou, and Yaqoob.
1. Introduction
Let be a real Banach space, and
be its dual space. Let
denote the unit sphere of
.
is said to be uniformly convex if for each
there exists a constant
such that for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ1_HTML.gif)
The norm on is said to be Gâteaux differentiable if the limit
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ2_HTML.gif)
exists for each and in this case
is said to have a uniformly Frechet differentiable norm if the limit (1.2) is attained uniformly for
and in this case
is said to be uniformly smooth. We define a function
, called the modulus of smoothness of
, as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ3_HTML.gif)
It is known that is uniformly smooth if and only if
. Let
be a fixed real number with
. Then a Banach space
is said to be
-uniformly smooth if there exists a constant
such that
for all
. For
, the generalized duality mapping
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ4_HTML.gif)
In particular, if , the mapping
is called the normalized duality mapping and usually, we write
. If
is a Hilbert space, then
. Further, we have the following properties of the generalized duality mapping
:
(1) for all
with
,
(2) for all
and
,
(3) for all
.
It is known that if is smooth, then
is single-valued, which is denoted by
. Recall that the duality mapping
is said to be weakly sequentially continuous if for each
with
weakly, we have
weakly-
. We know that if
admits a weakly sequentially continuous duality mapping, then
is smooth. For the details, see the work of Gossez and Lami Dozo in [1].
Let be a nonempty closed convex subset of a smooth Banach space
. Recall that a mapping
is said to be accretive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ5_HTML.gif)
for all . A mapping
is said to be
-strongly accretive if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ6_HTML.gif)
for all . A mapping
is said to be
-inverse strongly accretive if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ7_HTML.gif)
for all . A mapping
is said to be nonexpansive if
for all
. The fixed point set of
is denoted by
.
Let be a nonempty subset of
. A mapping
is said to be sunny if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ8_HTML.gif)
whenever for
and
. A mapping
is called a retraction if
for all
. Furthermore,
is a sunny nonexpansive retraction from
onto
if
is a retraction from
onto
which is also sunny and nonexpansive.
A subset of
is called a sunny nonexpansive retraction of
if there exists a sunny nonexpansive retraction from
onto
. It is well known that if
is a Hilbert space, then a sunny nonexpansive retraction
is coincident with the metric projection from
onto
.
Conveying an idea of the classical variational inequality, denoted by , is to find an
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ9_HTML.gif)
where is a Hilbert space and
is a mapping from
into
. The variational inequality has been widely studied in the literature; see, for example, the work of Chang et al. in [2], Zhao and He [3], Plubtieng and Punpaeng [4], Yao et al. [5] and the references therein.
Let be two mappings. In 2008, Ceng et al. [6] considered the following problem of finding
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ10_HTML.gif)
which is called a general system of variational inequalities, where and
are two constants. In particular, if
, then problem (1.10) reduces to finding
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ11_HTML.gif)
which is defined by Verma [7] and is called the new system of variational inequalities. Further, if we add up the requirement that , then problem (1.11) reduces to the classical variational inequality
.
In 2006, Aoyama et al. [8] first considered the following generalized variational inequality problem in Banach spaces. Let be an accretive operator. Find a point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ12_HTML.gif)
The problem (1.12) is very interesting as it is connected with the fixed point problem for nonlinear mapping and the problem of finding a zero point of an accretive operator in Banach spaces, see [9–11] and the references therein.
Aoyama et al. [8] introduced the following iterative algorithm in Banach spaces:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ13_HTML.gif)
where is a sunny nonexpansive retraction from
onto
. Then they proved a weak convergence theorem which is generalized simultaneously theorems of Browder and Petryshyn [12] and Gol'shteÄn and Tret'yakov [13]. In 2008, Hao [14] obtained a strong convergence theorem by using the following iterative algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ14_HTML.gif)
where ,
are two sequences in
and
.
Very recently, in 2009, Yao et al. [5] introduced the following system of general variational inequalities in Banach spaces. For given two operators , they considered the problem of finding
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ15_HTML.gif)
which is called the system of general variational inequalities in a real Banach space. They proved a strong convergence theorem by using the following iterative algorithm:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ16_HTML.gif)
where ,
, and
are three sequences in
and
.
In this paper, motivated and inspired by the idea of Yao et al. [5] and Cheng et al. [6]. First, we introduce the following system of variational inequalities in Banach spaces.
Let be a nonempty closed convex subset of a real Banach space
. Let
for all
be three mappings. We consider the following problem of finding
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ17_HTML.gif)
which is called a new general system of variational inequalities in Banach spaces, where for all
. In particular, if
,
, and
for
, then problem (1.17) reduces to problem (1.15). Further, if
,
, then problem (1.17) reduces to the problem (1.10) in a real Hilbert space. Second, we introduce iteration process for finding a solution of a new general system of variational inequalities in a real Banach space. Starting with arbitrary points
and let
,
, and
be the sequences generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ18_HTML.gif)
where for all
and
,
are two sequences in
. Using the demiclosedness principle for nonexpansive mapping, we will show that the sequence
converges strongly to a solution of a new general system of variational inequalities in Banach spaces under some control conditions.
2. Preliminaries
In this section, we recall the well known results and give some useful lemmas that will be used in the next section.
Lemma 2.1 (see [15]).
Let be a
-uniformly smooth Banach space with
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ19_HTML.gif)
for all , where
is the
-uniformly smooth constant of
.
The following lemma concerns the sunny nonexpansive retraction.
Let be a closed convex subset of a smooth Banach space
. Let
be a nonempty subset of
and
be a retraction. Then
is sunny and nonexpansive if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ20_HTML.gif)
for all and
.
The first result regarding the existence of sunny nonexpansive retractions on the fixed point set of a nonexpansive mapping is due to Bruck [18].
Remark 2.3.
If is strictly convex and uniformly smooth and if
is a nonexpansive mapping having a nonempty fixed point set
, then there exists a sunny nonexpansive retraction of
onto
.
Lemma 2.4 (see [19]).
Assume is a sequence of nonnegative real numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ21_HTML.gif)
where is a sequence in
and
is a sequence such that
(i);
(ii) or
.
Then .
Lemma 2.5 (see [20]).
Let and
be bounded sequences in a Banach space
and let
be a sequence in
with
. Suppose
for all integers
and
. Then,
.
Lemma 2.6 (see [21]).
Let be a uniformly convex Banach space,
a nonempty closed convex subset of
, and
be an nonexpansive mapping. Then
is demiclosed at 0, that is, if
weakly and
strongly, then
.
3. Main Results
In this section, we establish the equivalence between the new general system of variational inequalities (1.17) and some fixed point problem involving a nonexpansive mapping. Using the demiclosedness principle for nonexpansive mapping, we prove that the iterative scheme (1.18) converges strongly to a solution of a new general system of variational inequalities (1.17) in a Banach space under some control conditions. In order to prove our main result, the following lemmas are needed.
The next lemmas are crucial for proving the main theorem.
Lemma 3.1.
Let be a nonempty closed convex subset of a real 2-uniformly smooth Banach space
. Let the mapping
be
-inverse strongly accretive. Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ22_HTML.gif)
where is the 2-uniformly smooth constant of
. In particular, if
, then
is a nonexpansive mapping.
Proof.
Indeed, for all , from Lemma 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ23_HTML.gif)
It is clear that, if , then
is a nonexpansive mapping.
Lemma 3.2.
Let be a nonempty closed convex subset of a real 2-uniformly smooth Banach space
. Let
be the sunny nonexpansive retraction from
onto
. Let
be an
-inverse strongly accretive mapping for
. Let
be a mapping defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ24_HTML.gif)
If for all
, then
is nonexpansive.
Proof.
For all , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ25_HTML.gif)
From Lemma 3.1, we have is nonexpansive which implies by (3.4) that
is nonexpansive.
Lemma 3.3.
Let be a nonempty closed convex subset of a real smooth Banach space
. Let
be the sunny nonexpansive retraction from
onto
. Let
be three nonlinear mappings. For given
,
is a solution of problem (1.17) if and only if
,
and
, where
is the mapping defined as in Lemma 3.2.
Proof.
Note that we can rewrite (1.17) as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ26_HTML.gif)
From Lemma 2.2, we can deduce that (3.5) is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ27_HTML.gif)
It is easy to see that (3.6) is equivalent to ,
and
.
From now on we denote by the set of all fixed points of the mapping
. Now we prove the strong convergence theorem of algorithm (1.18) for solving problem (1.17).
Theorem 3.4.
Let be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space
which admits a weakly sequentially continuous duality mapping. Let
be the sunny nonexpansive retraction from
onto
. Let the mappings
be
-inverse strongly accretive with
, for all
and
. For given
, let the sequence
be generated iteratively by (1.18). Suppose the sequences
and
are two sequences in
such that
(C1) and
;
(C2).
Then converges strongly to
where
is the sunny nonexpansive retraction of
onto
.
Proof.
Let and
, it follows from Lemma 3.3 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ28_HTML.gif)
Put and
. Then
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ29_HTML.gif)
From Lemma 3.1, we have is nonexpansive. Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ30_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ31_HTML.gif)
By induction, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ32_HTML.gif)
Therefore, is bounded. Hence
,
,
,
,
, and
are also bounded. By nonexpansiveness of
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ33_HTML.gif)
Let ,
. Then
for all
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ34_HTML.gif)
By (3.12) and (3.13), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ35_HTML.gif)
This together with (C1) and (C2), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ36_HTML.gif)
Hence, by Lemma 2.5, we get as
. Consequently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ37_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ38_HTML.gif)
therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ39_HTML.gif)
Furthermore, by Lemma 3.2, we have is nonexpansive. Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ40_HTML.gif)
which implies as
.
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ41_HTML.gif)
therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ42_HTML.gif)
Let be the sunny nonexpansive retraction of
onto
. Now we show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ43_HTML.gif)
To prove (3.22), since is bounded, we can choose a subsequence
of
which converges weakly to
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ44_HTML.gif)
From Lemma 2.6 and (3.21), we obtain . Now, from Lemma 2.2, (3.23), and the weakly sequential continuity of the duality mapping
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ45_HTML.gif)
From (3.9), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ46_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ47_HTML.gif)
It follows from Lemma 2.4, (3.24), and (3.26) that converges strongly to
. This completes the proof.
Letting and
for
in Theorem 3.4, we obtain the following result.
Corollary 3.5 (see [5, Theorem 3.1]).
Let be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space
which admits a weakly sequentially continuous duality mapping. Let
be the sunny nonexpansive retraction from
onto
. Let the mappings
be
-inverse strongly accretive with
, for all
and
. For given
, and let
,
be the sequences generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F246808/MediaObjects/13663_2010_Article_1236_Equ48_HTML.gif)
where ,
are two sequences in
such that
(C1) and
;
(C2).
Then converges strongly to
where
is the sunny nonexpansive retraction of
onto
.
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Acknowledgments
The authors wish to express their gratitude to the referees for careful reading of the manuscript and helpful suggestions. The authors would like to thank the Commission on Higher Education, the Thailand Research Fund, the Thaksin university, the Centre of Excellence in Mathematics, and the Graduate School of Chiang Mai University, Thailand for their financial support.
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Imnang, S., Suantai, S. Strong Convergence Theorem for a New General System of Variational Inequalities in Banach Spaces. Fixed Point Theory Appl 2010, 246808 (2011). https://doi.org/10.1155/2010/246808
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DOI: https://doi.org/10.1155/2010/246808