Journal of Function Spaces
Volume 2017 (2017), Article ID 5847096, 10 pages
https://doi.org/10.1155/2017/5847096

Research Article

Convergence Analysis of Parallel -Iteration Process for System of Generalized Variational Inequalities

D. R. Sahu,¹ Shin Min Kang,2,3 and Ajeet Kumar¹

¹Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi 221005, India
²Department of Mathematics and the RINS, Gyeongsang National University, Jinju 52828, Republic of Korea
³Center for General Education, China Medical University, Taichung 40402, Taiwan

Correspondence should be addressed to Shin Min Kang; smkang@gnu.ac.kr

Received 8 July 2017; Accepted 27 August 2017; Published 12 October 2017

Academic Editor: Tomonari Suzuki

Copyright © 2017 D. R. Sahu et al. 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.

Abstract.

We consider a new system of generalized variational inequalities (SGVI) defined on two closed convex subsets of a real Hilbert space. To find the solution of considered SGVI, a parallel Mann iteration process and a parallel -iteration process have been proposed and the strong convergence of the sequences generated by these parallel iteration processes is discussed. Numerical example illustrates that the proposed parallel -iteration process has an advantage over parallel Mann iteration process in computing altering points of some mappings.

1. Introduction

Variational inequalities are the most interesting and important mathematical problems and have been studied intensively in the past years. The variational inequality problem was first introduced and studied by Stampacchia [1] in 1964, which is defined as follows.

Let be a nonempty closed convex subset of a real Hilbert space and let be a nonlinear mapping. Then the classical variational inequality problem is to find a point such that

The problem (1) is denoted by and the set of solutions of (1) is denoted and defined by for all . We denote by the set of fixed points of . It is well known that the variational inequality problem (1) is equivalent to the following fixed point problem:

where is the metric projection from onto , is a constant, and is the identity mapping from into itself. It is well known that if the mapping is -Lipschitzian and -strongly monotone, then the operator is a contraction on provided that . In this case, the Banach contraction principle guarantees that has a unique solution and the sequence of Picard iteration method given by

converges strongly to . This method is called the projected gradient method [2]. This method has been widely used in many practical problems, due partially to its fast convergence.

In 2007, Agarwal et al. [3] posed the following query.

Question 1. Is it possible to develop an iterative method whose rate of convergence is faster than the Picard iteration method for contraction mappings?

They introduced the following iteration process known as -iteration process as an answer to Question 1: let be a nonempty convex subset of a normed linear space , and let be an operator. Then, for arbitrary , the -iteration process is defined by

where and are real sequences in satisfying some suitable conditions. In [4], Sahu proved that the rate of convergence of -iteration process for contraction mappings is faster than that of Picard [5] and Mann [6] iteration processes by providing a numerical example. The -iteration process is more applicable than the Picard [5], Mann [6], and Ishikawa [7] iteration processes because it converges faster than these iteration processes for contraction mappings and also works for nonexpansive mappings. Due to the super rate of convergence of above iteration process, Agarwal et al. [3] called it the -iteration process. Due to its fastness, in recent years, the -iteration process attracted many researchers as an alternate iteration process and is used for solving fixed point problems, common fixed point problems, convex minimization problems, the problem of solving nonlinear operator equations, and other allied areas (see [8–10]). Moreover, the idea of -iteration process is applied by Cholamjiak et al. [11] for finding a minimizer of a convex function and fixed points of nonexpansive mappings in CAT(0) space setting. Sahu [4] also introduced the notion of -operator of a mapping generated by and and normal -iteration process in the following way: let be a nonempty convex subset of a normed linear space and let be an operator. Then, for arbitrary , the normal -iteration process is defined by

where is a sequence of real numbers in . In 2017, Verma and Shukla [12] designed some new algorithms based on -iteration processes and named them as -iteration-based forward-backward algorithm (SFBA) and normal -iteration-based forward-backward algorithm (NSFBA) and performed the nice experiments of the high-dimensional real datasets for SFBA, NSFBA, and others.

On the other hand, in Hilbert spaces, projection type methods have played a very crucial role in the numerical resolution of variational inequalities depending on their convergence analysis. By virtue of the projection, in 2011, Ceng et al. [13] proposed the following iterative method:

where is -Lipschitzian and -strongly monotone operator with , , is an -Lipschitzian mapping with , is a nonexpansive mapping with , , and an arbitrary initial point. They proved that the sequence generated by the iterative method (6) converges strongly to a fixed point of which solves the following variational inequality problem:

In 2001, Verma [14] generalized the concept of variational inequalities to a system of nonlinear variational inequalities (SNVI) in the following way: find such that

where is any mapping and and are constants. To solve (8), he introduced the following iterative method:

and he proved that the sequences and generated by (9) converge to and , respectively. In 2005, Verma [15] also introduced the general model for two-step projection methods for applying the approximation solvability of SNVI in Hilbert space setting as follows: let be a nonempty closed convex subset of a real Hilbert space and let be a nonlinear mapping. For arbitrary chosen initial point , let and be the sequences in defined by

where , and , . Further, problem (8) is equivalent to the following projection formulas:

for a monotone mapping . The problem of finding the solutions of (11) by using iterative methods has been studied by many authors (see [15–22]). A more general case has been studied in [23].

Parallel iteration processes have their own advantages. A variety of problems have been dealt with in these iteration processes (see [24, 25] and the references therein). Recently, Sahu [26] introduced the notion of altering points of nonlinear mappings and following the idea of -operator and normal -iteration process, he [26] introduced a parallel -iteration process for finding altering points of nonlinear mappings as follows.

Let and be two nonempty closed convex subsets of a Banach space and let and be two mappings. Then, for and arbitrary , the parallel normal -iteration process is defined by

The following convergence result is given in [26].

Theorem 1 (see [26]). Let and be two nonempty closed convex subsets of a Banach space . Let and be two Lipschitz continuous mappings with Lipschitz constants and , respectively. Then the sequence in generated by the parallel -iteration process (12) converges strongly to a unique point such that and are altering points of mappings and .

In this paper, motivated by the work of Ceng et al. [13], Verma [14, 15], Hao et al. [23], and Sahu [26], we consider a new SGVI defined on two closed convex subsets of a real Hilbert space and propose a parallel Mann and a more general parallel -iteration process for solving considered SGVI in the context of altering points and study the strong convergence of the sequences generated by the proposed algorithms to altering points of some nonlinear mappings.

2. Preliminaries

Throughout this paper, the symbol stands for the set of all natural numbers.

Let be a nonempty subset of a real Hilbert space with inner product and norm , respectively. A mapping is called

(1)monotone if

(2)-strongly monotone if there exists a positive real number such that

(3)-Lipschitzian if there exists a constant such that

(4)-contraction if there exists a constant such that

(5)nonexpansive if

Definition 2 (see [26]). Let and be two nonempty subsets of a metric space . Then and are altering points of mappings and if

The set of altering points of mappings and is denoted and defined by

We now give some numerical examples in support of the definition of altering points of some nonlinear mappings as follows.

Example 3 (see [26]). Let , , and . Define and by for . Note that is defined by . Thus, each point of is a fixed point of . Then altering points and of and are given by the relation . Indeed,

Example 4. Let , , and . Let and be two mappings defined, respectively, by

Note that is defined by . Clearly and are fixed points of and , respectively. Therefore, and are altering points of mappings and .

Let be a nonempty closed convex subset of . Then, for any , there exists a unique nearest point of such that

The mapping is called the metric projection [27] from onto . It is remarkable that the metric projection mapping is nonexpansive from onto (see Agarwal et al. [28]).

We need the following technical lemmas.

Lemma 5 (see [28]). Let be a nonempty closed convex subset of a real Hilbert space and let be the metric projection from onto . Given and , then if and only if for all .

Lemma 6 (see [29]). Let be a nonempty subset of a real Hilbert space . Suppose that and . Let be a -Lipschitzian and -strongly monotone operator. Define the mapping by

Then is a contraction provided . More precisely, for ,

where .

Lemma 7 (see [17]). Let , , and be three nonnegative real sequences satisfying the following conditions:

where is some nonnegative integer, with , , and . Then .

Lemma 8 (see [26, Theorem  3.1]). Let and be two nonempty closed subsets of a complete metric space . Suppose that and be two Lipschitz continuous mappings with Lipschitz constants and , respectively, such that . Then the following holds:

(a) There exists a unique point such that and are altering points of mappings and .(b)For arbitrary , a sequence generated by

 converges to .

3. Main Results

In this section, we introduce a new system of generalized variational inequalities and new iterative algorithms for solving the proposed system of generalized variational inequalities in the framework of real Hilbert spaces.

Let and be nonempty closed convex subsets of a real Hilbert space and let and be some mappings. Let be mappings. Consider a general system of generalized variational inequalities (SGVI) defined on and as follows.

Find such that

where and are constants.

Remark 9. If , , and , then the system of generalized variational inequalities (SGVI) (27) reduces to SNVI (8) studied by Verma [14].

The system of generalized variational inequalities (27) is more general in nature. One can find various systems of generalized variational inequalities from SGVI (27).

We now discuss some special cases of (27) as follows.

Let be single-valued -strongly monotone, -Lipschitz continuous, let be -Lipschitzian and -strongly monotone operator with constants , and let be -Lipschitzian mapping with constant for . Suppose that and , where for .

If for , then the system of generalized variational inequalities (27) reduces to the following system of generalized variational inequalities (SGVI).

Find such that

Define the mappings and by

where and are some constants in . Using Lemma 5, one can easily observe that the SGVI (28) is equivalent to the following altering point formulation:

First we introduce parallel Mann iteration process to solve system of generalized variational inequalities (28) as follows.

Algorithm 10. For any given , let be an iterative sequence in defined by

where is a sequence in and and are defined by (29) and (30), respectively.

Motivated by Sahu [26] and equivalent formulation (31), we now propose a more general parallel -iteration process to solve SGVI (28) as follows.

Algorithm 11. For any given , let be an iterative sequence in defined by

where is a sequence in and and are defined by (29) and (30), respectively.

Before proving our main results, we will prove the following proposition which will be used in sequel.

Proposition 12. Let and be nonempty closed convex subsets of a real Hilbert space . Let be single-valued -strongly monotone, -Lipschitz continuous, let be -Lipschitzian and -strongly monotone operator with constants , and let be -Lipschitzian mapping with constant for . Suppose that and , where for . Let and let and be real constants defined by

Then the mappings and defined by (29) and (30) are Lipschitz continuous with Lipschitz constants and , respectively.

Proof. Let . Then, we have

From (35), we have

Thus is -Lipschitz continuous.
Similarly, we can show that is -Lipschitz continuous.

Now we are ready to present our main results. First we establish the convergence analysis of Algorithm 10 for solving SGVI (28).

Theorem 13. Let and be nonempty closed convex subsets of a real Hilbert space . Let be single-valued -strongly monotone, -Lipschitz continuous, let be -Lipschitzian and -strongly monotone operator with constants , and let be -Lipschitzian mapping with constant for . Suppose that and , where for . Let and let and be real constants defined by (34). Let and be defined by (29) and (30), respectively. For given initial point , let be an iterative sequence defined by parallel Mann iteration process (32), where is a sequence in such that . Assume that the following condition is satisfied:

Then we have the following:

(i)There exists a unique point , which solves SGVI (28).(ii)The sequence generated by parallel Mann iteration process (32) converges strongly to the point .

Proof. (i) It follows from Lemma 8 and (31).
(ii) By (31), (32), and Proposition 12, we have

Again, by Proposition 12 that is -Lipschitz continuous and using (31) and (32), we have

Set

From (38) and (39), we get

Now, we define the norm on by for all . Therefore, using (41), we have

Noticing that and . Therefore, from Lemma 7, we have . Thus, we get and hence and converge to and , respectively.

Corollary 14. Let and be nonempty closed convex subsets of a real Hilbert space . Let be -Lipschitzian and -strongly monotone operator with constants and let be -Lipschitzian mapping with constant for . Suppose that and , where for . Let . Define mappings and by

For given initial point , let be an iterative sequence in defined by

where is a real sequence in such that . Assume that condition (37) of Theorem 13 is satisfied. Then the sequence generated by (44) converges strongly to the unique point , which solves system of generalized variational inequalities

Proof. The proof follows from Theorem 13 by taking .

Now we study the convergence analysis of Algorithm 11, that is, the parallel -iteration process defined by (33) for solving SGVI (28).

Theorem 15. Let and be nonempty closed convex subsets of a real Hilbert space . Let be single-valued -strongly monotone, -Lipschitz continuous, let be -Lipschitzian and -strongly monotone operator with constants , and let be -Lipschitzian mapping with constant for . Suppose that and , where for . Let and let and be real constants defined by (34). Let and be defined by (29) and (30), respectively. For given initial point , let be an iterative sequence in defined by parallel -iteration process (33), where is a sequence in . Assume that condition (37) of Theorem 13 is satisfied. Then we have the following:

(i)There exists a unique point , which solves SGVI (28).(ii)The sequence generated by parallel -iteration process (33) converges strongly to the point .

Proof. (i) It follows from Lemma 8 and (31).
(ii) From (31), (33), and Proposition 12, we have

Similarly

Set

From (46) and (47), we get

Now, we define the norm on by for all . Therefore, using (49), we have

Since , we obtain that . Thus, we get and hence and converge to and , respectively.

Corollary 16. Let and be nonempty closed convex subsets of a real Hilbert space . Let be -Lipschitzian and -strongly monotone operator with constants and let be -Lipschitzian mapping with constant for . Suppose that and , where for . Let and let and be defined by (43). For given initial point , let be an iterative sequence in defined by

where is a real sequence in . Assume that condition (37) of Theorem 13 is satisfied. Then the sequence generated by (51) converges strongly to the unique point , which solves system of generalized variational inequalities

Proof. The proof follows from Theorem 15 by taking .

4. Numerical Example

In this section, we discuss an example which leads to Theorems 13 and 15. The graphs are also presented for showing how the sequences and generated by both the algorithms, Algorithms 10 and 11, converge to the solutions of SGVI (28).

Example 17. Let , , and . Let and be two mappings from onto itself defined by for all and for all , respectively. Let and be two mappings defined by for all and for all , respectively. Let and be two mappings defined by for all and for all , respectively. Then is -strongly monotone and -Lipschitzian mapping for . We have and . Also is -strongly monotone and -Lipschitzian mapping for . We have and . Moreover is -Lipschitzian mapping for . We have and . We take , , and , . Define in by , , and . Then and .
Therefore and can be expressed as

where and are constants.

Let and . Then,

It can be easily seen that is -Lipschitzian and is -Lipschitzian. Also

One can observe that all the conditions of Theorems 13 and 15 are satisfied.

Now we will find the general term of the sequences and generated by the iteration process (32). For arbitrary and ,

Hence

Also, we will find the general term of the sequences and generated by the iteration process (33). For arbitrary and ,

Hence

It is clear from (57) and (59) that the sequences and generated by the proposed iterative algorithms converge to the altering points and of the mappings and , where and . The numerical values of and have been calculated for different starting values of and in Tables 1 and 2, respectively, and the convergence of both the sequences is shown in Figures 1 and 2, respectively.

Table 1: Numerical values of and .

Parallel Mann iteration process
Parallel -iteration process

Table 2: Numerical values of and .

Parallel Mann iteration process
Parallel -iteration process

Figure 1: Convergence of sequences and generated by parallel Mann iteration process (32).

Figure 2: Convergence of sequences and generated by parallel -iteration process (33).

5. Conclusions

In this paper, we have considered a new system of generalized variational inequalities (SGVI) defined on closed convex subsets of a real Hilbert space. It has been shown that the considered SGVI is equivalent to altering points problem of some nonlinear mappings. We have proposed two algorithms, Algorithms 10 and 11, for solving considered SGVI. An example is given in support of our main results. We observed that the sequence generated by Algorithm 11 converges faster than Algorithm 10 to altering points of some nonlinear mappings and .

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The third author is supported by the Council of Scientific and Industrial Research (CSIR), New Delhi, India, through Grant 09/013(0584)/2015-EMR-I.

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