# Convergence of Some Iterative Algorithms for System of Generalized Set-Valued Variational Inequalities

Authors: M. Akram; Aysha Khan; M. Dilshad
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this article, we consider and study a system of generalized set-valued variational inequalities involving relaxed cocoercive mappings in Hilbert spaces. Using the projection method and Banach contraction principle, we prove the existence of a solution for the considered problem. Further, we propose an iterative algorithm and discuss its convergence. Moreover, we establish equivalence between the system of variational inequalities and altering points problem. Some parallel iterative algorithms are proposed, and the strong convergence of the sequences generated by these iterative algorithms is discussed. Finally, a numerical example is constructed to illustrate the convergence analysis of the proposed parallel iterative algorithms.


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## Body

## 1. Introduction

The theory of variational inequality was planted in the early 1960’s by Stampacchia [1] in the framework of obstacle constraint minimization problems. The first evolutionary variational inequality was solved in the seminal paper of Lions and Stampacchia [2]. Since its inception, it has enjoyed a vigorous development for the last few decades. This subject has developed in multiple directions using innovative techniques to solve fundamental problems to be insurmountable previously. This field is influential and experiencing an explosive growth in theory as well as applications. Consequently, some of these developments in this area enriched other mutual areas of mathematical and engineering sciences such as economics, transportation, nonlinear programming, and operations research. It has been shown that this theory provides the most natural, direct, simple, unified, and efficient framework for a general treatment of a wide class of unrelated linear and nonlinear problems; see, for example, [3–9] and the references cited therein.Recently, fixed-point methods have been extensively investigated for solving monotone variational inequalities. Among the fixed point algorithms, Mann-like iterative algorithms are useful for solving several nonlinear problems. The proximal point algorithm is a widely used tool for solving a variety of (single objective) convex optimization problems such as finding zeros of maximal monotone operators and fixed points of nonexpansive mappings, as well as minimizing convex functions. In 2007, Agarwal et al. [10] introduced the following iteration process. Let C be a nonempty convex subset of a normed linear space X and let T:C→C be an operator. Then, for arbitrary x0∈C, estimate xn∈C by the following S-iterative scheme:(1)xn+1=1−αnTxn+αnTyn,yn=1−βnxn+βnTxn,where αn and βn are real sequences in (0,1) satisfying some suitable conditions. Sahu [11] proved that the S-iteration process is more applicable than the Picard [12], Mann [13], and Ishikawa [14] iteration algorithms because it converges faster than these iteration processes for contraction mappings and also works for nonexpansive mappings. Recently, Cholamjiak et al. [15] applied the S-iteration process for finding a minimizer of a convex function and fixed points of nonexpansive mappings in CAT(0) spaces. In [16], Amir et al. studied the convex constraint multiobjective optimization problem as the constrained set of fixed points of the nonexpansive mapping.On the other hand, a number of numerical methods such as auxiliary principle, projection method, Wiener-Hopf equations, dynamical systems, and decomposition have been developed for solving the variational inequalities and related optimization problems. Among these methods, the projection method and its variant forms have been proved innovative and important tools for finding the approximate solutions of variational inequalities. In this technique, the concept of projection is used for the fixed-point formulation of variational inequality. This alternative formulation has played a significant role in developing various projection-type methods for solving variational inequalities. It is well known that the convergence of the projection methods requires that the operator must be strongly monotone and Lipschitz continuous. It is also known that the relaxed cocoercive mappings are more general than strongly monotone mappings, and under some mild conditions, relaxed cocoercive mappings can be reduced to strongly monotone mappings. For some related works, see, [17–24].Following the facts and discussion mentioned above, in this paper, we consider a system of generalized set-valued variational inequalities defined over closed and convex subsets of a real Hilbert space. We establish an equivalence between the system of generalized set-valued variational inequalities and nonlinear projection equations using the projection technique. Further, by virtue of the projection method and Banach contraction principle, we prove an existence result. Furthermore, we propose an iterative algorithm and show that the approximate solution generated by the proposed algorithm converges strongly to the unique solution of the system of generalized set-valued variational inequalities involving relaxed cocoercive and relaxed monotone mappings in Hilbert space. Moreover, we establish equivalence between the system of variational inequalities and altering points problem. Parallel Mann and parallelS-iterative algorithms [11] have been proposed for solving the considered system of variational inequalities. Finally, the convergence analysis of the proposed parallel iterative algorithms is discussed. A numerical example is constructed to illustrate the convergence analysis of the proposed parallel iterative algorithms. The results presented in this paper can be viewed as generalizations and refinements of several results existing in the literature and include general variational inequality and some other classes of variational inequalities as special cases.Now, we enumerate some basic notions, definitions, and results that are worthwhile tools in succeeding analysis and will be utilized in the rest of this paper.Throughout the paper, unless otherwise specified, letH be a real Hilbert space with inner product ·,· and induced norm ∥·∥.Definition 1 [25].
A mappingT:H→H is said to be(i)
γ-strongly monotone, if there exists a constant γ>0 such that(2)Tx−Ty,x−y≥γ∥x−y∥2,∀x,y∈H.(ii)
μ1-cocoercive, if there exists a constant μ1>0 such that(3)Tx−Ty,x−y≥μ1∥Tx−Ty∥2,∀x,y∈H.(iii)
Relaxedμ2-cocoercive, if there exists a constant μ2>0 such that(4)Tx−Ty,x−y≥−μ2∥Tx−Ty∥2,∀x,y∈H.(iv)
Relaxedμ,γ-cocoercive, if there exist constants μ,γ>0 such that(5)Tx−Ty,x−y≥−μ∥Tx−Ty∥2+γ∥x−y∥2,∀x,y∈H.(v)
L-Lipschitz continuous, if there exists a constant L>0 such that(6)∥Tx−Ty∥≤L∥x−y∥,∀x,y∈H.(vi)
α-contraction, if there exists a constant 0<α<1 such that(7)∥Tx−Ty∥≤α∥x−y∥,∀x,y∈H.T is called nonexpansive, if α=1.Remark 2.
We remark that everyγ-strongly monotone mapping is relaxed μ,γ-cocoercive mapping and every γ-cocoercive mapping is 1/γ-Lipschitz continuous.
Define the norm∥·∥∗ on H×H by(8)∥p,q∥∗=∥p∥+∥q∥,∀p,q∈H.
Note thatH×H,∥·∥∗ is a Banach space.Definition 3 [26].
LetT:H→H be a single-valued mapping. A set-valued mapping G:H→2H is said to be a relaxed monotone with respect to T if and only if there exists a constant c>0 such that(9)Tu−Tv,x−y≥−c∥x−y∥2,∀x,y∈Handforsomeu∈Gx,v∈Gy.
LetΩ be a nonempty closed and convex subset of a real Hilbert space H. Then, for any x∈H, there exists a unique nearest point PΩx of Ω such that(10)∥x−PΩx∥≤∥x−y∥,∀y∈Ω.
The mappingPΩ is called the metric projection [27] from H onto Ω. Note that the metric projection mapping PΩ is nonexpansive from H onto Ω (see, Agarwal et al. [28]), i.e.,(11)∥PΩx−PΩy∥≤∥x−y∥,∀x,y∈H.Lemma 4 [27].
LetΩ be a closed and convex subset in H. Then, for any s∈H, the projection PΩ:H→Ω of H onto Ω satisfies(12)t−s,r−t≥0,∀r∈Ω,if and only if(13)t=PΩs,∀t∈Ω.Lemma 5.
[29]. Let ωn be a nonnegative real sequence satisfying the following condition(14)ωn+1≤1−αnωn+εn,∀n≥n0,where αn∈0,1, ∑n=0∞αn=∞, and εn=oαn. Then, limn→∞ωn=0.


## 2. System of Generalized Set-Valued Variational Inequalities

LetH be a real Hilbert space and Ω1 and Ω2 be the nonempty closed and convex subsets of H. Let I=1,2 be an index set; for each i∈I, let φi,ψi:H→H;Mi:H×H→H be nonlinear single-valued mappings and Pi:H→CBH be a set-valued mapping. We consider the problem of finding x,y∈Ω1×Ω2 with u∈P1x and v∈P2y such that(15)ϱ1M1u,v+φ1y−ψ1x,ψ1y∗−φ1y≥0,∀y∗∈Ω2,ϱ2M2v,u+φ2x−ψ2y,ψ2x∗−φ2x≥0,∀x∗∈Ω1,where ϱ1 and ϱ2 are positive constants. We call problem (15) a system of generalized set-valued variational inequalities. Some special cases of problem (15) are listed below.(i)
IfPi:H→H are the single-valued mappings and φi=I are the identity mappings, then problem (15) reduces to the equivalent problem of finding x,y∈Ω1×Ω2 such that(16)ϱ1M1y,x+y−ψ1x,ψ1y∗−y≥0,∀y∗∈Ω2,ϱ2M2x,y+x−ψ2y,ψ2x∗−x≥0,∀x∗∈Ω1.Problem (16) is called the system of variational inequalities.(ii)
IfMi·,·=Mi·, then problem (16) coincides with the following problem of finding x,y∈H×H such that(17)ϱ1M1x+y−ψ1x,ψ1y∗−y≥0,∀y∗∈Ω2,ϱ2M2y+x−ψ2y,ψ2x∗−x≥0,∀x∗∈Ω1.Problem (17) was studied by Sahu et al. [11].(iii)
IfM1=M2=T,ψ1=ψ2=I and Ω1=Ω2=C, then problem (17) reduces to the following system of nonlinear variational inequalities of finding x,y∈C such that(18)ϱTy+x−y,x∗−x≥0,∀x∗∈C,ηTx+y−x,x∗−y≥0,∀x∗∈C.Problem (18) was studied by Verma [30]. He extended the concept of variational inequalities to the system of nonlinear variational inequalities.(iv)
IfMi·,·=Mi·,ψ1x=φ1y,φ2x=ψ2y,ϱ1=ϱ2=1 and Pi:H→H are the single-valued mappings, then problem (15) coincides to the following system of extended general variational inequalities of finding x,y∈H such that(19)M1x,ψ1y∗−φ1y≥0,∀ψ1y∗∈Ω1,M2y,ψ2x∗−φ2x≥0,∀ψ2x∗∈Ω2.An equivalent form of problem (19) was studied by Noor et al. [31].(v)
Ifψ1y∗=ψ2x∗=gv,φ1=φ2=φ and Ω1=Ω2=Ω, then problem (19) reduces to the following system of variational inequalities of finding x,y∈H, φx,φy∈Ω such that(20)M1x,gv−φy≥0,∀gv∈Ω,M2y,gv−φx≥0,∀gv∈Ω.The problem of type (20) is called a system of extended general variational inequalities with four nonlinear operators.Next, we establish the following lemma which plays a crucial role to prove the existence result for the unique solution of the system of generalized set-valued variational inequalities (15) and to propose an iterative algorithm for studying convergence analysis.Lemma 6.
LetΩ1 and Ω2 be the nonempty closed convex subsets of a real Hilbert space H. Let I=1,2 be an index set; for each i∈I, let φi,ψi:H→H and Mi:H×H→H be the nonlinear single-valued mappings. Let Pi:H→CBH be a set-valued mapping. Then, the system of generalized set-valued variational inequalities (15) has a solution x,y,u,v, if and only if x,y,u,v,x∈Ω1,y∈Ω2,u∈P1x, and v∈P2y satisfies(21)φ1y=PΩ1ψ1x−ϱ1M1u,v,(22)φ2x=PΩ2ψ2y−ϱ2M2v,u,where ϱ1 and ϱ2 are positive constants.Proof.
Letx,y∈Ω1×Ω2,u∈P1x and v∈P2y is a solution of the system of generalized set-valued variational inequalities (15). Then(23)ϱ1M1u,v+φ1y−ψ1x,ψ1y∗−φ1y≥0,∀y∗∈Ω2,ϱ2M2v,u+φ2x−ψ2y,ψ2x∗−φ2x≥0,∀x∗∈Ω1.
Then, from Lemma4, we have(24)φ1y=PΩ1ψ1x−ϱ1M1u,v,φ2x=PΩ2ψ2y−ϱ2M2v,u.
Conversely, suppose thatx,y,u,v,x∈Ω1,y∈Ω2,u∈P1x and v∈P2y satisfies(25)φ1y=PΩ1ψ1x−ϱ1M1u,v,φ2x=PΩ2ψ2y−ϱ2M2v,u.
Again, it follows from Lemma4 that(26)ϱ1M1u,v+φ1y−ψ1x,ψ1y∗−φ1y≥0,∀y∗∈Ω2,ϱ2M2v,u+φ2x−ψ2y,ψ2x∗−φ2x≥0,∀x∗∈Ω1.
Thus,x,y,u,v,x∈Ω1,y∈Ω2,u∈P1x, and v∈P2y is solution of the system of generalized set-valued variational inequalities (15).Now, by virtue of the Banach contraction principle, we shall show the existence of the unique solution for the system of generalized set-valued variational inequalities (15).Theorem 7.
LetH be a real Hilbert space and I=1,2 be an index set, for each i∈I; let Ωi be a closed and convex subset of H. Let φi,ψi:H→H and Mi:H×H→H be nonlinear single-valued mappings such that φi is relaxed ∈φi,εφi-cocoercive and Lφi-Lipschitz type continuous, and ψi is relaxed ∈ψi,εψi-cocoercive and Lψi-Lipschitz type continuous. Let Pi:H→CBH be a set-valued mapping such that Pi is a relaxed monotone with respect to Mi with constant ci, D-Lipschitz type continuous with constant ςi, and Mi is LMi-Lipschitz type continuous in the first argument and LMi′-Lipschitz type continuous in the second argument. In addition, μi>0 satisfies(27)0<μ1,μ2<1,where μ1=κ1+ι1+ω2,μ2=κ2+ι2+ω1, and(28)κ1=1−2εψ1+1+2∈ψ1Lψ12,ι1=1+2ϱ1c1+ϱ12LM12ς12,κ2=1−2εψ2+1+2∈ψ2Lψ22,ι2=1+2ϱ2c2+ϱ22LM22ς22,ω1=1−2εφ1+1+2∈φ1Lφ12+LM1′ς2,ω2=1−2εφ2+1+2∈φ2Lφ22+LM2′ς1.
Then, the system of generalized set-valued variational inequalities (15) admits unique solution.Proof.
For any givenx∈Ω1,y∈Ω2, we define a mapping ϑ:H×H→H×H by(29)ϑx,y=Φx,y,Yx,y,where Φ,Y:Ω1×Ω2→H are the single-valued mappings defined by(30)Φx,y=x−φ2x+PΩ2ψ2y−ϱ2M2v,u,(31)Yx,y=y−φ1y+PΩ1ψ1x−ϱ1M1u,v,where u∈P1x,v∈P2y and ϱ1,ϱ2 are positive constants. Utilizing the fact that the projection mapping PΩ1 is nonexpansive, it follows from (31) that(32)∥Yx1,y1−Yx2,y2∥=∥y1−φ1y1+PΩ1ψ1x1−ϱ1M1u1,v1−y2−φ1y2+PΩ1ψ1x2−ϱ1M1u2,v2∥≤∥y1−y2−φ1y1−φ1y2∥+∥PΩ1ψ1x1−ϱ1M1u1,v1−PΩ1ψ1x2−ϱ1M1u2,v2∥≤∥y1−y2−φ1y1−φ1y2∥+∥ψ1x1−ϱ1M1u1,v1−ψ1x2−ϱ1M1u2,v2∥≤∥y1−y2−φ1y1−φ1y2∥+∥x1−x2−ψ1x1−ψ1x2∥+∥x1−x2−ϱ1M1u1,v1−M1u2,v1∥+ϱ1∥M1u2,v1−M1u2,v2∥.
Since the mappingφ1 is relaxed ∈φ1,εφ1-cocoercive and Lφ1-Lipschitz type continuous, then we have(33)y1−y2−φ1y1−φ1y22=y1−y22−2φ1y1−φ1y2,y1−y2+φ1y1−φ1y22≤y1−y22+2∈φ1φ1y1−φ1y22−2εφ1y1−y22+φ1y1−φ1y22≤1−2εφ1+1+2∈φ1Lφ12y1−y22,which implies that(34)y1−y2−φ1y1−φ1y2≤1−2εφ1+1+2∈φ1Lφ12y1−y2.
Again, using the fact that the mappingψ1 is relaxed ∈ψ1,εψ1-cocoercive and Lψ1-Lipschitz type continuous, then we have(35)x1−x2−ψ1x1−ψ1x22=x1−x22−2ψ1x1−ψ1x2,x1−x2+ψ1x1−ψ1x22≤x1−x22+2∈ψ1ψ1x1−ψ1x22−2εψ1x1−x22+ψ1x1−ψ1x22≤1−2εψ1+1+2∈ψ1Lψ12x1−x22,which implies that(36)x1−x2−ψ1x1−ψ1x2≤1−2εψ1+1+2∈ψ1Lψ12x1−x2.
SinceP1 is relaxed monotone with respect to M1 with constant c1 in the first argument, D-Lipschitz type continuous with constant ς1 and M1 is LM1-Lipschitz type continuous in the first argument, then we have(37)∥x1−x2−ϱ1M1u1,v1−M1u2,v1∥2=∥x1−x2∥2−2ϱ1M1u1,v1−M1u2,v1,x1−x2+ϱ12∥M1u1,v1−M1u2,v1∥2≤∥x1−x2∥2+2ϱ1c1∥x1−x2∥2+ϱ12LM12∥u1−u2∥2≤∥x1−x2∥2+2ϱ1c1∥x1−x2∥2+ϱ12LM12ς12∥x1−x2∥2=1+2ϱ1c1+ϱ12LM12ς12∥x1−x2∥2,which implies that(38)∥x1−x2−ϱ1M1u1,v1−M1u2,v1∥≤1+2ϱ1c1+ϱ12LM12ς12∥x1−x2∥.
SinceM1 is LM1′-Lipschitz type continuous in the second argument and P2 is D-Lipschitz type continuous with constant ς2, then we have(39)∥M1u2,v1−M1u2,v2∥≤LM1′∥v1−v2∥≤LM1′ς2∥y1−y2∥.
Thus, from (34), (36), (38), and (39), (32) becomes(40)Yx1,y1−Yx2,y2≤1−2εφ1+1+2∈φ1Lφ12+LM1′ς2y1−y2+1−2εψ1+1+2∈ψ1Lψ12+1+2ϱ1c1+ϱ12LM12ς12x1−x2.
Again, utilizing the fact that the projection mappingPΩ2 is nonexpansive, it follows from (30) that(41)∥Φx1,y1−Φx2,y2∥=∥x1−φ2x1+PΩ2ψ2y1−ϱ2M2v1,u1−x2−φ2x2+PΩ2ψ2y2−ϱ2M2v2,u2∥≤∥x1−x2−φ2x1−φ2x2∥+∥PΩ2ψ2y1−ϱ2M2v1,u1−PΩ2ψ2y2−ϱ2M2v2,u2∥≤∥x1−x2−φ2x1−φ2x2∥+∥ψ2y1−ϱ2M2v1,u1−ψ2y2−ϱ2M2v2,u2∥≤∥x1−x2−φ2x1−φ2x2∥+∥y1−y2−ψ2y1−ψ2y2∥+∥y1−y2−ϱ2M2v1,u1−M2v2,u1∥+ϱ2∥M2v2,u1−M2v2,u2∥.
Since the mappingφ2 is relaxed ∈φ2,εφ2-cocoercive and Lφ2-Lipschitz type continuous, then we have(42)x1−x2−φ2x1−φ2x22=x1−x22−2φ2x1−φ2x2,x1−x2+φ2x1−φ2x22≤x1−x22+2∈φ2φ2x1−φ2x22−2εφ2x1−x22+φ2x1−φ2x22≤1−2εφ2+1+2∈φ2Lφ22x1−x22,which implies that(43)x1−x2−φ2x1−φ2x2≤1−2εφ2+1+2∈φ2Lφ22x1−x2.
Again, using the fact that the mappingψ2 is relaxed ∈ψ2,εψ2-cocoercive and Lψ2-Lipschitz type continuous, then we have(44)y1−y2−ψ2y1−ψ2y22=y1−y22−2ψ2y1−ψ2y2,y1−y2+ψ2y1−ψ2y22≤y1−y22+2∈ψ2ψ2y1−ψ2y22−2εψ2y1−y22+ψ2y1−ψ2y22≤1−2εψ2+1+2∈ψ2Lψ22y1−y22,which implies that(45)y1−y2−ψ2y1−ψ2y2≤1−2εψ2+1+2∈ψ2Lψ22y1−y2.
SinceP2 is relaxed monotone with respect to M2 with constant c2 in the first argument, D-Lipschitz type continuous with constant ς2 and M2 is LM2-Lipschitz type continuous in the first argument, then we have(46)y1−y2−ϱ2M2v1,u1−M2v2,u12=y1−y22−2ϱ2M2v1,u1−M2v2,u1,y1−y2+ϱ22∥M2v1,u1−M2v2,u1∥2≤∥y1−y2∥2+2ϱ2c2∥y1−y2∥2+ϱ22LM22∥v1−v2∥2≤∥y1−y2∥2+2ϱ2c2∥y1−y2∥2+ϱ22LM22ς22∥y1−y2∥2=1+2ϱ2c2+ϱ22LM22ς22∥y1−y2∥2,which implies that(47)y1−y2−ϱ2M2v1,u1−M2v2,u1≤1+2ϱ2c2+ϱ22LM22ς22∥y1−y2∥.
SinceM2 is LM2′-Lipschitz type continuous in the second argument and P1 is D-Lipschitz type continuous with constant ς1, then we have(48)∥M2v2,u1−M2v2,u2∥≤LM2′∥u1−u2∥≤LM2′ς1∥x1−x2∥.
Thus, from (43), (45), (47), and (48), (41) becomes(49)Φx1,y1−Φx2,y2≤1−2εφ2+1+2∈φ2Lφ22+LM2′ς1x1−x2+1−2εψ2+1+2∈ψ2Lψ22+1+2ϱ2c2+ϱ22LM22ς22y1−y2.
Now, it follows from (40) and (49) that(50)∥Φx1,y1−Φx2,y2∥+∥Yx1,y1−Yx2,y2∥≤μ1∥x1−x2∥+μ2∥y1−y2∥≤maxμ1,μ2∥x1−x2∥+∥y1−y2∥,where μ1=κ1+ι1+ω2,μ2=κ2+ι2+ω1, and(51)κ1=1−2εψ1+1+2∈ψ1Lψ12,ι1=1+2ϱ1c1+ϱ12LM12ς12,κ2=1−2εψ2+1+2∈ψ2Lψ22,ι2=1+2ϱ2c2+ϱ22LM22ς22,ω1=1−2εφ1+1+2∈φ1Lφ12+LM1′ς2,ω2=1−2εφ2+1+2∈φ2Lφ22+LM2′ς1.
It follows from (8), (29), and (50) that(52)∥ϑx1,y1−ϑx2,y2∥∗≤maxμ1,μ2∥x1,y1−x2,y2∥.
It follows from conditions (27) and (52) that ϑ is a contraction mapping. Therefore, there exists unique x,y∈Ω1×Ω2 such that ϑx,y=x,y. Thus, we have(53)φ1y=PΩ1ψ1x−ϱ1M1u,v,φ2x=PΩ2ψ2y−ϱ2M2v,u.
Therefore, by Lemma6, one can conclude that x,y,u,v,x∈Ω1,y∈Ω2,u∈P1x, and v∈P2y is the unique solution of the system of generalized set-valued variational inequalities (15).


## 3. Iterative Algorithm and Convergence Result

In this section, we suggest an iterative algorithm to analyse the convergence of the system of generalized set-valued variational inequalities (15).By utilizing (21) and (22) of Lemma 6, we can offer the following iterative forms:(54)y=1−γny+γny−φ1y+PΩ1ψ1x−ϱ1M1u,v,(55)x=1−δnx+δnx−φ2x+PΩ2ψ2y−ϱ2M2v,u,where ϱ1 and ϱ2 are positive constants and the real sequences γn,δn∈0,1. Now, we propose the following iterative algorithm.Algorithm 8.
For any givenx0,y0∈Ω1×Ω2, we choose u0∈P1x0 and v0∈P2y0. For ϱ1,ϱ2>0 and γn,δn∈0,1 and from (54) and (55), let(56)y1=1−γny0+γny0−φ1y0+PΩ1ψ1x0−ϱ1M1u0,v0,x1=1−δnx0+δnx0−φ2x0+PΩ2ψ2y0−ϱ2M2v0,u0.
Sinceu0∈P1x0 and v0∈P2y0, by Nadler’s theorem [32], there exist u1∈P1x1 and v1∈P2y1 such that(57)∥u0−u1∥≤1+1DP1x0,P1x1,∥v0−v1∥≤1+1DP2y0,P2y1,where D is the Hausdorff metric. Let(58)y2=1−γny1+γny1−φ1y1+PΩ1ψ1x1−ϱ1M1u1,v1,x2=1−δnx1+δnx1−φ2x1+PΩ2ψ2y1−ϱ2M2v1,u1.
Again, it follows from Nadler’s theorem [32] that there exist u2∈P1x2 and v2∈P2y2 such that(59)∥u1−u2∥≤1+2−1DP1x1,P1x2,∥v1−v2∥≤1+2−1DP2y1,P2y2.
Continuing in the same manner, we can figure out the sequencesxn,yn,un, and vn by the following iterative process:(60)yn+1=1−γnyn+γnyn−φ1yn+PΩ1ψ1xn−ϱ1M1un,vn,(61)xn+1=1−δnxn+δnxn−φ2xn+PΩ2ψ2yn−ϱ2M2vn,un,and for n=0,1,,2,⋯, choose un+1∈P1xn+1 and vn+1∈P2yn+1 such that(62)∥un+1−un∥≤1+n+1−1DP1xn+1,P1xn,(63)∥vn+1−vn∥≤1+n+1−1DP2yn+1,P2yn.
Now, we are accessible to study the convergence of the proposed iterative algorithm for the system of generalized set-valued variational inequalities (15).Theorem 9.
LetH be a real Hilbert space and I=1,2 be an index set, for each i∈I; let Ωi be a closed convex subset of H. Let φi,ψi:H→H and Mi:H×H→H be nonlinear single-valued mappings such that φi is relaxed ∈φi,εφi-cocoercive, Lφi-Lipschitz type continuous; ψi is relaxed ∈ψi,εψi-cocoercive, Lψi-Lipschitz type continuous; and Mi is LMi-Lipschitz type continuous in the first argument and LMi′-Lipschitz type continuous in the second argument. Let Pi:H→CBH be a set-valued mapping such that Pi is a relaxed monotone with respect to Mi with constant ci and D-Lipschitz type continuous with constant ςi. In addition, for each i=1,2, ρi,ci>0 satisfy the following conditions:(i)
0<ni,li<1,(ii)
2εψi−1+2∈ψiLψi2<1,(iii)
δn1−l1−γnm2+n2≥0,γn1−l2−δnm1+n1≥0, such that(64)∑n=0∞δn1−l1−γnm2+n2=∞,∑n=0∞γn1−l2−δnm1+n1=∞,where(65)l1=1−2εφ2+1+2∈φ2Lφ22+ϱ2LM2′ς1,m1=1−2εψ2+1+2∈ψ2Lψ22,l2=1−2εφ1+1+2∈φ1Lφ12+ϱ1LM1′ς2,m2=1−2εψ1+1+2∈ψ1Lψ12,n1=1+2ϱ2c2+ϱ22LM22ς22,n2=1+2ϱ1c1+ϱ12LM12ς12.
Then, the approximate solutionxn,yn,un,vn generated by Algorithm 8 converges strongly to the unique solution x,y,u,v of the system of generalized set-valued variational inequalities (15).Proof.
Letx,y,u,v,x∈Ω1,y∈Ω2,u∈P1x and v∈P2y be a solution of (15); then, from (55) and (61) of Algorithm 8 and utilizing the fact that the projection mapping PΩ2 is nonexpansive, we have(66)∥xn+1−x∥=∥1−δnxn+δnxn−φ2xn+PΩ2ψ2yn−ϱ2M2vn,un−1−δnx+δnx−φ2x+PΩ2ψ2y−ϱ2M2v,u∥≤1−δn∥xn−x∥+δn∥xn−x−φ2xn−φ2x∥+δn∥PΩ2ψ2yn−ϱ2M2vn,un−PΩ2ψ2y−ϱ2M2v,u∥≤1−δn∥xn−x∥+δn∥xn−x−φ2xn−φ2x∥+δn∥ψ2yn−ϱ2M2vn,un−ψ2y−ϱ2M2v,u∥≤1−δn∥xn−x∥+δn∥xn−x−φ2xn−φ2x∥+δn∥yn−y−ψ2yn−ψ2y∥+δn∥yn−y−ϱ2M2vn,un−M2v,un∥+δnϱ2∥M2v,un−M2v,u∥.
Since the mappingφ2 is relaxed ∈φ2,εφ2-cocoercive and Lφ2-Lipschitz type continuous, then we have(67)xn−x−φ2xn−φ2x2=xn−x2−2φ2xn−φ2x,xn−x+φ2xn−φ2x2≤xn−x2+2εφ2φ2xn−φ2x2−2εφ2xn−x2+φ2xn−φ2x2≤1−2εφ2+1+2εφ2Lφ22xn−x2,which implies that(68)∥xn−x−φ2xn−φ2x∥≤1−2εφ2+1+2εφ2Lφ22∥xn−x∥.
Again, using the fact that mappingψ2 is relaxed εψ2,εψ2-cocoercive and Lψ2-Lipschitz type continuous, then we have(69)yn−y−ψ2yn−ψ2y2=yn−y2−2ψ2yn−ψ2y,yn−y+ψ2yn−ψ2y2≤yn−y2+2∈ψ2ψ2yn−ψ2y2−2εψ2yn−y2+ψ2yn−ψ2y2≤1−2εψ2+1+2∈ψ2Lψ22yn−y2,which implies that(70)∥yn−y−ψ2yn−ψ2y∥≤1−2εψ2+1+2∈ψ2Lψ22∥yn−y∥.
SinceP2 is relaxed monotone with respect to M2 with constant c2 in the first argument, D-Lipschitz type continuous with constant ς2 and M2 is LM2-Lipschitz type continuous in the first argument, then we have(71)∥yn−y−ϱ2M2vn,un−M2v,un∥2=∥yn−y∥2−2ϱ2M2vn,un−M2v,un,yn−y+ϱ22∥M2vn,un−M2v,un∥2≤∥yn−y∥2+2ϱ2c2∥yn−y∥2+ϱ22LM22∥vn−v∥2≤∥yn−y∥2+2ϱ2c2∥yn−y∥2+ϱ22LM22ς22∥yn−y∥2=1+2ϱ2c2+ϱ22LM22ς22∥yn−y∥2,which implies that(72)∥yn−y−ϱ2M2vn,un−M2v,un∥≤1+2ϱ2c2+ϱ22LM22ς22∥yn−y∥.
SinceM2 is LM2′-Lipschitz type continuous in the second argument and P1 is D-Lipschitz type continuous with constant ς1, then, we have(73)∥M2v,un−M2v,u∥≤LM2′∥un−u∥≤LM2′ς1∥xn−x∥.
Thus, from (68), (70), (72), and (73), (66) becomes(74)xn+1−x≤1−δn1−1−2εφ2+1+2∈φ2Lφ22−ϱ2LM2′ς1xn−x+δn1−2εψ2+1+2∈ψ2Lψ22+1+2ϱ2c2+ϱ22LM22ς22yn−y.
Again, from (54) and (60) of Algorithm 8 and utilizing the fact that the projection mapping PΩ1 is nonexpansive, we have(75)∥yn+1−y∥=∥1−γnyn+γnyn−φ1yn+PΩ1ψ1xn−ϱ1M1un,vn−1−γny+γny−φ1y+PΩ1ψ1x−ϱ1M1u,v∥≤1−γn∥yn−y∥+γn∥yn−y−φ1yn−φ1y∥+γn∥PΩ1ψ1xn−ϱ1M1un,vn−PΩ1ψ1x−ϱ1M1u,v∥≤1−γn∥yn−y∥+γn∥yn−y−φ1yn−φ1y∥+γn∥ψ1xn−ϱ1M1un,vn−ψ1x−ϱ1M1u,v∥≤1−γn∥yn−y∥+γn∥yn−y−φ1yn−φ1y∥+γn∥xn−x−ψ1xn−ψ1x∥+γn∥xn−x−ϱ1M1un,vn−M1u,vn∥+γnϱ1∥M1u,vn−M1u,v∥.
Since the mappingφ1 is relaxed ∈φ1,εφ1-cocoercive and Lφ1-Lipschitz type continuous, then we have(76)yn−y−φ1yn−φ1y2=yn−y2−2φ1yn−φ1y,yn−y+φ1yn−φ1y2≤yn−y2+2∈φ1φ1yn−φ1y2−2εφ1yn−y2+φ1yn−φ1y2≤1−2εφ1+1+2∈φ1Lφ12yn−y2,which implies that(77)∥yn−y−φ1yn−φ1y∥≤1−2εφ1+1+2∈φ1Lφ12∥yn−y∥.
Again, using the fact that mappingψ1 is relaxed ∈ψ1,εψ1-cocoercive and Lψ1-Lipschitz type continuous, then we have(78)xn−x−ψ1xn−ψ1x2=xn−x2−2ψ1xn−ψ1x,xn−x+ψ1xn−ψ1x2≤xn−x2+2∈ψ1ψ1xn−ψ1x2−2εψ1xn−x2+ψ1xn−ψ1x2≤1−2εψ1+1+2∈ψ1Lψ12xn−x2.which implies that(79)xn−x−ψ1xn−ψ1x≤1−2εψ1+1+2∈ψ1Lψ12xn−x.
SinceP1 is relaxed monotone with respect to M1 with constant c1 in the first argument, D-Lipschitz type continuous with constant ς1 and M1 is LM1-Lipschitz type continuous in the first argument, then we have(80)∥xn−x−ϱ1M1un,vn−M1u,vn∥2=∥xn−x∥2−2ϱ1M1un,vn−M1u,vn,xn−x+ϱ12∥M1un,vn−M1u,vn∥2≤∥xn−x∥2+2ϱ1c1∥xn−x∥2+ϱ12LM12∥un−u∥2≤∥xn−x∥2+2ϱ1c1∥xn−x∥2+ϱ12LM12ς12∥xn−x∥2=1+2ϱ1c1+ϱ12LM12ς12∥xn−x∥2,which implies that(81)∥xn−x−ϱ1M1un,vn−M1u,vn∥≤1+2ϱ1c1+ϱ12LM12ς12∥xn−x∥.
SinceM1 is LM1′-Lipschitz type continuous in the second argument and P2 is D-Lipschitz type continuous with constant ς2, then, we have(82)∥M1u,vn−M1u,v∥≤LM1′∥vn−v∥≤LM1′ς2∥yn−y∥.
Thus, from (77), (79), (81), and (82), (75) becomes(83)∥yn+1−y∥≤1−γn1−1−2εφ1+1+2∈φ1Lφ12−ϱ1LM1′ς2∥yn−y∥+γn1−2εψ1+1+2∈ψ1Lψ12+1+2ϱ1c1+ϱ12LM12ς12∥xn−x∥.
For the sake of simplicity, we put(84)l1=1−2εφ2+1+2∈φ2Lφ22+ϱ2LM2′ς1,m1=1−2εψ2+1+2∈ψ2Lψ22,l2=1−2εφ1+1+2∈φ1Lφ12+ϱ1LM1′ς2,m2=1−2εψ1+1+2∈ψ1Lψ12,n1=1+2ϱ2c2+ϱ22LM22ς22,n2=1+2ϱ1c1+ϱ12LM12ς12.
Then, it follows from (74) and (83) that(85)∥xn+1−x∥+∥yn+1−y∥≤1−δn1−l1∥xn−x∥+δnm1+n1∥yn−y∥+1−γn1−l2∥yn−y∥+γnm2+n2∥xn−x∥=1−δn1−l1+γnm2+n2∥xn−x∥+1−γn1−l2+δnm1+n1∥yn−y∥,which implies that(86)∥xn+1−x∥+∥yn+1−y∥≤τ1∥xn−x∥+τ2∥yn−y∥≤ω∥xn−x∥+∥yn−y∥,where
ω=maxτ1,τ2,τ1=1−δn1−l1+γnm2+n2 and τ2=1−γn1−l2+δnm1+n1.
Thus, from (86), Lemma 5, and conditions, we have(87)limn→∞∥xn−x∥+∥yn−y∥=0,which implies that(88)limn→∞∥xn−x∥=limn→∞∥yn−y∥=0.
Hence,xn→x and yn→y. It follows from (62) and (63) that un and vn are Cauchy sequences; we can assume that un→u and vn→v, strongly. Next, we show that u∈P1x and v∈P2y. Since un∈P1xn, then we have(89)du,P1x≤∥u−un∥+dun,P1x≤∥u−un∥+1+n−1DP1xn,P1x≤∥u−un∥+1+n−1ς1∥xn−x∥→0asn→∞.
Hence,du,P1x→0, so u∈P1x as P1x∈CBH. Similarly, it is easy to show that v∈P2y. Thus, by Lemma 6, one can deduce that x,y,u,v,x∈Ω1,y∈Ω2,u∈P1x, and v∈P2y is a solution to the system of generalized set-valued variational inequalities (15).


## 4. Altering Points Problem

In this section, the concept of altering points problem is used to find the solution of the considered system of variational inequalities (16). We propose parallel Mann and parallel S-iterative algorithms, and the strong convergence of the sequences generated by these parallel iterative algorithms is discussed.Definition 10.
[11]. Let Ω1 and Ω2 be nonempty subsets of a metric space X. Then, the points x∈Ω1 and y∈Ω2 are the altering points of mappings A1:Ω1→Ω2 and A2:Ω2→Ω1, if(90)A1x=y,A2y=x.
AltA1,A2=x,y∈Ω1×Ω2:A1x=yandA2y=x is called the set of altering points of the mappings A1 and A2.Lemma 11 [33].
LetΩ1 and Ω2 be nonempty closed subsets of a complete metric space X. Let A1:Ω1→Ω2 and A2:Ω2→Ω1 be Lipschitz continuous mappings with constants κ1 and κ2, respectively, such that κ1κ2<1. Then, the following conditions hold:(i)
There exists unique pointx,y∈Ω1×Ω2 such that x and y are altering points of the mappings A1 and A2.(ii)
For arbitraryx1∈Ω1, the sequence xn,yn∈Ω1×Ω2 generated by(91)yn=A1xn,xn+1=A2yn,n∈ℕ,converges to x,y.Lemma 12.
LetΩ1 and Ω2 be nonempty closed convex subsets of a real Hilbert space H. Let I=1,2 be an index set, for each i∈I; let φi,ψi:H→H and Mi:H×H→H be nonlinear single-valued mappings. Then, x,y∈Ω1×Ω2 is the solution to the system of variational inequalities (16), if and only if x,y satisfies(92)y=PΩ2ψ1x−ϱ1M1y,x,x=PΩ1ψ2y−ϱ2M2x,y,where ϱ1 and ϱ2 are positive constants.
Define the mappingsA1:Ω1→Ω2 and A2:Ω2→Ω1 as follows:(93)A1≔PΩ2ψ1−ϱ1M1y,·,(94)A2≔PΩ1ψ2−ϱ2M2x,·,where ϱ1,ϱ2 are constants in 0,1. By virtue of Lemma 4, we can formulate the system of variational inequalities (16) into the following equivalent altering points problem.
Findx,y∈Ω1×Ω2 such that(95)x=PΩ1ψ2−ϱ2M2x,·y,y=PΩ2ψ1−ϱ1M1y,·x.
Now, we propose the following parallel Mann iterative algorithm to solve the system of variational inequalities (16) as follows.Algorithm 13.
LetΩ1,Ω2 be nonempty closed convex subsets of a real Hilbert space H. For any x0,y0∈Ω1×Ω2, let xn,yn be an iterative sequence in Ω1×Ω2 defined by(96)xn+1=1−αnxn+αnA2yn,(97)yn+1=1−αnyn+αnA1xn,where αn∈0,1,n∈ℕ and A1,A2 are the mappings defined by (93) and (94), respectively.Also, we propose the following parallelS-iterative algorithm to solve the system of variational inequalities (16). Note that the parallel S-iterative algorithm is more general than the parallel Mann iterative algorithm.Algorithm 14.
LetΩ1 and Ω2 be nonempty closed convex subsets of a real Hilbert space H. For any x0,y0∈Ω1×Ω2, let xn,yn be an iterative sequence in Ω1×Ω2 defined by(98)xn+1=A21−αnyn+αnA1xn,(99)yn+1=A11−αnxn+αnA2yn,where αn∈0,1,n∈ℕ and A1,A2 are the mappings defined by (93) and (94), respectively.Theorem 15 [33].
LetΩ1 and Ω2 be nonempty closed convex subsets of a Banach space X. Let A1:Ω1→Ω2 and A2:Ω2→Ω1 be Lipschitz continuous mappings with constants κ1<1 and κ2<1, respectively. Then, the sequence xn,yn∈Ω1×Ω2 generated by the parallel S-iterative algorithm (98)–(99) converges strongly to the unique point x,y∈Ω1×Ω2 such that x and y are altering points of the mappings A1 and A2.Next, we prove the following proposition, which plays a crucial role to prove the convergence of parallel iterative algorithms.Proposition 16.
LetΩ1 and Ω2 be nonempty closed convex subsets of a real Hilbert space H. For each i=1,2, let ψi:H→H be the single valued mappings such that ψi is δψi-strongly monotone and Lψi-Lipschitz type continuous. Let M1:Ω2×Ω1→H and M2:Ω1×Ω2→H be the single-valued mappings such that Mi is δMi-strongly monotone and LMi-Lipschitz type continuous with respect to the second argument. Then, the mappings A1 and A2 defined by (93) and (94) are Lipschitz type continuous with constant ℧1+Θ1 and ℧2+Θ2, respectively, where ℧i=1−2δψi+Lψi2 and Θi=1−2ϱiδMi+ρi2LMi2.Proof.
For any givenx1,x2∈Ω1, (93), and using the nonexpansiveness of the projection mapping PΩ2, we have(100)∥A1x1−A1x2∥=∥PΩ2ψ1−ϱ1M1y,·x1−PΩ2ψ1−ϱ1M1y,·x2∥≤∥ψ1x1−ϱ1M1y,x1−ψ1x2−ϱ1M1y,x2∥≤∥ψ1x1−ψ1x2−x1−x2∥+∥x1−x2−ϱ1M1y,x1−M1y,x2∥.
Sinceψ1 is δψ1-strongly monotone and Lψ1-Lipschitz type continuous, then, we get(101)∥x1−x2−ψ1x1−ψ1x2∥2=∥x1−x2∥2−2ψ1x1−ψ1x2,x1−x2+∥ψ1x1−ψ1x2∥2≤∥x1−x2∥2−2δψ1∥x1−x2∥2+Lψ12∥x1−x2∥2=1−2δψ1+Lψ12∥x1−x2∥2=℧12∥x1−x2∥2,which implies that(102)∥x1−x2−ψ1x1−ψ1x2∥≤℧1∥x1−x2∥.
Again, using the fact thatM1 is δM1-strongly monotone and LM1-Lipschitz type continuous with respect to the second argument, we have(103)∥x1−x2−ϱ1M1y,x1−M1y,x2∥2=∥x1−x2∥2−2ϱ1M1y,x1−M1y,x2,x1−x2+ρ12∥M1y,x1−M1y,x2∥2≤∥x1−x2∥2−2ϱ1δM1∥x1−x2∥2+ρ12LM12∥x1−x2∥2=1−2ϱ1δM1+ϱ12LM12∥x1−x2∥2=Θ12∥x1−x2∥2,which implies that(104)∥x1−x2−ϱ1M1y,x1−M1y,x2∥≤Θ1∥x1−x2∥.
Utilizing (102) and (104), (100) becomes(105)∥A1x1−A1x2∥≤℧1+Θ1∥x1−x2∥,i.e., A1 is ℧1+Θ1-Lipschitz continuous, where ℧1=1−2δψ1+Lψ12 and Θ1=1−2ϱ1δψ1+ϱ12LM12. Similarly, one can prove that A2 is ℧2+Θ2-Lipschitz continuous.Now, we prove the convergence of the parallel Mann iterative scheme (96)–(97).Theorem 17.
LetΩ1 and Ω2 be nonempty closed convex subsets of a real Hilbert space H. For each i=1,2, let ψi:H→H be the single-valued mapping such that ψi is δψi-strongly monotone and Lψi-Lipschitz type continuous. Let M1:Ω2×Ω1→H and M2:Ω1×Ω2→H be the single-valued mappings such that Mi is δMi-strongly monotone and LMi-Lipschiz type continuous with respect to the second argument. For any x0,y0∈Ω1×Ω2, let xn,yn be an iterative sequence in Ω1×Ω2 generated by Algorithm 13, where αn∈0,1 satisfying ∑n=0∞αn=∞ and A1,A2 are the mappings defined by (93) and (94), respectively. In addition, the following condition holds:(106)0<℧i+Θi<1,where ℧i=1−2δψi+Lψi2 and Θi=1−2ϱiδMi+ϱi2LMi2, for i=1,2. Then(i)
There exists unique pointx,y∈Ω1×Ω2, which is the solution of the system of variational inequalities (16).(ii)
The sequencexn,yn generated by parallel Mann iterative Algorithm 13 converges strongly to the point x,y.Proof.
(i)
Conclusion follows from Lemma11 and (95).(ii)
It follows from (94), (95), (96), and Lipschitz continuity of mapping A2 that(107)∥xn+1−x∥=∥1−αnxn+αnA2yn−x∥=∥1−αnxn+αnA2yn−1−αnx−αnx∥≤1−αn∥xn−x∥+αn∥A2yn−x∥=1−αn∥xn−x∥+αn∥A2yn−A2y∥≤1−αn∥xn−x∥+αn℧2+Θ2∥yn−y∥.
Again, utilizing (93), (95), (97), and Lipschitz continuity of mapping A1, we get(108)∥yn+1−y∥=∥1−αnyn+αnA1xn−y∥=∥1−αnyn+αnA1xn−1−αny−αny∥≤1−αn∥yn−y∥+αn∥A1xn−y∥=1−αn∥yn−y∥+αn∥A1xn−A1x∥≤1−αn∥yn−y∥+αn℧1+Θ1∥xn−x∥.
LettingΘ=max℧1+Θ1,℧2+Θ2; then, from (107) and (108), we have(109)xn+1−x+yn+1−y≤1−αnxn−x+αn℧2+Θ2yn−y+1−αnyn−y+αn℧1+Θ1xn−x≤1−αnxn−x+αnΘyn−y+1−αnyn−y+αnΘxn−x=1−αn1−Θxn−x+yn−y.
Thus, from (8) and (109), we have(110)∥xn+1,yn+1−x,y∥∗=∥xn+1−x,yn+1−y∥∗=∥xn+1−x∥+∥yn+1−y∥≤1−αn1−Θ∥xn−x∥+∥yn−y∥≤1−αn1−Θ∥xn,yn−x,y∥∗.
Since∑n=0∞αn=∞ and from condition (106), we have Θ∈0,1. Thus, from Lemma 5, we can conclude that limn→∞∥xn,yn−x,y∥∗=0. Therefore, limn→∞∥xn−x∥=0=limn→∞∥yn−y∥. Hence, the sequences xn and yn generated by parallel Mann iterative algorithm converge strongly to x and y, respectively.Now, we prove the convergence of the parallelS-iterative algorithm (98)–(99).Theorem 18.
LetΩ1 and Ω2 be nonempty closed and convex subsets of a real Hilbert space H. For each i=1,2, let ψi:H→H be the single-valued mapping such that ψi is δψi-strongly monotone and Lψi-Lipschitz type continuous. Let M1:Ω2×Ω1→H and M2:Ω1×Ω2→H be the single-valued mappings such that Mi is δMi-strongly monotone and LMi-Lipschitz type continuous with respect to the second argument. For any x0,y0∈Ω1×Ω2, let xn,yn be an iterative sequence in Ω1×Ω2 generated by Algorithm 14, where αn∈0,1, A1, and A2 are the mappings defined by (93) and (94), respectively. In addition, the following condition holds:(111)0<℧i+Θi<1,where ℧i=1−2δψi+Lψi2 and Θi=1−2ρiδMi+ρi2LMi2, for i=1,2. Then(i)
There exists unique pointx,y∈Ω1×Ω2, which is the solution of the system of variational inequalities (16).(ii)
The sequencexn,yn generated by parallel S-iterative Algorithm 14 converges strongly to the point x,y.Proof.
(i)
The conclusion follows from Lemma11 and (95).(ii)
It follows from (94), (95), (98), and Lipschitz continuity of A2, we get(112)∥xn+1−x∥=∥A21−αnyn+αnA1xn−x∥=∥A21−αnyn+αnA1xn−A2y∥≤℧2+Θ2∥1−αnyn+αnA1xn−y∥≤℧2+Θ21−αn∥yn−y∥+αn∥A1xn−y∥=℧2+Θ21−αn∥yn−y∥+αn∥A1xn−A1x∥≤℧2+Θ21−αn∥yn−y∥+αn℧1+Θ1∥xn−x∥.
Using the same facts, we have(113)∥yn+1−y∥≤℧1+Θ11−αn∥xn−x∥+αn℧2+Θ2∥yn−y∥.
LettingΘ=max℧1+Θ1,℧2+Θ2, then from (112) and (113), we have(114)∥xn+1−x∥+∥yn+1−y∥≤℧2+Θ21−αn∥yn−y∥+αn℧1+Θ1∥xn−x∥+℧1+Θ11−αn∥xn−x∥+αn℧2+Θ2∥yn−y∥≤Θ1−αn∥yn−y∥+αnΘ∥xn−x∥+Θ1−αn∥xn−x∥+αnΘ∥yn−y∥=Θ1−αn1−Θ∥xn−x∥+∥yn−y∥.
Thus, from (8) and (114), we have(115)∥xn+1,yn+1−x,y∥∗=∥xn+1−x,yn+1−y∥∗=∥xn+1−x∥+∥yn+1−y∥≤Θ1−αn1−Θ∥xn−x∥+∥yn−y∥=Θ1−αn1−Θ∥xn,yn−x,y∥∗.
Notice thatΘ1−αn1−Θ<Θ and from condition (111), we have Θ∈0,1. Thus, we can conclude that limn→∞∥xn,yn−x,y∥∗=0. Therefore, limn→∞∥xn−x∥=0=limn→∞∥yn−y∥. Hence, the sequences xn and yn generated by parallel S-iterative algorithm converge strongly to x and y, respectively.Finally, we discuss an example which illustrates the convergence analysis of parallel Mann and parallelS-iterative algorithms.Example 1.
LetH=ℝ,Ω1=−∞,0 and Ω2=0,∞. Let ψ1,ψ2:H→H be the single-valued mappings defined by(116)ψ1x=2x−13,ψ2x=5x−26,∀x∈H.
Suppose that the mappingsM1,M2:H×H→H are defined by(117)M1x,y=4x+3y−23,M2x,y=17x+17y+112,∀x,y∈H.Then, it is easy to check that for eachi=1,2, ψi is δψi-strongly monotone and Lψi-Lipschitz continuous with δψ1=2/3=Lψ1 and δψ2=5/6=Lψ2 and Mi is δMi-strongly monotone and LMi-Lipschitz continuous in the second argument with δM1=4/3=LM1 and δM2=17/12=LM2. Let the mappings A1:Ω1→Ω2 and A2:Ω2→Ω1 be defined as follows:(118)A1≔PΩ2ψ1−ϱ1M1y,·,A2≔PΩ1ψ2−ϱ2M2x,·.Then, forϱ1,ϱ2=1,(119)A1x=PΩ2ψ1x−M1y,x=PΩ22x−13−4x+3y−23=−2x−3y+13,∀x,y∈H,A2y=PΩ1ψ1y−M2x,y=PΩ15y−26−17x+17y+112=−17x−7y−512,∀x,y∈H.It is easy to verify thatA1 is 2/3-Lipschitz continuous and A2 is 7/12-Lipschitz continuous. Also,(120)℧1+Θ1=1−2δψ1+Lψ12+1−2ρ1δM1+ρ12LM12=1−43+49+1−83+169=23<1,℧2+Θ2=1−2δψ2+Lψ22+1−2ρ2δM2+ρ22LM22=1−106+2536+1−3412+289144=712<1.Thus, all the conditions of Theorem17 and Theorem 18 are satisfied. Hence, by using Proposition 16, Algorithm 13, and Algorithm 14, the conclusions of Theorem 17 and Theorem 18 follow.


## 5. Concluding Remarks

In this paper, a system of generalized set-valued variational inequalities involving relaxed cocoercive mappings in Hilbert spaces is considered. Using the projection method and Banach contraction principle, an existence result is proved. Also, we proposed an iterative algorithm, and its convergence is discussed. Moreover, we established an equivalence between the system of variational inequalities and altering points problem. Some parallel iterative algorithms are proposed, and the strong convergence of the sequences generated by these iterative algorithms is discussed. Finally, a numerical example is constructed to illustrate the convergence analysis of the proposed parallel iterative algorithms.



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