Variational inequality theory was introduced by Stampacchia [1] in the early 1960s. The birth of variational inequality problem coincides with Signorini problem, see [[2], p.282]. The Signorini problem consists of finding the equilibrium of a spherically shaped elastic body resting on the rigid frictionless plane. Let H be a real Hilbert space whose inner product and norm are denoted by \u3008\cdot ,\cdot \u3009 and \parallel \cdot \parallel, respectively. A variational inequality involving the nonlinear bifurcation, which characterized the Signorini problem with nonlocal friction is: find x\in H such that
\u3008Tx,yx\u3009+\phi (y,x)\phi (x,x)\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\in H,
where T:H\to H is a nonlinear operator and \phi (\cdot ,\cdot ):H\times H\to \mathbb{R}\cup \{+\mathrm{\infty}\} is a continuous bifunction.
Inequality above is called mixed variational inequality problem. It is an useful and important generalization of variational inequalities. This type of variational inequality arise in the study of elasticity with nonlocal friction laws, fluid flow through porus media and structural analysis. Mixed variational inequalities have been generalized and extended in many directions using novel and innovative techniques. One interesting problem is to find common solution of a system of variational inequalities. The existence problem for solutions of a system of variational inequalities has been studied by Husain and Tarafdar [3]. System of variational inequalities arises in double porosity models and diffusion through a composite media, description of parallel membranes, etc.; see [4] for details.
In this paper, we consider the following system of generalized mixed variational inequalities (SGMVI). Find {x}^{\ast},{y}^{\ast}\in H such that
\{\begin{array}{c}\u3008{\rho}_{1}{T}_{1}({y}^{\ast},{x}^{\ast})+{g}_{1}({x}^{\ast}){g}_{1}({y}^{\ast}),x{g}_{1}({x}^{\ast})\u3009+\phi (x)\phi ({g}_{1}({x}^{\ast}))\ge 0,\hfill \\ \u3008{\rho}_{2}{T}_{2}({x}^{\ast},{y}^{\ast})+{g}_{2}({y}^{\ast}){g}_{2}({x}^{\ast}),x{g}_{2}({y}^{\ast})\u3009+\phi (x)\phi ({g}_{2}({y}^{\ast}))\ge 0\hfill \end{array}
(1.1)
for all x\in H and {\rho}_{1},{\rho}_{2}>0, where {T}_{1},{T}_{2}:H\times H\to H are nonlinear mappings and {g}_{1},{g}_{2}:H\to H are any mappings.
If {T}_{1},{T}_{2}:H\to H are univariate mappings then the problem (SGMVI) reduced to the following. Find {x}^{\ast},{y}^{\ast}\in H such that
\{\begin{array}{c}\u3008{\rho}_{1}{T}_{1}({y}^{\ast})+{g}_{1}({x}^{\ast}){g}_{1}({y}^{\ast}),x{g}_{1}({x}^{\ast})\u3009+\phi (x)\phi ({g}_{1}({x}^{\ast}))\ge 0,\hfill \\ \u3008{\rho}_{2}{T}_{2}({x}^{\ast})+{g}_{2}({y}^{\ast}){g}_{2}({x}^{\ast}),x{g}_{2}({y}^{\ast})\u3009+\phi (x)\phi ({g}_{2}({y}^{\ast}))\ge 0\hfill \end{array}
(1.2)
for all x\in H and {\rho}_{1},{\rho}_{2}>0.
If {T}_{1}={T}_{2}=T and {g}_{1}={g}_{2}=I, then the problem (SGMVI) reduces to the following system of mixed variational inequalities considered by [5, 6]. Find {x}^{\ast},{y}^{\ast}\in H such that
\{\begin{array}{c}\u3008{\rho}_{1}T({y}^{\ast},{x}^{\ast})+{x}^{\ast}{y}^{\ast},x{x}^{\ast}\u3009+\phi (x)\phi ({x}^{\ast})\ge 0,\hfill \\ \u3008{\rho}_{2}T({x}^{\ast},{y}^{\ast})+{y}^{\ast}{x}^{\ast},x{y}^{\ast}\u3009+\phi (x)\phi ({y}^{\ast})\ge 0\hfill \end{array}
(1.3)
for all x\in H and {\rho}_{1},{\rho}_{2}>0.
If K is closed convex set in H and \phi (x)={\delta}_{K}(x) for all x\in K, where {\delta}_{K} is the indicator function of K defined by
{\delta}_{K}(x)=\{\begin{array}{cc}0,\hfill & \text{if}x\in K;\hfill \\ +\mathrm{\infty},\hfill & \text{otherwise},\hfill \end{array}
then the problem (1.1) reduces to the following system of general variational inequality problem: Find {x}^{\ast},{y}^{\ast}\in K such that
\{\begin{array}{c}\u3008{\rho}_{1}{T}_{1}({y}^{\ast},{x}^{\ast})+{g}_{1}({x}^{\ast}){g}_{1}({y}^{\ast}),x{g}_{1}({x}^{\ast})\u3009\ge 0,\hfill \\ \u3008{\rho}_{2}{T}_{2}({x}^{\ast},{y}^{\ast})+{g}_{2}({y}^{\ast}){g}_{2}({x}^{\ast}),x{g}_{2}({y}^{\ast})\u3009\ge 0\hfill \end{array}
(1.4)
for all x\in K and {\rho}_{1},{\rho}_{2}>0. The problem (1.4) with {g}_{1}={g}_{2} has been studied by [7].
If {T}_{1}={T}_{2}=T and {g}_{1}={g}_{2}=I, then the problem (1.4) reduces to the following system of general variational inequality problem. Find {x}^{\ast},{y}^{\ast}\in K such that
\{\begin{array}{c}\u3008{\rho}_{1}T({y}^{\ast},{x}^{\ast})+{x}^{\ast}{y}^{\ast},x{x}^{\ast}\u3009\ge 0,\hfill \\ \u3008{\rho}_{2}T({x}^{\ast},{y}^{\ast})+{y}^{\ast}{x}^{\ast},x{y}^{\ast}\u3009\ge 0\hfill \end{array}
(1.5)
for all x\in K and {\rho}_{1},{\rho}_{2}>0. The problem (1.5) is studied by Verma [8, 9] and Chang et al. [10].
In the study of variational inequalities, projection methods and its variant form has played an important role. Due to presence of the nonlinear term φ, the projection method and its variant forms cannot be extended to suggest iterative methods for solving mixed variational inequalities. If the nonlinear term φ in the mixed variational inequalities is a proper, convex and lower semicontinuous function, then the variational inequalities involving the nonlinear term φ are equivalent to the fixed point problems and resolvent equations. Hassouni and Moudafi [11] used the resolvent operator technique to study a new class of mixed variational inequalities.
For a multivalued operator T:H\to H, the domain of T, the range of T and the graph of T denote by
D(T)=\{u\in H:T(u)\ne \mathrm{\varnothing}\},\phantom{\rule{2em}{0ex}}R(T)=\bigcup _{u\in H}T(u)
and
Graph(T)=\{(u,{u}^{\ast})\in H\times H:u\in D(T)\text{and}{u}^{\ast}\in T(u)\},
respectively.
Definition 1.1 T is called monotone if and only if for each u\in D(T), v\in D(T) and {u}^{\ast}\in T(u), {v}^{\ast}\in T(v), we have
\u3008{v}^{\ast}{u}^{\ast},vu\u3009\ge 0.
T is maximal monotone if it is monotone and its graph is not properly contained in the graph of any other monotone operator.
{T}^{1} is the operator defined by v\in {T}^{1}(u)\iff u\in T(v).
Definition 1.2 ([12])
For a maximal monotone operator T, the resolvent operator associated with T, for any \sigma >0, is defined as
{J}_{T}(u)={(I+\sigma T)}^{1}(u),\phantom{\rule{1em}{0ex}}\mathrm{\forall}u\in H.
It is known that a monotone operator is maximal if and only if its resolvent operator is defined everywhere. Furthermore, the resolvent operator is singlevalued and nonexpansive i.e., \parallel {J}_{T}(x){J}_{T}(y)\parallel \le \parallel xy\parallel for all x,y\in H. In particular, it is well known that the subdifferential ∂φ of φ is a maximal monotone operator; see [13].
Lemma 1.3 ([12])
For a given
z,u\in H
satisfies the inequality
\u3008uz,xu\u3009+\sigma \phi (x)\sigma \phi (u)\ge 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x\in H
if and only if u={J}_{\phi}(z), where {J}_{\phi}={(I+\sigma \partial \phi )}^{1} is the resolvent operator and \sigma >0 is a constant.
Using Lemma 1.3, we will establish following important relation.
Lemma 1.4 The variational inequality problem (1.1) is equivalent to finding {x}^{\ast},{y}^{\ast}\in H such that
\{\begin{array}{c}{x}^{\ast}={x}^{\ast}{g}_{1}({x}^{\ast})+{J}_{\phi}({g}_{1}({y}^{\ast}){\rho}_{1}{T}_{1}({y}^{\ast},{x}^{\ast})),\hfill \\ {y}^{\ast}={y}^{\ast}{g}_{2}({y}^{\ast})+{J}_{\phi}({g}_{2}({x}^{\ast}){\rho}_{2}{T}_{2}({x}^{\ast},{y}^{\ast})),\hfill \end{array}
(1.6)
where {J}_{\phi}={(I+\partial \phi )}^{1} is the resolvent operator and {\rho}_{1},{\rho}_{2}>0.
Proof Let {x}^{\ast},{y}^{\ast}\in H be a solution of (1.1). Then for all x\in H, we have
\{\begin{array}{c}\u3008{\rho}_{1}{T}_{1}({y}^{\ast},{x}^{\ast})+{g}_{1}({x}^{\ast}){g}_{1}({y}^{\ast}),x{g}_{1}({x}^{\ast})\u3009+\phi (x)\phi ({g}_{1}({x}^{\ast}))\ge 0,\hfill \\ \u3008{\rho}_{2}{T}_{2}({x}^{\ast},{y}^{\ast})+{g}_{2}({y}^{\ast}){g}_{2}({x}^{\ast}),x{g}_{2}({y}^{\ast})\u3009+\phi (x)\phi ({g}_{2}({y}^{\ast}))\ge 0,\hfill \end{array}
which can be written as
\{\begin{array}{c}\u3008{g}_{1}({x}^{\ast})({g}_{1}({y}^{\ast}){\rho}_{1}{T}_{1}({y}^{\ast},{x}^{\ast})),x{g}_{1}({x}^{\ast})\u3009+\phi (x)\phi ({g}_{1}({x}^{\ast}))\ge 0,\hfill \\ \u3008{g}_{2}({y}^{\ast})({g}_{2}({x}^{\ast}){\rho}_{2}{T}_{2}({x}^{\ast},{y}^{\ast})),x{g}_{2}({y}^{\ast})\u3009+\phi (x)\phi ({g}_{2}({y}^{\ast}))\ge 0,\hfill \end{array}
using Lemma 1.3 for \sigma =1, we get
\{\begin{array}{c}{g}_{1}({x}^{\ast})={J}_{\phi}({g}_{1}({y}^{\ast}){\rho}_{1}{T}_{1}({y}^{\ast},{x}^{\ast})),\hfill \\ {g}_{2}({y}^{\ast})={J}_{\phi}({g}_{2}({x}^{\ast}){\rho}_{2}{T}_{2}({x}^{\ast},{y}^{\ast})),\hfill \end{array}
i.e.,
\{\begin{array}{c}{x}^{\ast}={x}^{\ast}{g}_{1}({x}^{\ast})+{J}_{\phi}({g}_{1}({y}^{\ast}){\rho}_{1}{T}_{1}({y}^{\ast},{x}^{\ast})),\hfill \\ {y}^{\ast}={y}^{\ast}{g}_{2}({y}^{\ast})+{J}_{\phi}({g}_{2}({x}^{\ast}){\rho}_{2}{T}_{2}({x}^{\ast},{y}^{\ast})).\hfill \end{array}
This completes the proof. □
Definition 1.5 An operator g:H\to H is said to be

(1)
ζstrongly monotone if for each x,{x}^{\prime}\in H, there exists a constant \zeta >0 such that
\u3008g(x)g\left({x}^{\prime}\right),x{x}^{\prime}\u3009\ge \zeta {\parallel x{x}^{\prime}\parallel}^{2}
for all y,{y}^{\prime}\in H;

(2)
ηLipschitz continuous if for each x,{x}^{\prime}\in H, there exists a constant \eta >0 such that
\parallel g(x)g\left({x}^{\prime}\right)\parallel \le \eta \parallel x{x}^{\prime}\parallel .
An operator T:H\times H\to H is said to be

(3)
relaxed (\omega ,t)cocoercive with respect to the first argument if for each x,{x}^{\prime}\in H, there exist constants t>0 and \omega >0 such that
\u3008T(x,\cdot )T({x}^{\prime},\cdot ),x{x}^{\prime}\u3009\ge \omega {\parallel T(x,\cdot )T({x}^{\prime},\cdot )\parallel}^{2}+t{\parallel x{x}^{\prime}\parallel}^{2};

(4)
μLipschitz continuous with respect to the first argument if for each x,{x}^{\prime}\in H, there exists a constant \mu >0 such that
\parallel T(x,\cdot )T({x}^{\prime},\cdot )\parallel \le \mu \parallel x{x}^{\prime}\parallel ;

(5)
γLipschitz continuous with respect to the second argument if for each y,{y}^{\prime}\in H, there exists a constant \gamma >0 such that
\parallel T(\cdot ,y)T(\cdot ,{y}^{\prime})\parallel \le \gamma \parallel y{y}^{\prime}\parallel .
Lemma 1.6 ([14])
Let \{{a}_{n}\} and \{{b}_{n}\} be two nonnegative real sequences satisfying the following conditions:
{a}_{n+1}\le (1{d}_{n}){a}_{n}+{b}_{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall}n\ge {n}_{0},
where {n}_{0} is some nonnegative integer, {d}_{n}\in (0,1) with {\sum}_{n=0}^{\mathrm{\infty}}{d}_{n}=\mathrm{\infty} and {b}_{n}=o({d}_{n}), then {a}_{n}\to 0 as n\to \mathrm{\infty}.
Several iterative algorithms have been devised to study existence and approximation of different classes of variational inequalities. Most of them are sequential iterative methods, when we implement such algorithms on computers, then only one processor is used at a time. Availability of multiprocessor computers enabled researchers to develop iterative algorithms having the parallel characteristics. Lions [15] studied a parallel algorithm for a solution of parabolic variational inequalities. Bertsekas and Tsitsiklis [16, 17] developed parallel algorithm using the metric projection. Recently, Yang et al. [7] studied parallel projection algorithm for a system of nonlinear variational inequalities.