- Research Article
- Open access
- Published:
A General Iterative Process for Solving a System of Variational Inclusions in Banach Spaces
Journal of Inequalities and Applications volume 2010, Article number: 190126 (2010)
Abstract
The purpose of this paper is to introduce a general iterative method for finding solutions of a general system of variational inclusions with Lipschitzian relaxed cocoercive mappings. Strong convergence theorems are established in strictly convex and 2-uniformly smooth Banach spaces. Moreover, we apply our result to the problem of finding a common fixed point of a countable family of strict pseudo-contraction mappings.
1. Introduction
Let . A Banach space
is said to be uniformly convex if, for any
, there exists
such that, for any
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ1_HTML.gif)
It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space is said to be smooth if the limit
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ2_HTML.gif)
exists for all . It is also said to be uniformly smooth if the limit is attained uniformly for all
. The norm of
is said to be Fréchet differentiable if, for any
, the above limit is attained uniformly for all
. The modulus of smoothness of
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ3_HTML.gif)
where . It is known that
is uniformly smooth if and only if
. Let
be a fixed real number with
. A Banach space
is said to be
-uniformly smooth if there exists a constant
such that
for all
.
From [1], we know the following property.
Let be a real number with
and let
be a Banach space. Then
is
-uniformly smooth if and only if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ4_HTML.gif)
The best constant in the above inequality is called the
-uniformly smoothness constant of
(see [1] for more details).
Let be a real Banach space and
the dual space of
. Let
denote the pairing between
and
. For
, the generalized duality mapping
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ5_HTML.gif)
In particular, is called the normalized duality mapping. It is known that
for all
. If
is a Hilbert space, then
is the identity. Note the following.
(1) is a uniformly smooth Banach space if and only if
is single-valued and uniformly continuous on any bounded subset of
.
(2) All Hilbert spaces, (or
) spaces (
), and the Sobolev spaces
(
) are
-uniformly smooth, while
(or
) and
spaces (
) are
-uniformly smooth.
(3) Typical examples of both uniformly convex and uniformly smooth Banach spaces are , where
. More precisely,
is
-uniformly smooth for any
.
Further, we have the following properties of the generalized duality mapping :
(i) for all
with
,
(ii) for all
and
,
(iii) for all
.
It is known that, if is smooth, then
is single valued. Recall that the duality mapping
is said to be weakly sequentially continuous if, for each sequence
with
weakly, we have
weakly-
. We know that, if
admits a weakly sequentially continuous duality mapping, then
is smooth. For the details, see [2].
Let be a nonempty closed convex subset of a smooth Banach space
. Recall the following definitions of a nonlinear mapping
, the following are mentioned.
Definition 1.1.
Given a mapping .
(i) is said to be accretive if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ6_HTML.gif)
for all .
(ii) 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%2F190126/MediaObjects/13660_2010_Article_2079_Equ7_HTML.gif)
for all .
(iii) is said to be
-inverse-strongly accretive or
-cocoercive if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ8_HTML.gif)
for all .
(iv) is said to be
-relaxed cocoercive if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ9_HTML.gif)
for all .
(v) is said to be
-relaxed cocoercive if there exist positive constants
and
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ10_HTML.gif)
for all .
Remark 1.2.
() Every
-strongly accretive mapping is an accretive mapping.
() Every
-strongly accretive mapping is a
-relaxed cocoercive mapping for any positive constant
but the converse is not true in general. Then the class of relaxed cocoercive operators is more general than the class of strongly accretive operators.
() Evidently, the definition of the inverse-strongly accretive operator is based on that of the inverse-strongly monotone operator in real Hilbert spaces (see, e.g., [3]).
() The notion of the cocoercivity is applied in several directions, especially for solving variational inequality problems using the auxiliary problem principle and projection methods [4]. Several classes of relaxed cocoercive variational inequalities have been studied in [5, 6].
Next, we consider a system of quasivariational inclusions as follows.
Find such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ11_HTML.gif)
where and
are nonlinear mappings for each
As special cases of problem (1.11), we have the following.
(1) If and
, then problem (1.11) is reduced to the following.
Find such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ12_HTML.gif)
(2) Further, if in problem (1.12), then problem (1.12) is reduced to the following
Find such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ13_HTML.gif)
In 2006, Aoyama et al. [7] considered the following problem.
Find such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ14_HTML.gif)
They proved that the variational inequality (1.14) is equivalent to a fixed point problem. The element is a solution of the variational inequality (1.14) if and only if
satisfies the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ15_HTML.gif)
where is a constant and
is a sunny nonexpansive retraction from
onto
, see the definition below.
Let be a subset of
, and
be a mapping of
into
. Then
is said to be sunny if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ16_HTML.gif)
whenever for
and
. A mapping
of
into itself is called a retraction if
. If a mapping
of
into itself is a retraction, then
for all
, where
is the range of
. A subset
of
is called a sunny nonexpansive retract of
if there exists a sunny nonexpansive retraction from
onto
.
The following results describe a characterization of sunny nonexpansive retractions on a smooth Banach space.
Proposition 1.3 (see [8]).
Let be a smooth Banach space and
a nonempty subset of
. Let
be a retraction and
the normalized duality mapping on
. Then the following are equivalent:
(1) is sunny and nonexpansive,
(2)
Recall that a mapping is called contractive if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ17_HTML.gif)
A mapping is said to be
-strictly pseudocontractive if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ18_HTML.gif)
Note that the class of -strictly pseudocontractive mappings strictly includes the class of nonexpansive mappings which are mappings
on
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ19_HTML.gif)
for all . That is,
is nonexpansive if and only if
is
-strict pseudocontractive. We denote by
the set of fixed points of
.
Proposition 1.4 (see [9]).
Let be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space
and
a nonexpansive mapping of
into itself with
. Then the set
is a sunny nonexpansive retract of
.
Definition 1.5.
A countable family of mapping is called a family of uniformly
-strict pseudocontractions if there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ20_HTML.gif)
For the class of nonexpansive mappings, one classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping [10, 11]. More precisely, take and define a contraction
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ21_HTML.gif)
where is a fixed point and
is a nonexpansive mapping. Banach's contraction mapping principle guarantees that
has a unique fixed point
in
; that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ22_HTML.gif)
It is unclear, in general, what the behavior of is as
even if
has a fixed point. However, in the case of
having a fixed point, Ceng et al. [12] proved that, if
is a Hilbert space, then
converges strongly to a fixed point of
. Reich [11] extended Browder's result to the setting of Banach spaces and proved that, if
is a uniformly smooth Banach space, then
converges strongly to a fixed point of
, and the limit defines the (unique) sunny nonexpansive retraction from
onto
.
Reich [11] showed that, if is uniformly smooth and
is the fixed point set of a nonexpansive mapping from
into itself, then there is a unique sunny nonexpansive retraction from
onto
and it can be constructed as follows.
Proposition 1.6 (see [11]).
Let be a uniformly smooth Banach space and
a nonexpansive mapping such that
. For each fixed
and every
, the unique fixed point
of the contraction
converges strongly as
to a fixed point of
. Define
by
. Then
is the unique sunny nonexpansive retract from
onto
; that is,
satisfies the property.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ23_HTML.gif)
Notation 1.
We use to denote strong convergence to
of the net
as
.
Definition 1.7 (see [13]).
Let be a multivalued maximal accretive mapping. The single-valued mapping
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ24_HTML.gif)
is called the resolvent operator associated with , where
is any positive number and
is the identity mapping.
Recently, many authors have studied the problems of finding a common element of the set of fixed points of a nonexpansive mapping and one of the sets of solutions to the variational inequalities (1.11)–(1.14) by using different iterative methods (see, e.g., [7, 14–16]).
Very recently, Qin et al. [16] considered the problem of finding the solutions of a general system of variational inclusion (1.11) with -inverse strongly accretive mappings. To be more precise, they obtained the following results.
Lemma 1.8 (see [16]).
For any where
,
is a solution of the problem (1.11) if and only if
is a fixed point of the mapping
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ25_HTML.gif)
Theorem QCCK (see [16, Theorem ]).
Let be a uniformly convex and
-uniformly smooth Banach space with the smoothness constant
. Let
be a maximal monotone mapping and
a
-inverse-strongly accretive mapping, respectively, for each
. Let
be a
-strict pseudocontraction such that
. Define a mapping
by
,
. Assume that
, where
is defined as in Lemma 1.8. Let
and let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ26_HTML.gif)
where ,
,
, and
and
are sequences in
. If the control consequences
and
satisfy the following restrictions:
(C1),
(C2) and
,
then converges strongly to
, where
is the sunny nonexpansive retraction from
onto
and
, where
, is a solution to problem (1.11).
On the other hand, we recall the following well-known definitions and results.
In a smooth Banach space, a mapping is called strongly positive [17] if there exists a constant
with property
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ27_HTML.gif)
where is the identity mapping and
is the normalized duality mapping.
In [18], Moudafi introduced the viscosity approximation method for nonexpansive mappings (see [19] for further developments in both Hilbert and Banach spaces). Let be a contraction on
. Starting with an arbitrary initial point
, define a sequence
recursively by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ28_HTML.gif)
where is a sequence in
. It is proved [18, 19] that, under certain appropriate conditions imposed on
, the sequence
generated by (1.28) strongly converges to the unique solution
in
of the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ29_HTML.gif)
Recently, Marino and Xu [20] introduced the following general iterative method:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ30_HTML.gif)
where is a strongly positive bounded linear operator on a Hilbert space
. They proved that, if the sequence
of parameters satisfies appropriate conditions, then the sequence
generated by (1.30) converges strongly to the unique solution of the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ31_HTML.gif)
which is the optimality condition for the minimization problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ32_HTML.gif)
where is a potential function for
for
).
Recently, Qin et al. [21] introduce the following iterative algorithm scheme:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ33_HTML.gif)
where is nonself-
-strict pseudo-contraction,
is a contraction, and
is a strongly positive bounded linear operator on a Hilbert space
. They proved, under certain appropriate conditions imposed on the sequences
and
, that
defined by (1.33) converges strongly to a fixed point of
, which solves some variational inequality.
In this paper, motivated by Qin et al. [16], Moudafi [18], Marino and Xu [20], and Qin et al. [21], we introduce a general iterative approximation method for finding common elements of the set of solutions to a general system of variational inclusions (1.11) with Lipschitzian and relaxed cocoercive mappings and the set common fixed points of a countable family of strict pseudocontractions. We prove the strong convergence theorems of such iterative scheme for finding a common element of such two sets which is a unique solution of some variational inequality and is also the optimality condition for some minimization problems in strictly convex and -uniformly smooth Banach spaces. The results presented in this paper improve and extend the corresponding results announced by Qin et al. [16], Moudafi [18], Marino and Xu [20], Qin et al. [21], and many others.
2. Preliminaries
Now we collect some useful lemmas for proving the convergence result of this paper.
Lemma 2.1 (see [22]).
The resolvent operator associated with
is single valued and nonexpansive for all
.
Lemma 2.2 (see [13]).
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_IEq316_HTML.gif)
is a solution of variational inclusion (1.13) if and only if ; that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ34_HTML.gif)
where denotes the set of solutions to problem (1.13).
Lemma 2.3 (see [23]).
Let be a strictly convex Banach space. Let
and
be two nonexpansive mappings from
into itself with a common fixed point. Define a mapping
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ35_HTML.gif)
where is a constant in
. Then
is nonexpansive and
.
Lemma 2.4 (see [24]).
Let be a nonempty closed convex subset of reflexive Banach space
which satisfies Opial's condition, and suppose that
is nonexpansive. Then the mapping
is demiclosed at zero, that is,
imply that
.
Lemma 2.5 (see [25]).
Assume that is a sequence of nonnegative real numbers such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ36_HTML.gif)
where is a sequence in
and
is a sequence such that
(a),
(b) or
Then
Lemma 2.6 (see [26]).
Let and
be bounded sequences in a Banach space
and
a sequence in
with
. Suppose that
for all
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ37_HTML.gif)
Then
Definition 2.7 (see [27]).
Let be a family of mappings from a subset
of a Banach space
into
with
. We say that
satisfies the AKTT-condition if, for each bounded subset
of
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ38_HTML.gif)
Remark 2.8.
The example of the sequence of mappings satisfying AKTT-condition is supported by Example 3.11.
Lemma 2.9 (see [27, Lemma ]).
Suppose that satisfies AKTT-condition. Then, for each
,
converses strongly to a point in
. Moreover, let the mapping
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ39_HTML.gif)
Then for each bounded subset of
,
Lemma 2.10 (see [28]).
Let be a real
-uniformly smooth Banach space and
a
-strict pseudocontraction. Then
is nonexpansive and
.
Lemma 2.11 (see [29]).
Let be a real
-uniformly smooth Banach space with the best smoothness constant
. Then the following inequality holds:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ40_HTML.gif)
Lemma 2.12 (see [17, Lemma ]).
Assume that is a strongly positive linear bounded operator on a smooth Banach space
with coefficient
and
Then
3. Main Results
In this section, we prove that the strong convergence theorem for a countable family of uniformly -strict pseudocontractions in a strictly convex and
-uniformly smooth Banach space admits a weakly sequentially continuous duality mapping. Before proving it, we need the following theorem.
Theorem 3.1 (see [17, Lemma ]).
Let be a nonempty closed convex subset of a reflexive, smooth Banach space
which admits a weakly sequentially continuous duality mapping
from
to
. Let
be a nonexpansive mapping such that
is nonempty, let
be a contraction with coefficient
, and let
be a strongly positive bounded linear operator with coefficient
and
. Then the net
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ41_HTML.gif)
converges strongly as to a fixed point
of
which solves the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ42_HTML.gif)
Lemma 3.2.
Let be a nonempty closed convex subset of a real
-uniformly smooth Banach space
with the smoothness constant
. Let
be an
-Lipschitzian and relaxed
-cocoercive mapping. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ43_HTML.gif)
If , then
is nonexpansive.
Proof.
Using Lemma 2.11 and the cocoercivity of the mapping , we have, for all
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ44_HTML.gif)
Hence (3.3) is proved. Assume that . Then, we have
. This together with (3.3) implies that
is nonexpansive.
Lemma 3.3.
Let be a strictly convex and
-uniformly smooth Banach space admiting a weakly sequentially continuous duality mapping with the smoothness constant
. Let
be a maximal monotone mapping and
a
-Lipschitzian and relaxed
-cocoercive mapping with
, respectively, for each
. Let
be a countable family of uniformly
-strict pseudocontractions. Define a mapping
and
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ45_HTML.gif)
where is defined as in Lemma 1.8. Assume that
. Let
be an
-contraction; let
be a strongly positive linear bounded self-adjoint operator with coefficient
with
. Then the following hold.
(i)For each ,
is nonexpansive such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ46_HTML.gif)
(ii)Suppose that satisfies AKTT-condition. Let
be the mapping defined by
for all
and suppose that
. The net
defined by
converges strongly as
to a fixed point
of
, which solves the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ47_HTML.gif)
and is a solution of general system of variational inequality problem (1.11) such that
.
Proof.
It follows from Lemma 2.10 that is nonexpansive such that
for each
. Next, we prove that
is nonexpansive. Indeed, we observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ48_HTML.gif)
The nonexpansivity of ,
,
, and
implies that
is nonexpansive. By Lemma 2.3, we have that
is nonexpansive such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ49_HTML.gif)
Hence (i) is proved. It is well known that, if is uniformly smooth, then
is reflexive. Hence Theorem 3.1 implies that
converges strongly as
to a fixed point
of
, which solves the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ50_HTML.gif)
and is a solution of problem (1.11), where
. This completes the proof of (ii).
Theorem 3.4.
Let be a strictly convex and
-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and has the smoothness constant
. Let
be a maximal monotone mapping and
a
-Lipschitzian and relaxed
-cocoercive mapping with
, respectively, for each
. Let
be a countable family of uniformly
-strict pseudocontractions. Define a mapping
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ51_HTML.gif)
Assume that , where
is defined as in Lemma 1.8. Let
be an
-contraction; let
be a strongly positive linear bounded self adjoint operator with coefficient
with
. Let
and let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ52_HTML.gif)
where , and
and
are sequences in
. Suppose that
satisfies AKTT-condition. Let
be the mapping defined by
for all
and suppose that
. If the control consequences
and
satisfy the following restrictions
(C1),
(C2) and
,
then converges strongly to
, which solves the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ53_HTML.gif)
and is a solution of general system of variational inequality problem (1.11) such that
.
Proof.
First, we show that sequences ,
, and
are bounded.
By the control condition (C2), we may assume, with no loss of generality, that .
Since is a linear bounded operator on
, by (1.27), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ54_HTML.gif)
Observe that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ55_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ56_HTML.gif)
Therefore, taking one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ57_HTML.gif)
Putting one sees that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ58_HTML.gif)
It follows from Lemmas 2.1 and 3.2 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ59_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ60_HTML.gif)
Setting and applying Lemma 2.10, we have that
is a nonexpansive mapping such that
for all
and hence
. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ61_HTML.gif)
It follows from the last inequality that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ62_HTML.gif)
By induction, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ63_HTML.gif)
This shows that the sequence is bounded, and so are
,
, and
.
On the other hand, from the nonexpansivity of the mappings , one sees that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ64_HTML.gif)
In a similar way, one can obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ65_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ66_HTML.gif)
This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ67_HTML.gif)
Setting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ68_HTML.gif)
one sees that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ69_HTML.gif)
and so it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ70_HTML.gif)
which, combined, with (3.27) yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ71_HTML.gif)
Using the conditions (C1) and (C2) and AKTT-condition of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ72_HTML.gif)
Hence, from Lemma 2.6, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ73_HTML.gif)
From (3.28), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ74_HTML.gif)
By (3.33), one sees that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ75_HTML.gif)
On the other hand, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ76_HTML.gif)
It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ77_HTML.gif)
From the conditions (C1), (C2) and from (3.35), one sees that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ78_HTML.gif)
Define the mapping by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ79_HTML.gif)
where is defined as in Lemma 1.8. From Lemma 3.3(i), we see that
is nonexpansive such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ80_HTML.gif)
From (3.38), it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ81_HTML.gif)
Since satisfies AKTT-condition and
is the mapping defined by
for all
, we have that
satisfies AKTT-condition. Let the mapping
be the mapping defined by
for all
. It follows from the nonexpansivity of
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ82_HTML.gif)
that is nonexpansive such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ83_HTML.gif)
Next, we prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ84_HTML.gif)
where with
be the fixed point of the contraction
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ85_HTML.gif)
Then solves the fixed point equation
. It follows from Lemma 3.3(ii) that
, which solves the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ86_HTML.gif)
and is a solution of general system of variational inequality problem (1.11) such that
. Let
be a subsequence of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ87_HTML.gif)
If follows from reflexivity of and the boundedness of sequence
that there exists
which is a subsequence of
converging weakly to
as
. It follows from (3.41) and the nonexpansivity of
, we have
by Lemma 2.4. Since the duality map
is single valued and weakly sequentially continuous from
to
, we get that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ88_HTML.gif)
as required. Now from Lemma 2.11, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ89_HTML.gif)
which implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ90_HTML.gif)
where is an appropriate constant such that
. Put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ91_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ92_HTML.gif)
It follows from conditions (C1), (C2) and from (3.44) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ93_HTML.gif)
Apply Lemma 2.5 to (3.52) to conclude that as
. This completes the proof.
Setting , and
, we have the following result.
Theorem 3.5.
Let be a strictly convex and
-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and has the smoothness constant
. Let
be a maximal monotone mapping and
a
-Lipschitzian and relaxed
-cocoercive mapping with
, respectively, for each
. Let
be a countable family of uniformly
-strict pseudocontractions. Define a mapping
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ94_HTML.gif)
Assume that , where
is defined as in Lemma 1.8. Let
and let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ95_HTML.gif)
where , and
and
are sequences in
. Suppose that
satisfies AKTT-condition. Let
be the mapping defined by
for all
and suppose that
. If the control consequences
and
satisfy the following restrictions
(C1),
(C2) and
,
then converges strongly to
, which solves the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ96_HTML.gif)
and is a solution of general system of variational inequality problem (1.11) such that
.
Remark 3.6.
Theorem 3.4 mainly improves Theorem of Qin et al. [16], in the following respects:
(a)from the class of inverse-strongly accretive mappings to the class of Lipchitzian and relaxed cocoercive mappings,
(b)from a -strict pseudocontraction to the countable family of uniformly
-strict pseudocontractions,
(c)from a uniformly convex and -uniformly smooth Banach space to a strictly convex and
-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping.
Further, if is a countable family of nonexpansive mappings, then Theorem 3.4 is reduced to the following result.
Theorem 3.7.
Let be a strictly convex and
-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and has the smoothness constant
. Let
be a maximal monotone mapping and
a
-Lipschitzian and relaxed
-cocoercive mapping with
, respectively, for each
. Let
be a countable family of nonexpansive mappings. Assume that
, where
is defined as in Lemma 1.8. Let
be an
-contraction; let
be a strongly positive linear bounded self adjoint operator with coefficient
with
. Let
and let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ97_HTML.gif)
where , and
and
are sequences in
. Suppose that
satisfies AKTT-condition. Let
be the mapping defined by
for all
and suppose that
. If the control consequences
and
satisfy the following restrictions
(C1),
(C2) and
,
then converges strongly to
which solves the variational inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ98_HTML.gif)
and is a solution of general system of variational inequality problem (1.11) such that
.
Remark 3.8.
As in [27, Theorem ], we can generate a sequence
of nonexpansive mappings satisfying AKTT-condition; that is,
for any bounded subset
of
by using convex combination of a general sequence
of nonexpansive mappings with a common fixed point. To be more precise, they obtained the following lemma.
Lemma 3.9 (see [27]).
Let be a closed convex subset of a smooth Banach space
. Suppose that
is a sequence of nonexpansive mappings of
into inself with a common fixed point. For each
, define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ99_HTML.gif)
where is a family of nonnegative numbers with indices
with
such that
(i) for all
,
(ii) for every
,
(iii).
Then the following are given.
()Each is a nonexpansive mapping.
() satisfies AKTT-condition.
()If is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ100_HTML.gif)
then and
.
Theorem 3.10.
Let be a strictly convex and
-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and has the smoothness constant
. Let
be a maximal monotone mapping and
a
-Lipschitzian and relaxed
-cocoercive mapping with
, respectively, for each
. Let
be a countable family of nonexpansive mappings. Assume that
, where
is defined as in Lemma 1.8. Let
be an
-contraction; let
be a strongly positive linear bounded self adjoint operator with coefficient
with
. Let
and let
be a sequence generated by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ101_HTML.gif)
where satisfies conditions (i)–(iii) of Lemma 3.9,
, and
and
are sequences in
. Suppose that
satisfies AKTT-condition. Let
be the mapping defined by
for all
and suppose that
. If the control consequences
and
satisfy the following restrictions:
(C1),
(C2) and
,
then converges strongly to
, which solves the variational inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ102_HTML.gif)
and is a solution of general system of variational inequality problem (1.11) such that
.
Proof.
We write the iteration (3.61) as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ103_HTML.gif)
where is defined by (3.59). It is clear that each mapping
is nonexpansive. By Theorem 3.7 and Lemma 3.9, the conclusion follows.
The following example appears in [27] shows that there exists satisfying the conditions of Lemma 3.9.
Example 3.11.
Let
be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ104_HTML.gif)
for all with
In this case, the sequence
of mappings generated by
is defined as follows: For
.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F190126/MediaObjects/13660_2010_Article_2079_Equ105_HTML.gif)
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Acknowledgments
The first author is supported under grant from 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 "Centre of Excellence in Mathematics" under the Commission on Higher Education, Ministry of Education, Thailand. Finally, The authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.
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Kamraksa, U., Wangkeeree, R. A General Iterative Process for Solving a System of Variational Inclusions in Banach Spaces. J Inequal Appl 2010, 190126 (2010). https://doi.org/10.1155/2010/190126
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DOI: https://doi.org/10.1155/2010/190126