# Existence of solutions of a new system of generalized variational inequalities in Banach spaces

- Somyot Plubtieng
^{1}Email author and - Tipphawan Thammathiwat
^{1}

**2012**:8

https://doi.org/10.1186/1029-242X-2012-8

© Plubtieng and Thammathiwat; licensee Springer. 2012

**Received: **30 July 2011

**Accepted: **16 January 2012

**Published: **16 January 2012

## Abstract

In this article, we consider the solutions of the system of generalized variational inequality problems in Banach spaces. By employing the generalized projection operator, the well-known Fan's KKM theorem and Kakutani-Fan-Glicksberg fixed point theorem, we establish some new existence theorems of solutions for two classes of generalized set-valued variational inequalities in reflexive Banach spaces under some suitable conditions.

**AMS Subject classification: 47H04; 47H99; 47J20; 47J40**.

### Keywords

system of generalized variational inequalities generalized projection reflexive Banach spaces## 1 Introduction

Let *E* be a Banach space, *E** be the dual space of *E*, and let 〈·,·〉 denotes the duality pairing of *E** and *E*. If E is a Hilbert space and *K* is a nonempty, closed and convex subset of *E*, then it is well known that the metric projection operator *P*_{
K
}: *E* → *K* plays an important role in nonlinear functional analysis, optimization theory, fixed point theory, nonlinear programming, game theory, variational inequality problem, and complementarity problems, etc. (see example, [1–32] and the references therein.)

*K*be a nonempty, closed and convex subset of a Hilbert space

*H*and let

*A*:

*K*→

*H*be a mapping.

*The classical variational inequality problem*, denoted by

*VIP*(

*A, K*), is to find

*x** ∈

*K*such that

for all *z* ∈ *K*. The variational inequality has emerged as a fascinating and interesting branch of mathematical and engineering sciences with a wide range of applications in industry, finance, economics, social, ecology, regional, pure, and applied sciences; see, e.g., [3, 10, 11, 17, 21–24, 29] and the references therein. Related to the variational inequalities, we have the problem of finding the fixed points of the nonexpansive mappings, which is the current interest in functional analysis. It is natural to consider the unified approach to these different problems; see e.g. [17, 20, 22].

The system of variational inequality problems are the model of several equilibrium problems, namely, traffic equilibrium problem, the spatial equilibrium problem, the Nash equilibrium, the general equilibrium programming problem, etc. For further detail see [2, 6, 12, 13, 18, 33] and the references therein. In [6, 18], some solution methods are proposed. However, the existence of a solution of system of variational inequalities is studied in [2, 6, 12, 13, 33].

On the other hand, Verma [23–26] introduced and studied a two step model for some systems of variational inequalities which were difference from the sense of Pang [18] and developed some iterative algorithms for approximating the solutions of these systems in Hilbert spaces base on the convergence analysis of a two step projection method. In 2011, Yao et al. [30] extended the main results of Verma [26] from the Hilbert spaces to the Banach spaces.

*π*

_{ K }:

*E** →

*K*and Π

_{ K }:

*E*→

*K*from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces and studied their in detail. In [35], Alber presented some applications of the generalized projections to approximately solve variational inequalities (1.1) and von Neumann intersection problem in Banach spaces. Let

*A*:

*K*→

*E** be a mapping and let us find

*x**∈

*K*such that

where *ξ* ∈ *E**.

*π*

_{ K }:

*E** →

*K*from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces and studied some properties of generalized projection operator with applications to solve the variational inequality (1.1) in Banach spaces. Very recently,

*the generalized variational inequality problem (GVIP(A,K))*has been studied by many authors (for example, see [19, 28, 36, 37]). It is the problem to find

*x** ∈

*K*such that there exists

*u** ∈

*Ax** satisfying

where *A* : *K* → 2^{E*}is a multivalued mapping with nonempty values and 2^{E*}denotes the family of all subset of *E**.

*T*, a nonempty closed convex subset

*K*of a Banach space

*E*and

*b*∈

*E**is to find $\stackrel{\u0304}{x}\in K$ such that there exists $\u016b\in T\left(\stackrel{\u0304}{x}\right)$ satisfying

in reflexive and smooth Banach spaces by using generalized projection operator, Fan's KKM theorem and minimax theorem.

*K*be a nonempty, closed and convex subset of

*E*and

*A, B*:

*K*→ 2

^{E*}be two multivalued mappings with nonempty values, where 2

^{E*}denotes the family of all subset of

*E**.

*The system of generalized variational inequality problem*(SGVIP(A,B,K)) is to find (

*x*,y**) ∈

*K*×

*K*such that there exist

*u**∈

*Ay*, v**∈

*Bx**satisfying

*A*and

*B*are single-valued, then the system of generalized variational inequality problem is reduced to find (

*x*,y**) ∈

*K*×

*K*such that

which is called *a system of variational inequality problem* (SVIP(A,B,K)).

*Remark* 1.1. (i) *x** ∈ *GVIP*(*A, K*) if and only if (*x*, x**) ∈ *SGVIP*(*A, A, K*).

(ii) *x** ∈ *VIP*(*A, K*) if and only if (*x*, x**) ∈ *SVIP*(*A, A, K*).

The purpose of this article is to establish some existence results of solutions for the system variational inequalities (1.3) in reflexive Banach spaces by employing the properties of the generalized projection operator, the well-known Fan's KKM theorem and Kakutani-Fan-Glicksberg theorem.

## 2 Preliminaries

*E*be a real Banach space and let

*S*= {

*x*∈

*E*: ∥

*x*∥ = 1} be the unit sphere of

*E*. A Banach space

*E*is said to be

*strictly convex*if for any

*x, y*∈

*S*,

*uniformly convex*if for each

*ε*∈ (0, 2], there exists

*δ*> 0 such that for any

*x, y*∈

*S*,

*δ*: [0, 2] → [0,1] called the

*modulus of convexity of E*as follows:

Then *E* is uniformly convex if and only if *δ*(*ε*) > 0 for all *ε* ∈ (0, 2].

*E*is said to be

*locally uniformly convex*if for each

*ε*> 0 and

*x*∈

*S*, there exists

*δ*(

*ε, x*) > 0 for

*y*∈

*S*,

From the above definition, it is easy to see that the following implications are valid: *E* is uniformly convex ⇒ *E* is locally uniformly convex ⇒ *E* is strictly convex

*E*is said to be

*smooth*if the limit

*x, y*∈

*S*. It is also said to be

*uniformly smooth*if the limit (2.5) is attained uniformly for

*x, y*∈

*S*. We recall that

*E*is uniformly convex if and only if

*E**is uniformly smooth. It is well known that

*E*is smooth if and only if

*E**is strictly convex. The mapping

*J*:

*E*→

*E** defined by

is called the *duality mapping of E*. It is known that $J\left(x\right)=\partial \left(\frac{1}{2}{\u2225x\u2225}^{2}\right)$, where *∂ϕ* (*x*) denotes the subdifferential of *ϕ* at *x*. The following properties of duality mapping *J* which are useful for the rest of this work.

**Proposition 2.1**. [38] *Let E be a reflexive Banach space and E* be strictly convex*.

*(i) The duality mapping J* : *E* → *E* is single-valued, surjective and bounded*.

*(ii) If E and E* are locally uniformly convex, then J is a homeomorphism, that is, J and J*^{-1} *are continuous single-valued mappings*.

*V*:

*E**×

*E*→

**ℝ**defined as

It is clear that *V*(*φ, x*) is continuous and the map *x* ↦ *V*(*φ, x*) and *φ* ↦ *V*(*φ, x*) are convex and (∥*φ*∥ - ∥*x*∥)^{2} ≤ *V*(*φ, x*) ≤ (∥*φ*∥ + ∥*x*∥)^{2}. We remark that the main Lyapunov functional *V* was first introduced by Alber [35] and its properties were studied there. By using this functional, Alber defined a generalized projection operator on uniformly convex and uniformly smooth Banach spaces which was further extended by Li [16] on reflexive Banach spaces.

**Definition 2.2**. [16] Let

*E*be reflexive Banach space with its dual

*E**and

*K*be a nonempty, closed and convex subset of

*E*. The operator

*π*

_{ K }:

*E**→

*K*defined by

is said to be a generalized projection operator. For each *φ* ∈ *E**, the set *π*_{
K
}(*φ*) is called the generalized projection of *φ* on *K*.

We mention the following useful properties of the operator *π*_{
k
}(*φ*).

**Lemma 2.3**. [16] *Let E be a reflexive Banach space with its dual E* and K be a nonempty closed convex subset of E, then the following properties hold:*

*(i) The operator π*_{
K
}: *E** → 2^{
K
}*is single-valued if and only if E is strictly convex*.

*(ii) If E is smooth, then for any given φ* ∈ *E**, *x* ∈ *π*_{
K
}*φ if and only if* 〈*φ* - *J*(*x*), *x* - *y*〉 ≥ 0, ∀*y* ∈ *K*.

*(iii) If E is strictly convex, then the generalized projection operator π*_{
K
}: *E** → *K is continuous*.

Lemma 2.4. [5] *In every reflexive Banach space, an equivalent norm can be introduced so that E and E* are locally uniformly convex and thus also strictly convex with respect to the new norm on E and E**.

From Lemma 2.4, we can assume for the rest of this work that the norm ∥·∥ of the reflexive Banach space *E* is such that *E* and *E** are locally uniformly convex. In this case, we note that the generalized metric projection operator *π*_{
K
}and the duality mapping *J* are single-valued and continuous.

*Let A and B be convex subsets of some real topological vector space with B is compact and let p : A*×

*B*→ ℝ.

*If p*(·,

*b*)

*is lower semicontinuous and quasiconvex on A for all b*∈

*B and p*(

*a*, ·)

*is upper semicontinuous and quasiconcave on B for all a*∈

*A, then*

**Definition 2.6**(KKM mapping). Let

*K*be a nonempty subset of a linear space

*E*. A set-valued mapping

*G*:

*K*→ 2

^{ E }is said to be a

*KKM mapping*if for any finite subset {

*y*

_{1},

*y*

_{2},...,

*y*

_{ n }} of

*K*, we have

where *co*{*y*_{1}, *y*_{2}, ..., *y*_{
n
}} denotes the convex hull of {*y*_{1}, *y*_{2},..., *y*_{
n
}}.

**Lemma 2.7** (FanKKM Theorem). *Let K be a nonempty convex subset of a Hausdorff topological vector space E and let G* : *K* → 2^{
E
}*be a KKM mapping with closed values. If there exists a point y*_{0} ∈ *K such that G*(*y*_{0}) *is a compact subset of K, then* ${\cap}_{y\in K}G\left(y\right)\ne \varnothing $.

**Lemma 2.8**. [9] *Let K be a nonempty compact subset of a locally convex Hausdorff vector topology space E. If S* : *K* → 2^{
K
}*is upper semicontinuous and for any x* ∈ *K, S*(*x*) *is nonempty, convex and closed, then there exists an x** ∈ *K such that x** ∈ *S*(*x**).

**Lemma 2.9**. [39] *Let X and Y be two Hausdorff topological vector spaces and T* : *X* → 2^{
Y
}*be a set-valued mapping. Then the following properties hold:*

*(i) If T is closed and* $\overline{T\left(X\right)}$ *is compact, then T is upper semicontinuous, where T*(*X*) = ∪_{x∈X}*T*(*x*) *and* $\overline{T\left(X\right)}$ *denotes the closure of the set T*(*X*).

*(ii) If T is upper semicontinuous and for any x* ∈ *X, T*(*x*) *is closed, then T is closed*.

*(iii) T is lower semicontinuous at x* ∈ *X if and only if for any y* ∈ *T*(*x*) *and any net* {*x*_{
α
}}, *x*_{
α
}→ *x, there exists a net* {*y*_{
α
}} *such that y*_{
α
}∈ *T*(*x*_{
α
}) *and y*_{
α
}→ *y*.

## 3 Main result

**Proposition 3.1**.

*Let E be a reflexive and smooth Banach space and let K be a closed convex subset of E. Assume that α, β*> 0

*and A*:

*K*→ 2

^{E*},

*B*:

*K*→ 2

^{E*}

*are two multivalued mappings with nonempty values. Then*(

*x*, y**)

*is a solution of*(1.3)

*if and only if there exist u**∈

*Ay*, v**∈

*Bx* such that*

*Proof*. It follows from the definition of SGVIP(A,B,K) and Lemma 2.3, that (

*x*, y**) is a solution of (1.3) ⇔ ∃

*u**∈

*Ay*, v**∈

*Bx**such that

**Theorem 3.2**. *Let E be a reflexive and smooth Banach space such that E and E* are locally uniformly convex. Let K be a compact convex subset of E. Let A* : *K* → 2^{E*}*and B* : *K* → 2^{E*}*be two upper semicontinuous multivalued mappings with nonempty values such that A*(*x*) *and B*(*x*) *are weak* compact and convex for each x* ∈ *K. Then the problem* (1.3) *has a solution and the set of solutions* (1.3) *is closed*.

*Proof*. Step 1. Let

*α, β*> 0 and fixed

*x*∈

*K*, for each

*z*∈

*K*, the sets

*G*

_{ x }(

*z*) and

*H*

_{ x }(

*z*) define as follow

(a1) For each *z* ∈ *K, we* have *z* ∈ *G*_{
x
}(*z*) and *z* ∈ *H*_{
x
}(*z*). Hence *G*_{
x
}(*z*) and *H*_{
x
}(*z*) are nonempty subsets of *K*.

(a2) For any finite set {*z*_{1}, *z*_{2},..., *z*_{
n
}} ⊂ *K* we claim that $co\left\{{z}_{1},{z}_{2},...,{z}_{n}\right\}\subset {\bigcup}_{j=1}^{n}{G}_{x}\left({z}_{j}\right)$ and $co\left\{{z}_{1},{z}_{2},...,{z}_{n}\right\}\subset {\bigcup}_{j=1}^{n}{H}_{x}\left({z}_{j}\right)$.

*z*∈

*co*{

*z*

_{1}

*, z*

_{2},...,

*z*

_{ n }}. Then $z={\sum}_{j=1}^{n}{\lambda}_{j}{z}_{j}$ where λ

_{ j }∈ [0, 1] and ${\sum}_{j=1}^{n}{\lambda}_{j}=1$. We observe that

*j*> 0 such that

Therefore $z\in {G}_{x}\left({z}_{j}\right)\subset {\bigcup}_{j=1}^{n}{G}_{x}\left({z}_{j}\right)$. Similarly, we obtain that there exists *k* > 0 such that $z\in {H}_{x}\left({z}_{k}\right)\subset {\bigcup}_{j=1}^{n}{H}_{x}\left({z}_{j}\right)$. Hence we have the claim. This implies that *G*_{
x
}(·) and *H*_{
x
}(·) are KKM-mappings.

Step 2. Show that *G*_{
x
}(*z*) and *H*_{
x
}(*z*) are closed for all *z* ∈ *K*.

*x*

_{ n }} be a sequence in

*G*

_{ x }(

*z*) such that

*x*

_{ n }→

*x*

_{0}in a norm topology. Then there exists

*u*

_{ n }∈

*Ax*such that

*A*(

*x*) is compact, there exists a subsequence $\left\{{u}_{{n}_{j}}\right\}$ of {

*u*

_{ n }} such that ${u}_{{n}_{j}}\to {u}_{0}\in A\left(x\right)$. Thus without loss of generality, we may assume that

*u*

_{ n }→

*u*

_{0}and observe that

This implies that *x*_{0} ∈ *G*_{
x
}(*z*) and so *G*_{
x
}(*z*) is closed for all *z* ∈ *K*. Similarly, we obtain that *H*_{
x
}(*z*) is closed for all *z* ∈ *K*. Then ∩_{z ∈ K}*G*_{
y
}(*z*) and ∩_{z ∈ K}*H*_{
x
}(*z*) are also closed.

Step 3. Show that ${\bigcap}_{z\in K}{G}_{x}\left(z\right)\ne \varnothing \ne {\bigcap}_{z\in K}{H}_{z}\left(z\right)$.

Since *G*_{
x
}(*z*) and *H*_{
x
}(*z*) are closed subsets of *K* and *K* is compact, *G*_{
x
}(*z*) and *H*_{
x
}(*z*) are compact subsets of *K*. It follows from Steps 1, 2, and Lemma 2.7 that ${\bigcap}_{z\in K}{G}_{x}\left(z\right)\ne \varnothing \ne {\bigcap}_{z\in K}{H}_{z}\left(z\right)$.

Step 4. Show that the problem (1.3) has a solution.

*x, y*∈

*K*, we may choose $\stackrel{\u0304}{x}\in {\bigcap}_{z\in K}{G}_{y}\left(z\right)$ and $\u0233\in {\bigcap}_{z\in K}{H}_{x}\left(z\right)$ by Step 3. We define the set-valued mapping

*S*:

*K*×

*K*→ 2

^{K×K}by

*S*(

*x, y*) and Step 3, we obtain that

*S*(

*x, y*) is a nonempty closed convex subset of

*K*×

*K*for all (

*x, y*) ∈

*K*×

*K*. Since ${\bigcap}_{z\in K}{G}_{y}\left(z\right),{\bigcap}_{z\in K}{H}_{x}\left(z\right)\subset K$ and

*K*is compact, $\overline{{\bigcap}_{z\in K}{G}_{y}\left(z\right)}$ and $\overline{{\bigcap}_{z\in K}{H}_{y}\left(z\right)}$ are compact. We only show that

*S*is a closed mapping. Indeed, let {(

*x*

_{ n },

*y*

_{ n })} be a net in

*K*×

*K*such that (

*x*

_{ n },

*y*

_{ n }) → (

*x*

_{0},

*y*

_{0}) in the norm topology and let (

*u*

_{ n },

*v*

_{ n }) ∈

*S*(

*x*

_{ n },

*y*

_{ n }) such that (

*u*

_{ n },

*v*

_{ n }) → (

*u*

_{0},

*v*

_{0}). By definition of a mapping

*S*, we have $\left({u}_{n},{v}_{n}\right)\in \left(\left\{{\stackrel{\u0304}{x}}_{n}\right\},\left\{{\u0233}_{n}\right\}\right)$ where ${\stackrel{\u0304}{x}}_{n}\in {\bigcap}_{z\in K}{G}_{{y}_{n}}\left(z\right)$ and ${\u0233}_{n}\in {\bigcap}_{z\in K}{H}_{{x}_{n}}\left(z\right)$. That is for each $z\phantom{\rule{2.77695pt}{0ex}}\in K,\phantom{\rule{2.77695pt}{0ex}}{u}_{n}={\stackrel{\u0304}{x}}_{n}\in {G}_{{y}_{n}}\left(z\right)$ and ${v}_{n}={\u0233}_{n}\in {H}_{{x}_{n}}\left(z\right)$. It follows from (3.2) that there exist

*a*

_{ n }∈

*Ay*

_{ n }and

*b*

_{ n }∈

*Bx*

_{ n }such that

Now, we define two sets *T*_{1} := {*x*_{1}, *x*_{2},..., *x*_{
n
},...} ∪ {*x*_{0}} and

*T*_{2} := {*y*_{1}, *y*_{2},...,*y*_{
n
},...} ∪ {*y*_{0}}.

*A*(

*T*

_{2}) and

*B*(

*T*

_{1}) are compact. Thus there exist two subsequences $\left\{{a}_{{n}_{j}}\right\}$ of {

*a*

_{ n }} and $\left\{{b}_{{n}_{k}}\right\}$ of {

*b*

_{ n }} such that ${a}_{{n}_{j}}\to {a}_{0}\in A\left({T}_{2}\right)$ and ${b}_{{n}_{k}}\to {b}_{0}\in B\left({T}_{2}\right)$. Since

*A*and

*B*are upper semicontinuous,

*a*

_{0}∈

*Ay*

_{0}and

*b*

_{0}∈

*Bx*

_{0}. Taking

*j, k*→ ∞ in (3.7), we obtain that

*u*

_{0}∈

*G*

_{y 0}(

*z*) and ${v}_{0}\in {H}_{{x}_{0}}\left(z\right)$ for all

*z*∈

*K*. This implies that (

*u*

_{0},

*v*

_{0}) ∈ ({

*u*

_{0}}, {

*v*

_{0}}) =

*S*(

*x*

_{0},

*y*

_{0}). Thus,

*S*is a closed mapping. It follows from Lemma 2.9 that

*S*is upper semicontinuous. By Lemma 2.8, there exists a point $\left({x}^{*},{y}^{*}\right)\in S\left({x}^{*},{y}^{*}\right)=\left(\left\{\stackrel{\u0304}{x}\right\},\left\{\u0233\right\}\right)$ where $\stackrel{\u0304}{x}\in {\bigcap}_{z\in K}{G}_{{y}^{*}}\left(z\right)$ and $\u0233\in {\bigcap}_{z\in K}{H}_{{x}^{*}}\left(z\right)$. That is ${x}^{*}=\stackrel{\u0304}{x}\in {G}_{{y}^{*}}\left(z\right)$ and ${y}^{*}=\u0233\in {H}_{{x}^{*}}\left(z\right)$ for all

*z*∈

*K*. By definition of

*G*

_{ y* }(

*z*) and

*H*

_{ x* }.(

*z*), we get

*p*

_{1}(·,

*z*),

*p*

_{2}(·,

*z*) are lower semicontinuous and convex. Also the function

*p*

_{1}(

*u*, ·),

*p*

_{2}(

*v*, ·) are upper semicontinuous and concave. Apply minimax theorem, we have

*u*↦ sup

_{z∈ K}

*p*

_{1}(

*u, z*) and

*v*↦ sup

_{z∈ K}

*p*

_{2}(

*v, z*) are lower semicontinuous and

*A*(

*y**),

*B*(

*x**) are compact, there exist

*u**∈

*A*(

*y**) and

*v**∈

*B*(

*x**) such that

By definition of generalized projection operator, we get *x** = *π*_{
K
}(*J*(*x**) - *αu**) and *y** = *π*_{
K
}(*J*(*y**) - *βv**). It follows from Proposition 3.1 that (*x*, y**) is the solutions of problem (1.3).

Step 5. Show that the set of solutions (1.3) is closed.

*T*:= {(

*x, y*) ∈

*K*×

*K*: (

*x, y*) is a solution of (1.3)}. Let {(

*x*

_{ n },

*y*

_{ n })} be a net in

*T*such that (

*x*

_{ n }

*, y*

_{ n }) → (

*x*

_{0},

*y*

_{0}) in the norm topology. By definition (1.3) we obtain that there exist

*u*

_{ n }∈

*A*(

*y*

_{ n }) and

*v*

_{ n }∈

*B*(

*x*

_{ n }) such that

We define two sets *T*_{1} := {*x*_{1}, *x*_{2}, ..., *x*_{
n
},...} ∪ {*x*_{0}} and *T*_{2} := {*y*_{1}, *y*_{2}, ..., *y*_{
n
},...} ∪ {*y*_{0}}.

*A*(

*T*

_{2}) and

*B*(

*T*

_{1}) are compact. Thus there exist two subsequences $\left\{{u}_{{n}_{j}}\right\}$ of {

*u*

_{ n }} and $\left\{{v}_{{n}_{k}}\right\}$ of {

*v*

_{ n }} such that ${u}_{{n}_{j}}\to {u}_{0}\in A\left({T}_{2}\right)$ and ${v}_{{n}_{k}}\to {v}_{0}\in B\left({T}_{2}\right)$. Since

*A*and

*B*are upper semicontinuous,

*u*

_{0}∈

*Ay*

_{0}and

*v*

_{0}∈

*Bx*

_{0}. Taking

*j, k*→ ∞ in (3.16), we obtain that

Thus (*x*_{0}, *y*_{0}) ∈ *T* and so *T* is closed. This completes the proof.

If *A* and *B* are two single-valued mappings, then from Theorem 3.2, we derive the following result.

**Corollary 3.3**. *Let E be a reflexive and smooth Banach space such that E and E* are locally uniformly convex. Let K be a compact convex subset of E. Let A* : *K* → *E* and B:K* → *E* be two continuous single-valued mappings. Then the problem* (1.4) *has a solution and the set of solutions* (1.4) *is closed*.

## Declarations

### Acknowledgements

The authors would like to thanks The Thailand Research Fund for financial support. Moreover, TT is also supported by the Royal Golden Jubilee Program under Grant PHD/0117/2549, Thailand.

## Authors’ Affiliations

## References

- Baiocchi C, Capelo A:
*Variational and Quasi-Variational Inequalietis, Application to Free Boundary Problem.*Wiley, New York/London; 1984.Google Scholar - Bianchi M: Pseudo P-monotone operators and variational inequalities. In
*Report 6 Istituto di econometria e Matematica per le decisioni economiche*. Universita Cattolica del Sacro Cuore, Milan, Italy; 1993.Google Scholar - Chang SS, Lee Joseph HW, Chan CK: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization.
*Nonlinear Anal*2009, 70: 3307–3319. 10.1016/j.na.2008.04.035MATHMathSciNetView ArticleGoogle Scholar - Cho YJ, Fang YP, Huang NJ, Hwang HJ: Algorithms for systems of nonlinear variational inequalities.
*J Korean Math Soc*2004, 41: 489–499.MATHMathSciNetView ArticleGoogle Scholar - Cioranescu I:
*Geometry of Banach Spaces Duality Mappings and Nonlinear Problem.*Kluwer Acadimic Publishers, Dordrecht; 1990.View ArticleGoogle Scholar - Cohen G, Chaplais F: Nested monotony for variational inequalities over a product of spaces and convergence of iterative algorithms.
*J Optim Theory Appl*1988, 59: 360–390.MathSciNetView ArticleGoogle Scholar - Fan K: A generalization of Tychonoff's fixed point theorem.
*Math Ann*1961, 142: 305–310. 10.1007/BF01353421MATHMathSciNetView ArticleGoogle Scholar - Giannessi F, Maugeri A:
*Vairational Inequalities and Network Equilibrium Problems.*Plenum, New York; 1995.View ArticleGoogle Scholar - Holmes RB:
*Geometric Functional Analysis and its Application.*Springer, New York; 1975.View ArticleGoogle Scholar - Isac G: Complementarity problems. In
*Lecture Notes in Math*.*Volume 1528*. Springer, Berlin; 1992.Google Scholar - Isac G, Sehgal VM, Singh SP: An altenate version of a variational inequality.
*Indian J Math*1999, 41: 25–31.MATHMathSciNetGoogle Scholar - Kassay G, Kolumbán J: System of multi-valued variational variational inequalities.
*Publ Math Debrecen*1999, 54: 267–279.MATHMathSciNetGoogle Scholar - Kassay G, Kolumbán J, Páles Z: Factorization of Minty and Stampacchia variational inequality system.
*European J Oper Res*2002, 143: 377–389. 10.1016/S0377-2217(02)00290-4MATHMathSciNetView ArticleGoogle Scholar - Kim JK, Kim DS: A new system of generalized nonlinear mixed variational inequalities in Hilbert spaces.
*J Convex Anal*2004, 11: 235–243.MATHMathSciNetGoogle Scholar - Li JL: On the existence of solutions of variational inequalities in Banach space.
*J Math Anal Appl*2004, 295: 115–126. 10.1016/j.jmaa.2004.03.010MATHMathSciNetView ArticleGoogle Scholar - Li JL: The generalized projection operator on reflexive Banach spaces and its application.
*J Math Anal Appl*2005, 306: 55–71. 10.1016/j.jmaa.2004.11.007MATHMathSciNetView ArticleGoogle Scholar - Nadezhkina N, Takahashi W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings.
*J Optim Theory Appl*2006, 128: 191–201. 10.1007/s10957-005-7564-zMATHMathSciNetView ArticleGoogle Scholar - Pang JS: Asymmetric variational inequality problems over product sets: Applications and iterative methods.
*Math Program*1985, 31: 206–219. 10.1007/BF02591749MATHView ArticleGoogle Scholar - Shih M-H, Tan K-K: Browder-Hartman-Stempacchia variational inequalities for multi-valued monotone operators.
*J Math Anal Appl*1988, 134: 431–440. 10.1016/0022-247X(88)90033-9MATHMathSciNetView ArticleGoogle Scholar - Suzuki T: Strong convergence of krasnoselskii and mann's type sequences for one-parameter nonexpansive semigroups without bochner integrals.
*J Math Anal Appl*2005, 305: 227–239. 10.1016/j.jmaa.2004.11.017MATHMathSciNetView ArticleGoogle Scholar - Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces.
*J Math Anal Appl*2007, 311(1):506–515.View ArticleGoogle Scholar - Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings.
*J Optim Theory Appl*2003, 118: 417–428. 10.1023/A:1025407607560MATHMathSciNetView ArticleGoogle Scholar - Verma RU: On a new system of nonlinear variational inequalities and associated iterative algorithms.
*Math Sci Res Hot-Line*1999, 3(8):65–68.MATHMathSciNetGoogle Scholar - Verma RU: Iterative algorithms and a new system of nonlinear quasivariational inequalities.
*Adv Nonlinear Var Inequal*2001, 4(1):117–124.MATHMathSciNetGoogle Scholar - Verma RU: Projection methods, algorithms, and a new system of nonlinear variational inequalities.
*Comp Math Appl*2001, 41: 1025–1031. 10.1016/S0898-1221(00)00336-9MATHView ArticleGoogle Scholar - Verma RU: General convergence analysis for two-step projection methods and application to variational problems.
*Appl Math Lett*2005, 18: 1286–1292. 10.1016/j.aml.2005.02.026MATHMathSciNetView ArticleGoogle Scholar - Wong M-M, Ansari QH, Yao J-C: Existence of solutions of generalized variational inequalities in reflexive Banach spaces.
*Appl Math Lett*2009, 22: 197–201. 10.1016/j.aml.2008.03.009MATHMathSciNetView ArticleGoogle Scholar - Yao JC: Muti-valued variational inequalities with K-psuedomonotone opeerators.
*J Optim Theory Appl*1994, 83(2):391–403. 10.1007/BF02190064MATHMathSciNetView ArticleGoogle Scholar - Yao JC, Chadli O: Pseudomonotone Complementarity Problems and variational Inequalities. In
*Handbook of Generalized Convexity and Monotonicity*. Edited by: Crouzeix, JP, Haddjissas, N, Schaible, S. Springer, New York; 2005:501–558.View ArticleGoogle Scholar - Yao Y, Liou Y-C, Kang SM: Two step projdection methods for a system of variational inequality problems in Banach spaces.
*J Glob Optim*2011.Google Scholar - Yao Y, Shahzad N: Strong convergence of a proximal point algorithm with general errors.
*Optim Lett*2011.Google Scholar - Yuan GXZ:
*KKM Theory and Applications in Nonlinear Analysis.*Marcel-Dekker, New York; 1999.Google Scholar - Ansari QH, Yao JC: A fixed point theorem and its applications to a system of variational inequalities.
*Bull Aust Math Soc*1999, 59: 433–442. 10.1017/S0004972700033116MATHMathSciNetView ArticleGoogle Scholar - Alber Ya: Generalized projection operators in Banach spaces: properties and application.
*Proceedinng of the Israel Seminar Ariel Israel Funct Differential Equation vol*1994, 1: 1–21.Google Scholar - Alber Ya: Metric and Genernalized Projection Operators in Banach Space: Properties and Application. In
*Theory and Applications of Nonlinear Operators of Accretive and Monotone Type*. Edited by: Kartsatos, A. Marcel Dekker, New York; 1996:15–50.Google Scholar - Ansari QH, Lin YC, Yao JC: General KKM theorem with applications to minimax and variational inequalities.
*J Optim Theory Appl*2000, 104(1):41–57.MATHMathSciNetView ArticleGoogle Scholar - Zeng LC, Yao JC: Existence theorems for variational inequalities in Banach spaces.
*J Optim Theory Appl*2007, 132(2):321–337. 10.1007/s10957-006-9139-zMATHMathSciNetView ArticleGoogle Scholar - Zeidler E:
*Nonlinear Functional Analysis and Its Application, I Fixed-Point Theorems.*Springer, New York; 1993.Google Scholar - Aubin JP, Ekeland I:
*Applied Nonlinear Analysis.*Wiley, New York; 1984.Google Scholar

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