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A General Iterative Process for Solving a System of Variational Inclusions in Banach Spaces

  • Uthai Kamraksa
  • Rabian Wangkeeree
Open Access
Research Article

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.

Keywords

Banach Space Variational Inequality Nonexpansive Mapping Common Fixed Point Nonempty Closed Convex Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Let Open image in new window . A Banach space Open image in new window is said to be uniformly convex if, for any Open image in new window , there exists Open image in new window such that, for any Open image in new window ,
It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space Open image in new window is said to be smooth if the limit
exists for all Open image in new window . It is also said to be uniformly smooth if the limit is attained uniformly for all Open image in new window . The norm of Open image in new window is said to be Fréchet differentiable if, for any Open image in new window , the above limit is attained uniformly for all Open image in new window . The modulus of smoothness of Open image in new window is defined by

where Open image in new window . It is known that Open image in new window is uniformly smooth if and only if Open image in new window . Let Open image in new window be a fixed real number with Open image in new window . A Banach space Open image in new window is said to be Open image in new window -uniformly smooth if there exists a constant Open image in new window such that Open image in new window for all Open image in new window .

From [1], we know the following property.

Let Open image in new window be a real number with Open image in new window and let Open image in new window be a Banach space. Then Open image in new window is Open image in new window -uniformly smooth if and only if there exists a constant Open image in new window such that

The best constant Open image in new window in the above inequality is called the Open image in new window -uniformly smoothness constant of Open image in new window (see [1] for more details).

Let Open image in new window be a real Banach space and Open image in new window the dual space of Open image in new window . Let Open image in new window denote the pairing between Open image in new window and Open image in new window . For Open image in new window , the generalized duality mapping Open image in new window is defined by

In particular, Open image in new window is called the normalized duality mapping. It is known that Open image in new window for all Open image in new window . If Open image in new window is a Hilbert space, then Open image in new window is the identity. Note the following.

(1) Open image in new window is a uniformly smooth Banach space if and only if Open image in new window is single-valued and uniformly continuous on any bounded subset of Open image in new window .

(2) All Hilbert spaces, Open image in new window (or Open image in new window ) spaces ( Open image in new window ), and the Sobolev spaces Open image in new window ( Open image in new window ) are Open image in new window -uniformly smooth, while Open image in new window (or Open image in new window ) and Open image in new window spaces ( Open image in new window ) are Open image in new window -uniformly smooth.

(3) Typical examples of both uniformly convex and uniformly smooth Banach spaces are Open image in new window , where Open image in new window . More precisely, Open image in new window is Open image in new window -uniformly smooth for any Open image in new window .

Further, we have the following properties of the generalized duality mapping Open image in new window :

(i) Open image in new window for all Open image in new window with Open image in new window ,

(ii) Open image in new window for all Open image in new window and Open image in new window ,

(iii) Open image in new window for all Open image in new window .

It is known that, if Open image in new window is smooth, then Open image in new window is single valued. Recall that the duality mapping Open image in new window is said to be weakly sequentially continuous if, for each sequence Open image in new window with Open image in new window weakly, we have Open image in new window weakly- Open image in new window . We know that, if Open image in new window admits a weakly sequentially continuous duality mapping, then Open image in new window is smooth. For the details, see [2].

Let Open image in new window be a nonempty closed convex subset of a smooth Banach space Open image in new window . Recall the following definitions of a nonlinear mapping Open image in new window , the following are mentioned.

Definition 1.1.

Given a mapping Open image in new window .

(i) Open image in new window is said to be accretive if

for all Open image in new window .

(ii) Open image in new window is said to be Open image in new window -strongly accretive if there exists a constant Open image in new window such that

for all Open image in new window .

(iii) Open image in new window is said to be Open image in new window -inverse-strongly accretive or Open image in new window -cocoercive if there exists a constant Open image in new window such that

for all Open image in new window .

(iv) Open image in new window is said to be Open image in new window -relaxed cocoercive if there exists a constant Open image in new window such that

for all Open image in new window .

(v) Open image in new window is said to be Open image in new window -relaxed cocoercive if there exist positive constants Open image in new window and Open image in new window such that

for all Open image in new window .

Remark 1.2.

( Open image in new window ) Every Open image in new window -strongly accretive mapping is an accretive mapping.

( Open image in new window ) Every Open image in new window -strongly accretive mapping is a Open image in new window -relaxed cocoercive mapping for any positive constant Open image in new window 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.

( Open image in new window ) 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]).

( Open image in new window ) 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.

where Open image in new window and Open image in new window are nonlinear mappings for each Open image in new window

As special cases of problem (1.11), we have the following.

(1) If Open image in new window and Open image in new window , then problem (1.11) is reduced to the following.

(2) Further, if Open image in new window in problem (1.12), then problem (1.12) is reduced to the following

In 2006, Aoyama et al. [7] considered the following problem.

They proved that the variational inequality (1.14) is equivalent to a fixed point problem. The element Open image in new window is a solution of the variational inequality (1.14) if and only if Open image in new window satisfies the following equation:

where Open image in new window is a constant and Open image in new window is a sunny nonexpansive retraction from Open image in new window onto Open image in new window , see the definition below.

whenever Open image in new window for Open image in new window and Open image in new window . A mapping Open image in new window of Open image in new window into itself is called a retraction if Open image in new window . If a mapping Open image in new window of Open image in new window into itself is a retraction, then Open image in new window for all Open image in new window , where Open image in new window is the range of Open image in new window . A subset Open image in new window of Open image in new window is called a sunny nonexpansive retract of Open image in new window if there exists a sunny nonexpansive retraction from Open image in new window onto Open image in new window .

The following results describe a characterization of sunny nonexpansive retractions on a smooth Banach space.

Proposition 1.3 (see [8]).

Let Open image in new window be a smooth Banach space and Open image in new window a nonempty subset of Open image in new window . Let Open image in new window be a retraction and Open image in new window the normalized duality mapping on Open image in new window . Then the following are equivalent:

(1) Open image in new window is sunny and nonexpansive,

(2) Open image in new window

Recall that a mapping Open image in new window is called contractive if there exists a constant Open image in new window such that
A mapping Open image in new window is said to be Open image in new window -strictly pseudocontractive if there exists a constant Open image in new window such that
Note that the class of Open image in new window -strictly pseudocontractive mappings strictly includes the class of nonexpansive mappings which are mappings Open image in new window on Open image in new window such that

for all Open image in new window . That is, Open image in new window is nonexpansive if and only if Open image in new window is Open image in new window -strict pseudocontractive. We denote by Open image in new window the set of fixed points of Open image in new window .

Proposition 1.4 (see [9]).

Let Open image in new window be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space Open image in new window and Open image in new window a nonexpansive mapping of Open image in new window into itself with Open image in new window . Then the set Open image in new window is a sunny nonexpansive retract of Open image in new window .

Definition 1.5.

A countable family of mapping Open image in new window is called a family of uniformly Open image in new window -strict pseudocontractions if there exists a constant Open image in new window such that
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 Open image in new window and define a contraction Open image in new window by
where Open image in new window is a fixed point and Open image in new window is a nonexpansive mapping. Banach's contraction mapping principle guarantees that Open image in new window has a unique fixed point Open image in new window in Open image in new window ; that is,

It is unclear, in general, what the behavior of Open image in new window is as Open image in new window even if Open image in new window has a fixed point. However, in the case of Open image in new window having a fixed point, Ceng et al. [12] proved that, if Open image in new window is a Hilbert space, then Open image in new window converges strongly to a fixed point of Open image in new window . Reich [11] extended Browder's result to the setting of Banach spaces and proved that, if Open image in new window is a uniformly smooth Banach space, then Open image in new window converges strongly to a fixed point of Open image in new window , and the limit defines the (unique) sunny nonexpansive retraction from Open image in new window onto Open image in new window .

Reich [11] showed that, if Open image in new window is uniformly smooth and Open image in new window is the fixed point set of a nonexpansive mapping from Open image in new window into itself, then there is a unique sunny nonexpansive retraction from Open image in new window onto Open image in new window and it can be constructed as follows.

Proposition 1.6 (see [11]).

Let Open image in new window be a uniformly smooth Banach space and Open image in new window a nonexpansive mapping such that Open image in new window . For each fixed Open image in new window and every Open image in new window , the unique fixed point Open image in new window of the contraction Open image in new window converges strongly as Open image in new window to a fixed point of Open image in new window . Define Open image in new window by Open image in new window . Then Open image in new window is the unique sunny nonexpansive retract from Open image in new window onto Open image in new window ; that is, Open image in new window satisfies the property.

Notation 1.

We use Open image in new window to denote strong convergence to Open image in new window of the net Open image in new window as Open image in new window .

Definition 1.7 (see [13]).

Let Open image in new window be a multivalued maximal accretive mapping. The single-valued mapping Open image in new window defined by

is called the resolvent operator associated with Open image in new window , where Open image in new window is any positive number and Open image in new window 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, 15, 16]).

Very recently, Qin et al. [16] considered the problem of finding the solutions of a general system of variational inclusion (1.11) with Open image in new window -inverse strongly accretive mappings. To be more precise, they obtained the following results.

Lemma 1.8 (see [16]).

For any Open image in new window where Open image in new window , Open image in new window is a solution of the problem (1.11) if and only if Open image in new window is a fixed point of the mapping Open image in new window defined by

Theorem QCCK (see [16, Theorem Open image in new window ]).

Let Open image in new window be a uniformly convex and Open image in new window -uniformly smooth Banach space with the smoothness constant Open image in new window . Let Open image in new window be a maximal monotone mapping and Open image in new window a Open image in new window -inverse-strongly accretive mapping, respectively, for each Open image in new window . Let Open image in new window be a Open image in new window -strict pseudocontraction such that Open image in new window . Define a mapping Open image in new window by Open image in new window , Open image in new window . Assume that Open image in new window , where Open image in new window is defined as in Lemma 1.8. Let Open image in new window and let Open image in new window be a sequence generated by

where Open image in new window , Open image in new window , Open image in new window , and Open image in new window and Open image in new window are sequences in Open image in new window . If the control consequences Open image in new window and Open image in new window satisfy the following restrictions:

(C1) Open image in new window ,

(C2) Open image in new window and Open image in new window ,

then Open image in new window converges strongly to Open image in new window , where Open image in new window is the sunny nonexpansive retraction from Open image in new window onto Open image in new window and Open image in new window , where Open image in new window , 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 Open image in new window is called strongly positive [17] if there exists a constant Open image in new window with property

where Open image in new window is the identity mapping and Open image in new window 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 Open image in new window be a contraction on Open image in new window . Starting with an arbitrary initial point Open image in new window , define a sequence Open image in new window recursively by
where Open image in new window is a sequence in Open image in new window . It is proved [18, 19] that, under certain appropriate conditions imposed on Open image in new window , the sequence Open image in new window generated by (1.28) strongly converges to the unique solution Open image in new window in Open image in new window of the variational inequality
Recently, Marino and Xu [20] introduced the following general iterative method:
where Open image in new window is a strongly positive bounded linear operator on a Hilbert space Open image in new window . They proved that, if the sequence Open image in new window of parameters satisfies appropriate conditions, then the sequence Open image in new window generated by (1.30) converges strongly to the unique solution of the variational inequality
which is the optimality condition for the minimization problem

where Open image in new window is a potential function for Open image in new window for Open image in new window ).

Recently, Qin et al. [21] introduce the following iterative algorithm scheme:

where Open image in new window is nonself- Open image in new window -strict pseudo-contraction, Open image in new window is a contraction, and Open image in new window is a strongly positive bounded linear operator on a Hilbert space Open image in new window . They proved, under certain appropriate conditions imposed on the sequences Open image in new window and Open image in new window , that Open image in new window defined by (1.33) converges strongly to a fixed point of Open image in new window , 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 Open image in new window -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 Open image in new window associated with Open image in new window is single valued and nonexpansive for all Open image in new window .

Lemma 2.2 (see [13]).

is a solution of variational inclusion (1.13) if and only if Open image in new window Open image in new window ; that is,

where Open image in new window denotes the set of solutions to problem (1.13).

Lemma 2.3 (see [23]).

Let Open image in new window be a strictly convex Banach space. Let Open image in new window and Open image in new window be two nonexpansive mappings from Open image in new window into itself with a common fixed point. Define a mapping Open image in new window by

where Open image in new window is a constant in Open image in new window . Then Open image in new window is nonexpansive and Open image in new window .

Lemma 2.4 (see [24]).

Let Open image in new window be a nonempty closed convex subset of reflexive Banach space Open image in new window which satisfies Opial's condition, and suppose that Open image in new window is nonexpansive. Then the mapping Open image in new window is demiclosed at zero, that is, Open image in new window imply that Open image in new window .

Lemma 2.5 (see [25]).

Assume that Open image in new window is a sequence of nonnegative real numbers such that

where Open image in new window is a sequence in Open image in new window and Open image in new window is a sequence such that

(a) Open image in new window ,

(b) Open image in new window or Open image in new window

Then Open image in new window

Lemma 2.6 (see [26]).

Then Open image in new window

Definition 2.7 (see [27]).

Let Open image in new window be a family of mappings from a subset Open image in new window of a Banach space Open image in new window into Open image in new window with Open image in new window . We say that Open image in new window satisfies the AKTT-condition if, for each bounded subset Open image in new window of Open image in new window ,

Remark 2.8.

The example of the sequence of mappings Open image in new window satisfying AKTT-condition is supported by Example 3.11.

Lemma 2.9 (see [27, Lemma Open image in new window ]).

Suppose that Open image in new window satisfies AKTT-condition. Then, for each Open image in new window , Open image in new window converses strongly to a point in Open image in new window . Moreover, let the mapping Open image in new window be defined by

Then for each bounded subset Open image in new window of Open image in new window , Open image in new window

Lemma 2.10 (see [28]).

Let Open image in new window be a real Open image in new window -uniformly smooth Banach space and Open image in new window a Open image in new window -strict pseudocontraction. Then Open image in new window is nonexpansive and Open image in new window .

Lemma 2.11 (see [29]).

Let Open image in new window be a real Open image in new window -uniformly smooth Banach space with the best smoothness constant Open image in new window . Then the following inequality holds:

Lemma 2.12 (see [17, Lemma Open image in new window ]).

Assume that Open image in new window is a strongly positive linear bounded operator on a smooth Banach space Open image in new window with coefficient Open image in new window and Open image in new window Then Open image in new window

3. Main Results

In this section, we prove that the strong convergence theorem for a countable family of uniformly Open image in new window -strict pseudocontractions in a strictly convex and Open image in new window -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 Open image in new window ]).

Let Open image in new window be a nonempty closed convex subset of a reflexive, smooth Banach space Open image in new window which admits a weakly sequentially continuous duality mapping Open image in new window from Open image in new window to Open image in new window . Let Open image in new window be a nonexpansive mapping such that Open image in new window is nonempty, let Open image in new window be a contraction with coefficient Open image in new window , and let Open image in new window be a strongly positive bounded linear operator with coefficient Open image in new window and Open image in new window . Then the net Open image in new window defined by
converges strongly as Open image in new window to a fixed point Open image in new window of Open image in new window which solves the variational inequality:

Lemma 3.2.

Let Open image in new window be a nonempty closed convex subset of a real Open image in new window -uniformly smooth Banach space Open image in new window with the smoothness constant Open image in new window . Let Open image in new window be an Open image in new window -Lipschitzian and relaxed Open image in new window -cocoercive mapping. Then

If Open image in new window , then Open image in new window is nonexpansive.

Proof.

Using Lemma 2.11 and the cocoercivity of the mapping Open image in new window , we have, for all Open image in new window ,

Hence (3.3) is proved. Assume that Open image in new window . Then, we have Open image in new window . This together with (3.3) implies that Open image in new window is nonexpansive.

Lemma 3.3.

Let Open image in new window be a strictly convex and Open image in new window -uniformly smooth Banach space admiting a weakly sequentially continuous duality mapping with the smoothness constant Open image in new window . Let Open image in new window be a maximal monotone mapping and Open image in new window a Open image in new window -Lipschitzian and relaxed Open image in new window -cocoercive mapping with Open image in new window , respectively, for each Open image in new window . Let Open image in new window be a countable family of uniformly Open image in new window -strict pseudocontractions. Define a mapping Open image in new window and Open image in new window by

where Open image in new window is defined as in Lemma 1.8. Assume that Open image in new window . Let Open image in new window be an Open image in new window -contraction; let Open image in new window be a strongly positive linear bounded self-adjoint operator with coefficient Open image in new window with Open image in new window . Then the following hold.

(i)For each Open image in new window , Open image in new window is nonexpansive such that
(ii)Suppose that Open image in new window satisfies AKTT-condition. Let Open image in new window be the mapping defined by Open image in new window for all Open image in new window and suppose that Open image in new window . The net Open image in new window defined by Open image in new window converges strongly as Open image in new window to a fixed point Open image in new window of Open image in new window , which solves the variational inequality

and Open image in new window is a solution of general system of variational inequality problem (1.11) such that Open image in new window .

Proof.

It follows from Lemma 2.10 that Open image in new window is nonexpansive such that Open image in new window for each Open image in new window . Next, we prove that Open image in new window is nonexpansive. Indeed, we observe that
The nonexpansivity of Open image in new window , Open image in new window , Open image in new window , and Open image in new window implies that Open image in new window is nonexpansive. By Lemma 2.3, we have that Open image in new window is nonexpansive such that
Hence (i) is proved. It is well known that, if Open image in new window is uniformly smooth, then Open image in new window is reflexive. Hence Theorem 3.1 implies that Open image in new window converges strongly as Open image in new window to a fixed point Open image in new window of Open image in new window , which solves the variational inequality

and Open image in new window is a solution of problem (1.11), where Open image in new window . This completes the proof of (ii).

Theorem 3.4.

Let Open image in new window be a strictly convex and Open image in new window -uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and has the smoothness constant Open image in new window . Let Open image in new window be a maximal monotone mapping and Open image in new window a Open image in new window -Lipschitzian and relaxed Open image in new window -cocoercive mapping with Open image in new window , respectively, for each Open image in new window . Let Open image in new window be a countable family of uniformly Open image in new window -strict pseudocontractions. Define a mapping Open image in new window by
Assume that Open image in new window , where Open image in new window is defined as in Lemma 1.8. Let Open image in new window be an Open image in new window -contraction; let Open image in new window be a strongly positive linear bounded self adjoint operator with coefficient Open image in new window with Open image in new window . Let Open image in new window and let Open image in new window be a sequence generated by

where Open image in new window , and Open image in new window and Open image in new window are sequences in Open image in new window . Suppose that Open image in new window satisfies AKTT-condition. Let Open image in new window be the mapping defined by Open image in new window for all Open image in new window and suppose that Open image in new window . If the control consequences Open image in new window and Open image in new window satisfy the following restrictions

(C1) Open image in new window ,

(C2) Open image in new window and Open image in new window ,

then Open image in new window converges strongly to Open image in new window , which solves the variational inequality

and Open image in new window is a solution of general system of variational inequality problem (1.11) such that Open image in new window .

Proof.

First, we show that sequences Open image in new window , Open image in new window , and Open image in new window are bounded.

By the control condition (C2), we may assume, with no loss of generality, that Open image in new window .

Since Open image in new window is a linear bounded operator on Open image in new window , by (1.27), we have
Observe that
It follows that
Therefore, taking Open image in new window one has
It follows from Lemmas 2.1 and 3.2 that
This implies that
Setting Open image in new window and applying Lemma 2.10, we have that Open image in new window is a nonexpansive mapping such that Open image in new window for all Open image in new window and hence Open image in new window . Then
It follows from the last inequality that
By induction, we have

This shows that the sequence Open image in new window is bounded, and so are Open image in new window , Open image in new window , and Open image in new window .

On the other hand, from the nonexpansivity of the mappings Open image in new window , one sees that
In a similar way, one can obtain that
It follows that
This implies that
one sees that
and so it follows that
which, combined, with (3.27) yields that
Using the conditions (C1) and (C2) and AKTT-condition of Open image in new window , we have
Hence, from Lemma 2.6, it follows that
From (3.28), it follows that
By (3.33), one sees that
On the other hand, one has
It follows that
From the conditions (C1), (C2) and from (3.35), one sees that
where Open image in new window is defined as in Lemma 1.8. From Lemma 3.3(i), we see that Open image in new window is nonexpansive such that
From (3.38), it follows that
Since Open image in new window satisfies AKTT-condition and Open image in new window is the mapping defined by Open image in new window for all Open image in new window , we have that Open image in new window satisfies AKTT-condition. Let the mapping Open image in new window be the mapping defined by Open image in new window for all Open image in new window . It follows from the nonexpansivity of Open image in new window and
that Open image in new window is nonexpansive such that
Next, we prove that
where Open image in new window with Open image in new window be the fixed point of the contraction
Then Open image in new window solves the fixed point equation Open image in new window . It follows from Lemma 3.3(ii) that Open image in new window , which solves the variational inequality:
and Open image in new window is a solution of general system of variational inequality problem (1.11) such that Open image in new window . Let Open image in new window be a subsequence of Open image in new window such that
If follows from reflexivity of Open image in new window and the boundedness of sequence Open image in new window that there exists Open image in new window which is a subsequence of Open image in new window converging weakly to Open image in new window as Open image in new window . It follows from (3.41) and the nonexpansivity of Open image in new window , we have Open image in new window by Lemma 2.4. Since the duality map Open image in new window is single valued and weakly sequentially continuous from Open image in new window to Open image in new window , we get that
as required. Now from Lemma 2.11, we have
which implies that
where Open image in new window is an appropriate constant such that Open image in new window . Put
that is,
It follows from conditions (C1), (C2) and from (3.44) that

Apply Lemma 2.5 to (3.52) to conclude that Open image in new window as Open image in new window . This completes the proof.

Setting Open image in new window , and Open image in new window , we have the following result.

Theorem 3.5.

Let Open image in new window be a strictly convex and Open image in new window -uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and has the smoothness constant Open image in new window . Let Open image in new window be a maximal monotone mapping and Open image in new window a Open image in new window -Lipschitzian and relaxed Open image in new window -cocoercive mapping with Open image in new window , respectively, for each Open image in new window . Let Open image in new window be a countable family of uniformly Open image in new window -strict pseudocontractions. Define a mapping Open image in new window by
Assume that Open image in new window , where Open image in new window is defined as in Lemma 1.8. Let Open image in new window and let Open image in new window be a sequence generated by

where Open image in new window , and Open image in new window and Open image in new window are sequences in Open image in new window . Suppose that Open image in new window satisfies AKTT-condition. Let Open image in new window be the mapping defined by Open image in new window for all Open image in new window and suppose that Open image in new window . If the control consequences Open image in new window and Open image in new window satisfy the following restrictions

(C1) Open image in new window ,

(C2) Open image in new window and Open image in new window ,

then Open image in new window converges strongly to Open image in new window , which solves the variational inequality

and Open image in new window is a solution of general system of variational inequality problem (1.11) such that Open image in new window .

Remark 3.6.

Theorem 3.4 mainly improves Theorem Open image in new window 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 Open image in new window -strict pseudocontraction to the countable family of uniformly Open image in new window -strict pseudocontractions,

(c)from a uniformly convex and Open image in new window -uniformly smooth Banach space to a strictly convex and Open image in new window -uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping.

Further, if Open image in new window is a countable family of nonexpansive mappings, then Theorem 3.4 is reduced to the following result.

Theorem 3.7.

Let Open image in new window be a strictly convex and Open image in new window -uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and has the smoothness constant Open image in new window . Let Open image in new window be a maximal monotone mapping and Open image in new window a Open image in new window -Lipschitzian and relaxed Open image in new window -cocoercive mapping with Open image in new window , respectively, for each Open image in new window . Let Open image in new window be a countable family of nonexpansive mappings. Assume that Open image in new window , where Open image in new window is defined as in Lemma 1.8. Let Open image in new window be an Open image in new window -contraction; let Open image in new window be a strongly positive linear bounded self adjoint operator with coefficient Open image in new window with Open image in new window . Let Open image in new window and let Open image in new window be a sequence generated by

where Open image in new window , and Open image in new window and Open image in new window are sequences in Open image in new window . Suppose that Open image in new window satisfies AKTT-condition. Let Open image in new window be the mapping defined by Open image in new window for all Open image in new window and suppose that Open image in new window . If the control consequences Open image in new window and Open image in new window satisfy the following restrictions

(C1) Open image in new window ,

(C2) Open image in new window and Open image in new window ,

then Open image in new window converges strongly to Open image in new window which solves the variational inequality:

and Open image in new window is a solution of general system of variational inequality problem (1.11) such that Open image in new window .

Remark 3.8.

As in [27, Theorem Open image in new window ], we can generate a sequence Open image in new window of nonexpansive mappings satisfying AKTT-condition; that is, Open image in new window for any bounded subset Open image in new window of Open image in new window by using convex combination of a general sequence Open image in new window of nonexpansive mappings with a common fixed point. To be more precise, they obtained the following lemma.

Lemma 3.9 (see [27]).

Let Open image in new window be a closed convex subset of a smooth Banach space Open image in new window . Suppose that Open image in new window is a sequence of nonexpansive mappings of Open image in new window into inself with a common fixed point. For each Open image in new window , define Open image in new window by

where Open image in new window is a family of nonnegative numbers with indices Open image in new window with Open image in new window such that

(i) Open image in new window for all Open image in new window ,

(ii) Open image in new window for every Open image in new window ,

(iii) Open image in new window .

Then the following are given.

()Each Open image in new window is a nonexpansive mapping.

() Open image in new window satisfies AKTT-condition.

()If Open image in new window is defined by

then Open image in new window and Open image in new window .

Theorem 3.10.

Let Open image in new window be a strictly convex and Open image in new window -uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and has the smoothness constant Open image in new window . Let Open image in new window be a maximal monotone mapping and Open image in new window a Open image in new window -Lipschitzian and relaxed Open image in new window -cocoercive mapping with Open image in new window , respectively, for each Open image in new window . Let Open image in new window be a countable family of nonexpansive mappings. Assume that Open image in new window , where Open image in new window is defined as in Lemma 1.8. Let Open image in new window be an Open image in new window -contraction; let Open image in new window be a strongly positive linear bounded self adjoint operator with coefficient Open image in new window with Open image in new window . Let Open image in new window and let Open image in new window be a sequence generated by

where Open image in new window satisfies conditions (i)–(iii) of Lemma 3.9, Open image in new window , and Open image in new window and Open image in new window are sequences in Open image in new window . Suppose that Open image in new window satisfies AKTT-condition. Let Open image in new window be the mapping defined by Open image in new window for all Open image in new window and suppose that Open image in new window . If the control consequences Open image in new window and Open image in new window satisfy the following restrictions:

(C1) Open image in new window ,

(C2) Open image in new window and Open image in new window ,

then Open image in new window converges strongly to Open image in new window , which solves the variational inequality

and Open image in new window is a solution of general system of variational inequality problem (1.11) such that Open image in new window .

Proof.

We write the iteration (3.61) as

where Open image in new window is defined by (3.59). It is clear that each mapping Open image in new window is nonexpansive. By Theorem 3.7 and Lemma 3.9, the conclusion follows.

The following example appears in [27] shows that there exists Open image in new window satisfying the conditions of Lemma 3.9.

Example 3.11.

for all Open image in new window with Open image in new window In this case, the sequence Open image in new window of mappings generated by Open image in new window is defined as follows: For Open image in new window .

Notes

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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Copyright information

© Uthai Kamraksa and Rabian Wangkeeree. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceNaresuan UniversityPhitsanulokThailand
  2. 2.Centre of Excellence in Mathematics, CHEBangkokThailand

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