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

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## 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## 1. Introduction

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.

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).

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 .

for all 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 .

*-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 .

*-relaxed cocoercive*if there exists a constant Open image in new window such that

for all 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.

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,

*-strictly pseudocontractive*if there exists a constant 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.

*family of uniformly*Open image in new window

*-strict pseudocontractions*if there exists a constant Open image in new window such that

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]).

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]).

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]).

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

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.

*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.

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

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]).

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

Lemma 2.3 (see [23]).

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]).

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

Lemma 2.6 (see [26]).

Definition 2.7 (see [27]).

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 ]).

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]).

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 ]).

Lemma 3.2.

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

Proof.

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.

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.

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.

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.

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 ,

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 .

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 .

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.

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 ,

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.

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 ,

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]).

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.

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 ,

Proof.

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.

## 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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