# Convergence Analysis of ParallelS-Iteration Process for System of Generalized Variational Inequalities

Authors: D. R. Sahu; Shin Min Kang; Ajeet Kumar
Journal: Journal of Function Spaces
 (2017)

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## 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 parallelS-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 S-iteration process has an advantage over parallel Mann iteration process in computing altering points of some mappings.


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## Body

## 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.LetC be a nonempty closed convex subset of a real Hilbert space H and let T:C→H be a nonlinear mapping. Then the classical variational inequality problem is to find a point x∗∈C such that(1)Tx∗,x-x∗≥0∀x∈C.The problem (1) is denoted by VI(C,T) and the set of solutions of (1) is denoted and defined by ΩVIC,T={x∗∈C:Tx∗,x-x∗≥0 for all x∈C}. We denote by Fix(T) the set of fixed points of T. It is well known that the variational inequality problem (1) is equivalent to the following fixed point problem: (2)findx∗∈Csuch  thatx∗=PCI-μTx∗,where PC is the metric projection from H onto C, μ>0 is a constant, and I is the identity mapping from H into itself. It is well known that if the mapping T is k-Lipschitzian and η-strongly monotone, then the operator PC(I-μT) is a contraction on C provided that 0<μ<2η/k2. In this case, the Banach contraction principle guarantees that VI(C,T) has a unique solution x∗ and the sequence of Picard iteration method given by(3)xn+1=PCI-μTxn∀n∈Nconverges strongly to x∗. 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 asS-iteration process as an answer to Question 1: let C be a nonempty convex subset of a normed linear space X, and let T:C→C be an operator. Then, for arbitrary x1∈C, the S-iteration process is defined by(4)xn+1=1-αnTxn+αnTyn,yn=1-βnxn+βnTxn∀n∈N,where {αn} and {βn} are real sequences in (0,1) satisfying some suitable conditions. In [4], Sahu proved that the rate of convergence of S-iteration process for contraction mappings is faster than that of Picard [5] and Mann [6] iteration processes by providing a numerical example. The S-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 S-iteration process. Due to its fastness, in recent years, the S-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 S-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 S-operator of a mapping T generated by α∈(0,1) and T and normal S-iteration process in the following way: let C be a nonempty convex subset of a normed linear space X and let T:C→C be an operator. Then, for arbitrary x1∈C, the normal S-iteration process is defined by(5)xn+1=T1-αnxn+αnTxn∀n∈N,where {αn} is a sequence of real numbers in (0,1). In 2017, Verma and Shukla [12] designed some new algorithms based on S-iteration processes and named them as S-iteration-based forward-backward algorithm (SFBA) and normal S-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:(6)xn+1=PCαnγVxn+I-αnμFTxn∀n∈N,where F:C→H is k-Lipschitzian and η-strongly monotone operator with k>0, η>0, V:C→H is an L-Lipschitzian mapping with L≥0, T:C→C is a nonexpansive mapping with Fix(T)≠∅, {αn}⊂(0,1), and x1∈C an arbitrary initial point. They proved that the sequence {xn} generated by the iterative method (6) converges strongly to a fixed point x∗ of T which solves the following variational inequality problem: (7)μF-γVx∗,x-x∗≥0∀x∈C≔FixT.In 2001, Verma [14] generalized the concept of variational inequalities to a system of nonlinear variational inequalities (SNVI) in the following way: find x∗,y∗∈C such that(8)ρTy∗+x∗-y∗,x-x∗≥0,∀x∈C,ηTx∗+y∗-x∗,x-y∗≥0,∀x∈C,where T:C→H is any mapping and η>0 and ρ>0 are constants. To solve (8), he introduced the following iterative method:(9)yn=PCxn-ηTxn,xn+1=1-αnxn+αnPCyn-ρTyn,and he proved that the sequences {xn} and {yn} generated by (9) converge to x∗ and y∗, 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 C be a nonempty closed convex subset of a real Hilbert space H and let T:C→H be a nonlinear mapping. For arbitrary chosen initial point x1∈C, let {xn} and {yn} be the sequences in C defined by (10)yn=1-βnxn+βnPCxn-ηTxn,xn+1=1-αnxn+αnPCyn-ρTyn,where η>0, ρ>0 and 0≤αn, βn≤1. Further, problem (8) is equivalent to the following projection formulas:(11)x∗=PCI-ρTy∗,y∗=PCI-ηTx∗,for a monotone mapping T:C→H. 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 S-operator and normal S-iteration process, he [26] introduced a parallel S-iteration process for finding altering points of nonlinear mappings as follows.LetC1 and C2 be two nonempty closed convex subsets of a Banach space X and let T1:C1→C2 and T2:C2→C1 be two mappings. Then, for α∈(0,1) and arbitrary x1,y1∈C1×C2, the parallel normal S-iteration process is defined by(12)xn+1=T21-αyn+αT1xn,yn+1=T11-αxn+αT2yn∀n∈N.The following convergence result is given in [26].Theorem 1 (see [26]).
LetC1 and C2 be two nonempty closed convex subsets of a Banach space X. Let T1:C1→C2 and T2:C2→C1 be two Lipschitz continuous mappings with Lipschitz constants k1<1 and k2<1, respectively. Then the sequence {(xn,yn)} in C1×C2 generated by the parallel S-iteration process (12) converges strongly to a unique point (x∗,y∗)∈C1×C2 such that x∗ and y∗ are altering points of mappings T1 and T2.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 S-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 symbolN stands for the set of all natural numbers.LetC be a nonempty subset of a real Hilbert space H with inner product ·,· and norm ·, respectively. A mapping T:C→H is called(1)
monotone if (13)Tx-Ty,x-y≥0∀x,y∈C,(2)
η-strongly monotone if there exists a positive real number η such that(14)Tx-Ty,x-y≥ηx-y2∀x,y∈C,(3)
k-Lipschitzian if there exists a constant k≥0 such that (15)Tx-Ty≤kx-y∀x,y∈C,(4)
θ-contraction if there exists a constant θ∈0,1 such that (16)Tx-Ty≤θx-y∀x,y∈C,(5)
nonexpansive if (17)Tx-Ty≤x-y∀x,y∈C.Definition 2 (see [26]).
LetC1 and C2 be two nonempty subsets of a metric space X. Then x∗∈C1 and y∗∈C2 are altering points of mappings T1:C1→C2 and T2:C2→C1 if (18)T1x∗=y∗,T2y∗=x∗.
The set of altering points of mappingsT1 and T2 is denoted and defined by (19)AltT1,T2=x∗,y∗∈C1×C2:T1x∗=y∗,T2y∗=x∗.We now give some numerical examples in support of the definition of altering points of some nonlinear mappings as follows.Example 3 (see [26]).
LetX=[0,1], C1=[0,1/2], and C2=[1/2,1]. Define T1:C1→C2 and T2:C2→C1 by Ti(x)=1-x for i=1,2. Note that T2T1:C1→C1 is defined by T2T1x=T2(1-x)=x. Thus, each point of C1 is a fixed point of T2T1. Then altering points x∗∈C1 and y∗∈C2 of T1 and T2 are given by the relation x∗+y∗=1. Indeed, (20)AltT1,T2=x∗,y∗∈C1×C2:x∗+y∗=1.Example 4.
LetX=R2, C1={(x,y)∈R2:x≥0,y≥0}, and C2={(x,y)∈R2:x≤0,y≤0}. Let T1:C1→C2 and T2:C2→C1 be two mappings defined, respectively, by (21)T1x,y=-x,-y-1∀x,y∈C1,T2x,y=1-x2,1-y2∀x,y∈C2.Note that T2T1:C1→C1 is defined by T2T1(x,y)=T2(-x,-y-1)=((x+1)/2,(y+2)/2). Clearly (1,2)∈C1 and (-1,-3)∈C2 are fixed points of T2T1 and T1T2, respectively. Therefore, x∗=(1,2) and y∗=(-1,-3) are altering points of mappings T1 and T2.LetC be a nonempty closed convex subset of H. Then, for any x∈H, there exists a unique nearest point PC(x) of C such that (22)x-PCx≤x-y∀y∈C.The mapping PC is called the metric projection [27] from H onto C. It is remarkable that the metric projection mapping PC is nonexpansive from H onto C (see Agarwal et al. [28]).We need the following technical lemmas.Lemma 5 (see [28]).
LetC be a nonempty closed convex subset of a real Hilbert space H and let PC be the metric projection from H onto C. Given x∈H and z∈C, then z=PC(x) if and only if x-z,z-y≥0 for all y∈C.Lemma 6 (see [29]).
LetC be a nonempty subset of a real Hilbert space H. Suppose that λ∈0,1 and μ>0. Let F:C→H be a k-Lipschitzian and η-strongly monotone operator. Define the mapping Tλ:C→H by (23)Tλx=I-λμFx∀x∈C,λ∈0,1.Then Tλ is a contraction provided 0<μ<2η/k2. More precisely, for μ∈(0,2η/k2), (24)Tλx-Tλy≤1-λτx-y∀x,y∈C,where τ=1-1-μ(2η-μk2)∈(0,1].Lemma 7 (see [17]).
Let{an}, {bn}, and {cn} be three nonnegative real sequences satisfying the following conditions: (25)an+1≤1-αnan+bn+cn∀n≥n0,where n0 is some nonnegative integer, αn∈(0,1) with ∑n=0∞αn=∞, bn=∘(αn), and ∑n=0∞cn<∞. Then limn→∞⁡an=0.Lemma 8 (see [26, Theorem  3.1]).
LetC1 and C2 be two nonempty closed subsets of a complete metric space X. Suppose that S1:C1→C2 and S2:C2→C1 be two Lipschitz continuous mappings with Lipschitz constants k1 and k2, respectively, such that k1k2<1. Then the following holds:(a)
There exists a unique point(x∗,y∗)∈C1×C2 such that x∗ and y∗ are altering points of mappings S1 and S2.(b)
For arbitraryx1∈C1, a sequence {(xn,yn)}∈C1×C2 generated by (26)yn=S1xn,xn+1=S2yn∀n∈Nconverges to(x∗,y∗).


## 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.LetC1 and C2 be nonempty closed convex subsets of a real Hilbert space H and let T1:C1→C2 and T2:C2→C1 be some mappings. Let g1,g2:H→H be mappings. Consider a general system of generalized variational inequalities (SGVI) defined on C1 and C2 as follows.Find(x∗,y∗)∈C1×C2 such that(27)sT1x∗+y∗-g1x∗,g1y-y∗≥0,∀y∈C2,tT2y∗+x∗-g2y∗,g2x-x∗≥0,∀x∈C1,where s>0 and t>0 are constants.Remark 9.
IfT1=T2=T, g1=g2=I, and C1=C2=C, 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.Letgi:H→H be single-valued δi-strongly monotone, ηi-Lipschitz continuous, let Fi:Ci→H be ki-Lipschitzian and ξi-strongly monotone operator with constants ki,ξi>0, and let Vi:Ci→H be Li-Lipschitzian mapping with constant Li≥0 for i∈{1,2}. Suppose that 0<μi<2ξi/ki2 and 0≤γi<τi/Li, where τi=1-1-μi(2ξi-μiki2) for i∈{1,2}.IfTi=μiFi-γiVi for i=1,2, then the system of generalized variational inequalities (27) reduces to the following system of generalized variational inequalities (SGVI).Find(x∗,y∗)∈C1×C2 such that(28)sμ1F1-γ1V1x∗+y∗-g1x∗,g1y-y∗≥0,∀y∈C2,tμ2F2-γ2V2y∗+x∗-g2y∗,g2x-x∗≥0,∀x∈C1.Define the mappings S1:C1→C2 and S2:C2→C1 by(29)S1≔PC2g1-sμ1F1-γ1V1(30)S2≔PC1g2-tμ2F2-γ2V2,where s and t are some constants in 0,1. Using Lemma 5, one can easily observe that the SGVI (28) is equivalent to the following altering point formulation:(31)to  findx∗,y∗∈C1×C2such  thatx∗=PC1g2-tμ2F2-γ2V2y∗,y∗=PC2g1-sμ1F1-γ1V1x∗.First we introduce parallel Mann iteration process to solve system of generalized variational inequalities (28) as follows.Algorithm 10.
For any given(x1,y1)∈C1×C2, let {(xn,yn)} be an iterative sequence in C1×C2 defined by (32)xn+1=1-αnxn+αnS2yn,yn+1=1-αnyn+αnS1xn,∀n∈N,where {αn} is a sequence in [0,1] and S1 and S2 are defined by (29) and (30), respectively.Motivated by Sahu [26] and equivalent formulation (31), we now propose a more general parallel S-iteration process to solve SGVI (28) as follows.Algorithm 11.
For any given(x1,y1)∈C1×C2, let {(xn,yn)} be an iterative sequence in C1×C2 defined by (33)xn+1=S21-αnyn+αnS1xn,yn+1=S11-αnxn+αnS2yn,∀n∈N,where {αn} is a sequence in (0,1) and S1 and S2 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.
LetC1 and C2 be nonempty closed convex subsets of a real Hilbert space H. Let gi:H→H be single-valued δi-strongly monotone, ηi-Lipschitz continuous, let Fi:Ci→H be ki-Lipschitzian and ξi-strongly monotone operator with constants ki,ξi>0, and let Vi:Ci→H be Li-Lipschitzian mapping with constant Li≥0 for i∈{1,2}. Suppose that 0<μi<2ξi/ki2 and 0≤γi<τi/Li, where τi=1-1-μi(2ξi-μiki2) for i∈{1,2}. Let s,t∈(0,1] and let θ1 and θ2 be real constants defined by(34)θi=1-2δi+ηi2fori=1,2.Then the mappings S1 and S2 defined by (29) and (30) are Lipschitz continuous with Lipschitz constants [θ1+1-s(τ1-γ1L1)] and [θ2+1-t(τ2-γ2L2)], respectively.Proof.
Letx,y∈C1. Then, we have(35)S1x-S1y=PC2g1-sμ1F1-γ1V1x-PC2g1-sμ1F1-γ1V1y≤g1x-sμ1F1-γ1V1x-g1y+sμ1F1-γ1V1y=g1x-g1y-x-y+x-y-sμ1F1-γ1V1x-y≤g1x-g1y-x-y+x-y-sμ1F1-γ1V1x-y≤x-y-g1x-g1y+x-sμ1F1x-y+sμ1F1y+sγ1V1x-sγ1V1y≤x-y-g1x-g1y+I-sμ1F1x-I-sμ1F1y+sγ1V1x-V1y≤x-y-g1x-g1y+1-sτ1x-y+sγ1L1x-y=x-y-g1x-g1y+1-sτ1-γ1L1x-y,x-y-g1x-g1y2=x-y2-2x-y,g1x-g1y+g1x-g1y2≤x-y2-2δ1x-y2+η12x-y2=1-2δ1+η12x-y2=θ12x-y2.From (35), we have(36)S1x-S1y≤θ1x-y+1-sτ1-γ1L1x-y=θ1+1-sτ1-γ1L1x-y.Thus S1 is [θ1+1-s(τ1-γ1L1)]-Lipschitz continuous.
Similarly, we can show thatS2 is [θ2+1-t(τ2-γ2L2)]-Lipschitz continuous.Now we are ready to present our main results. First we establish the convergence analysis of Algorithm10 for solving SGVI (28).Theorem 13.
LetC1 and C2 be nonempty closed convex subsets of a real Hilbert space H. Let gi:H→H be single-valued δi-strongly monotone, ηi-Lipschitz continuous, let Fi:Ci→H be ki-Lipschitzian and ξi-strongly monotone operator with constants ki,ξi>0, and let Vi:Ci→H be Li-Lipschitzian mapping with constant Li≥0 for i∈{1,2}. Suppose that 0<μi<2ξi/ki2 and 0≤γi<τi/Li, where τi=1-1-μi(2ξi-μiki2) for i∈{1,2}. Let s,t∈(0,1] and let θ1 and θ2 be real constants defined by (34). Let S1 and S2 be defined by (29) and (30), respectively. For given initial point (x1,y1)∈C1×C2, let {(xn,yn)} be an iterative sequence defined by parallel Mann iteration process (32), where {αn} is a sequence in [0,1] such that ∑n=1∞αn=∞. Assume that the following condition is satisfied:(37)θ1s<τ1-γ1L1,θ2t<τ2-γ2L2.Then we have the following:(i)
There exists a unique point(x∗,y∗)∈C1×C2, which solves SGVI (28).(ii)
The sequence{(xn,yn)} generated by parallel Mann iteration process (32) converges strongly to the point (x∗,y∗).Proof.
(i) It follows from Lemma8 and (31).
(ii) By (31), (32), and Proposition 12, we have(38)xn+1-x∗=1-αnxn+αnS2yn-1-αnx∗-αnx∗≤1-αnxn-x∗+αnS2yn-x∗=1-αnxn-x∗+αnS2yn-S2y∗≤1-αnxn-x∗+αnθ2+1-tτ2-γ2L2yn-y∗.Again, by Proposition 12 that S1:C1→C2 is [θ1+1-s(τ1-γ1L1)]-Lipschitz continuous and using (31) and (32), we have(39)yn+1-y∗≤1-αnyn-y∗+αnθ1+1-sτ1-γ1L1xn-x∗.Set (40)θ=max⁡θ1+1-sτ1-γ1L1,θ2+1-tτ2-γ2L2.
From (38) and (39), we get(41)xn+1-x∗+yn+1-y∗≤1-αnxn-x∗+αnθ2+1-tτ2-γ2L2yn-y∗+1-αnyn-y∗+αnθ1+1-sτ1-γ1L1xn-x∗≤1-αnxn-x∗+αnθyn-y∗+1-αnyn-y∗+αnθxn-x∗=1-αn1-θxn-x∗+yn-y∗.
Now, we define the norm·1 on H×H by (x,y)1=x+y for all x,y∈H×H. Therefore, using (41), we have (42)xn+1,yn+1-x∗,y∗1=xn+1-x∗,yn+1-y∗1=xn+1-x∗+yn+1-y∗≤1-αn1-θxn-x∗+yn-y∗=1-αn1-θxn,yn-x∗,y∗1.Noticing that ∑n=1∞αn=∞ and θ∈(0,1). Therefore, from Lemma 7, we have limn→∞⁡(xn,yn)-(x∗,y∗)1=0. Thus, we get limn→∞⁡xn-x∗=limn→∞⁡yn-y∗=0 and hence {xn} and {yn} converge to x∗ and y∗, respectively.Corollary 14.
LetC1 and C2 be nonempty closed convex subsets of a real Hilbert space H. Let Fi:Ci→H be ki-Lipschitzian and ξi-strongly monotone operator with constants ki,ξi>0 and let Vi:Ci→H be Li-Lipschitzian mapping with constant Li≥0 for i∈{1,2}. Suppose that 0<μi<2ξi/ki2 and 0≤γi<τi/Li, where τi=1-1-μi(2ξi-μiki2) for i∈{1,2}. Let s,t∈(0,1]. Define mappings S1:C1→C2 and S2:C2→C1 by(43)S1≔PC2I-sμ1F1-γ1V1,S2≔PC1I-tμ2F2-γ2V2.For given initial point (x1,y1)∈C1×C2, let {(xn,yn)} be an iterative sequence in C1×C2 defined by(44)xn+1=1-αnxn+αnS2yn,yn+1=1-αnyn+αnS1xn,∀n∈N,where {αn} is a real sequence in [0,1] such that ∑n=1∞αn=∞. Assume that condition (37) of Theorem 13 is satisfied. Then the sequence {(xn,yn)} generated by (44) converges strongly to the unique point (x∗,y∗), which solves system of generalized variational inequalities (45)sμ1F1-γ1V1x∗+y∗-x∗,y-y∗≥0,∀y∈C2,tμ2F2-γ2V2y∗+x∗-y∗,x-x∗≥0,∀x∈C1.Proof.
The proof follows from Theorem13 by taking g1=g2=I.Now we study the convergence analysis of Algorithm11, that is, the parallel S-iteration process defined by (33) for solving SGVI (28).Theorem 15.
LetC1 and C2 be nonempty closed convex subsets of a real Hilbert space H. Let gi:H→H be single-valued δi-strongly monotone, ηi-Lipschitz continuous, let Fi:Ci→H be ki-Lipschitzian and ξi-strongly monotone operator with constants ki,ξi>0, and let Vi:Ci→H be Li-Lipschitzian mapping with constant Li≥0 for i∈{1,2}. Suppose that 0<μi<2ξi/ki2 and 0≤γi<τi/Li, where τi=1-1-μi(2ξi-μiki2) for i∈{1,2}. Let s,t∈(0,1] and let θ1 and θ2 be real constants defined by (34). Let S1 and S2 be defined by (29) and (30), respectively. For given initial point (x1,y1)∈C1×C2, let {(xn,yn)} be an iterative sequence in C1×C2 defined by parallel S-iteration process (33), where {αn} is a sequence in (0,1). Assume that condition (37) of Theorem 13 is satisfied. Then we have the following:(i)
There exists a unique point(x∗,y∗)∈C1×C2, which solves SGVI (28).(ii)
The sequence{(xn,yn)} generated by parallel S-iteration process (33) converges strongly to the point (x∗,y∗).Proof.
(i) It follows from Lemma8 and (31).
(ii) From (31), (33), and Proposition 12, we have(46)yn+1-y∗=S11-αnxn+αnS2yn-S1x∗≤θ1+1-sτ1-γ1L11-αnxn+αnS2yn-x∗≤θ1+1-sτ1-γ1L11-αnxn-x∗+αnS2yn-x∗=θ1+1-sτ1-γ1L11-αnxn-x∗+αnS2yn-S2y∗≤θ1+1-sτ1-γ1L11-αnxn-x∗+αnθ2+1-tτ2-γ2L2yn-y∗.Similarly(47)xn+1-x∗≤θ2+1-tτ2-γ2L21-αnyn-y∗+αnθ1+1-sτ1-γ1L1xn-x∗.Set (48)θ=max⁡θ1+1-sτ1-γ1L1,θ2+1-tτ2-γ2L2.From (46) and (47), we get(49)xn+1-x∗+yn+1-y∗≤θ2+1-tτ2-γ2L21-αnyn-y∗+αnθ1+1-sτ1-γ1L1xn-x∗+θ1+1-sτ1-γ1L11-αnxn-x∗+αnθ2+1-tτ2-γ2L2yn-y∗≤θ1-αnyn-y∗+αnθxn-x∗+θ1-αnxn-x∗+αnθyn-y∗=θ1-αn1-θxn-x∗+θ1-αn1-θyn-y∗=θ1-αn1-θxn-x∗+yn-y∗.Now, we define the norm ·1 on H×H by (x,y)1=x+y for all (x,y)∈H×H. Therefore, using (49), we have (50)xn+1,yn+1-x∗,y∗1=xn+1-x∗,yn+1-y∗1=xn+1-x∗+yn+1-y∗≤θ1-αn1-θxn-x∗+yn-y∗=θ1-αn1-θxn,yn-x∗,y∗1.Since θ(1-αn(1-θ))≤θ<1, we obtain that limn→∞⁡(xn,yn)-(x∗,y∗)1=0. Thus, we get limn→∞⁡xn-x∗=limn→∞⁡yn-y∗=0 and hence {xn} and {yn} converge to x∗ and y∗, respectively.Corollary 16.
LetC1 and C2 be nonempty closed convex subsets of a real Hilbert space H. Let Fi:Ci→H be ki-Lipschitzian and ξi-strongly monotone operator with constants ki,ξi>0 and let Vi:Ci→H be Li-Lipschitzian mapping with constant Li≥0 for i∈{1,2}. Suppose that 0<μi<2ξi/ki2 and 0≤γi<τi/Li, where τi=1-1-μi(2ξi-μiki2) for i∈{1,2}. Let s,t∈(0,1] and let S1 and S2 be defined by (43). For given initial point (x1,y1)∈C1×C2, let {(xn,yn)} be an iterative sequence in C1×C2 defined by(51)xn+1=S21-αnyn+αnS1xn,yn+1=S11-αnxn+αnS2yn,∀n∈N,where {αn} is a real sequence in (0,1). Assume that condition (37) of Theorem 13 is satisfied. Then the sequence {(xn,yn)} generated by (51) converges strongly to the unique point (x∗,y∗), which solves system of generalized variational inequalities (52)sμ1F1-γ1V1x∗+y∗-x∗,y-y∗≥0,∀y∈C2,tμ2F2-γ2V2y∗+x∗-y∗,x-x∗≥0,∀x∈C1.Proof.
The proof follows from Theorem15 by taking g1=g2=I.


## 4. Numerical Example

In this section, we discuss an example which leads to Theorems13 and 15. The graphs are also presented for showing how the sequences {xn} and {yn} generated by both the algorithms, Algorithms 10 and 11, converge to the solutions of SGVI (28).Example 17.
LetH=R, C1=(-∞,0], and C2=[0,∞). Let g1 and g2 be two mappings from H onto itself defined by g1(x)=(2x-3)/3 for all x∈H and g2(x)=(5x-10)/6 for all x∈H, respectively. Let F1:C1→H and F2:C2→H be two mappings defined by F1(x)=2x-3 for all x∈C1 and F2(x)=3x-2 for all x∈C2, respectively. Let V1:C1→H and V2:C2→H be two mappings defined by V1(x)=1-4x for all x∈C1 and V2(x)=6-6x for all x∈C2, respectively. Then gi is δi-strongly monotone and ηi-Lipschitzian mapping for i∈{1,2}. We have δ1=2/3=η1 and δ2=5/6=η2. Also Fi is ξi-strongly monotone and ki-Lipschitzian mapping for i∈{1,2}. We have ξ1=2=k1 and ξ2=3=k2. Moreover Vi is Li-Lipschitzian mapping for i∈{1,2}. We have L1=4 and L2=6. We take μ1=1/2, μ2=1/3, τ1=1=τ2 and γ1=1/12, γ2=5/72. Define {αn} in [0,1] by αn=n/(n+1), T1=μ1F1-γ1V1, and T2=μ2F2-γ2V2. Then T1(x)=(16x-19)/12 and T2(x)=(51x-39)/36.
ThereforeS1 and S2 can be expressed as (53)S1≔PC2g1-sT1,S2≔PC1g2-tT2,where s>0 and t>0 are constants.Lets=1 and t=1. Then, (54)S1x=PC2-8x+712=7-8x12∀x∈C1,S2x=PC1-7x-712=-7x+712∀x∈C2.It can be easily seen that S1:C1→C2 is (2/3)-Lipschitzian and S2:C2→C1 is (7/12)-Lipschitzian. Also (55)θ1s-τ1-γ1L1=1-2δ1+η12-τ1-γ1L1=-13<0,θ2t-τ2-γ2L2=1-2δ2+η22-τ2-γ2L2=-512<0.One can observe that all the conditions of Theorems 13 and 15 are satisfied.Now we will find the general term of the sequences{xn} and {yn} generated by the iteration process (32). For arbitrary x1∈C1 and y1∈C2, (56)xn+1=1-αnxn+αnS2yn=xnn+1-n7yn+712n+1=112n+112xn-7nyn-7n,yn+1=1-αnyn+αnS1xn=ynn+1+n7-8xn12n+1=112n+112yn-8nxn+7n.Hence(57)xn+1=112n+112xn-7nyn-7n,yn+1=112n+112yn-8nxn+7n∀n∈N.Also, we will find the general term of the sequences {xn} and {yn} generated by the iteration process (33). For arbitrary x1∈C1 and y1∈C2, (58)xn+1=S21-αnyn+αnS1xn=S2ynn+1+n7-8xn12n+1=1144n+156nxn-84yn-133n-84,yn+1=S11-αnxn+αnS2yn=S1xnn+1-n7yn+712n+1=1144n+156nyn-96xn+140n+84.Hence(59)xn+1=1144n+156nxn-84yn-133n-84,yn+1=1144n+156nyn-96xn+140n+84∀n∈N.It is clear from (57) and (59) that the sequences {xn} and {yn} generated by the proposed iterative algorithms converge to the altering points x∗∈C1 and y∗∈C2 of the mappings S1 and S2, where x∗=-1.511363636363637 and y∗=1.590909090909091. The numerical values of {xn} and {yn} have been calculated for different starting values of x1 and y1 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 ofxn and yn.
n
Parallel Mann iteration process
ParallelS-iteration process
x
n
y
n
x
n
y
n
1
-
4.000000000000000
4.000000000000000
-
4.000000000000000
4.000000000000000
7
-
1.847585060258671
1.950379984812855
-
1.531137975937710
1.612050811863993
14
-
1.529511272915085
1.610309552955496
-
1.511402781884560
1.590950939614087
21
-
1.512204443955688
1.591807954179574
-
1.511363702883595
1.590909162022088
28
-
1.511400117495092
1.590948090842850
-
1.511363636469492
1.590909091022255
35
-
1.511365162111111
1.590910722002483
-
1.511363636363799
1.590909090909265
42
-
1.511363698691403
1.590909157540259
-
1.511363636363637
1.590909090909091
49
-
1.511363638868444
1.590909093586843
-
1.511363636363637
1.590909090909091
56
-
1.511363636463094
1.590909091015415
-
1.511363636363636
1.590909090909091
63
-
1.511363636367549
1.590909090913274
-
1.511363636363637
1.590909090909091
70
-
1.511363636363790
1.590909090909254
-
1.511363636363637
1.590909090909091
77
-
1.511363636363642
1.590909090909097
-
1.511363636363637
1.590909090909091
84
-
1.511363636363637
1.590909090909091
-
1.511363636363636
1.590909090909091
91
-
1.511363636363637
1.590909090909091
-
1.511363636363637
1.590909090909091
98
-
1.511363636363637
1.590909090909091
-
1.511363636363637
1.590909090909091Table 2
Numerical values ofxn and yn.
n
Parallel Mann iteration process
ParallelS-iteration process
x
n
y
n
x
n
y
n
1
-
5.000000000000000
10.000000000000000
-
5.000000000000000
10.000000000000000
7
-
2.316760667744464
2.451261819028022
-
1.558731808819676
1.641509397106758
14
-
1.554814798532934
1.637363641212413
-
1.511457370603911
1.591009289899863
21
-
1.513376896656526
1.593061316754131
-
1.511363795641447
1.590909261181073
28
-
1.511450986841399
1.591002473172358
-
1.511363636617098
1.590909091180052
35
-
1.511367289637843
1.590912996409872
-
1.511363636364025
1.590909090909507
42
-
1.511363785601782
1.590909250451687
-
1.511363636363637
1.590909090909092
49
-
1.511363642361179
1.590909097320727
-
1.511363636363637
1.590909090909091
56
-
1.511363636601778
1.590909091163675
-
1.511363636363636
1.590909090909091
63
-
1.511363636373005
1.590909090919107
-
1.511363636363637
1.590909090909091
70
-
1.511363636364002
1.590909090909482
-
1.511363636363637
1.590909090909091
77
-
1.511363636363651
1.590909090909106
-
1.511363636363637
1.590909090909091
84
-
1.511363636363637
1.590909090909092
-
1.511363636363636
1.590909090909091
91
-
1.511363636363637
1.590909090909091
-
1.511363636363637
1.590909090909091
98
-
1.511363636363637
1.590909090909091
-
1.511363636363637
1.590909090909091Figure 1
Convergence of sequences{xn} and {yn} generated by parallel Mann iteration process (32).Figure 2
Convergence of sequences{xn} and {yn} generated by parallel S-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, Algorithms10 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 S1 and S2.



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