Implicit vector quasi-equilibrium problems with applications to variational inequalities

Nonlinear Analysis 63 (2005) e1823–e1831www.elsevier.com/locate/na

Implicit vector quasi-equilibrium problems withapplications to variational inequalities�

Abdul KhaliqDepartment of Mathematics, University of Jammu, Jammu and Kashmir-180 006, India

Abstract

In this paper, we introduce and study a class of implicit vector quasi-equilibrium problems, whichincludes implicit vector equilibrium problems, implicit vector quasi-variational inequalities and im-plicit vector quasi-complementarity problems as special cases. We establish some existence resultsunder compact and noncompact settings by using one person game theorems. As an application of ourresults, we have obtained existence results for variational inequalities. Our results extends and unifythe well known earlier works of many authors.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Vector quasi-equilibrium problem; Implicit vector equilibrium problem; Quasi-variationalinequalities; Upper semicontinuous mapping

1. Introduction

Vector equilibrium problems (VEP) have attracted increasing attention of many au-thors and have been proven to be significant in the study of vector optimization, vectorvariational inequalities and vector complementarity problems, see [2,3,12,24]. For compre-hensive bibliography we refer to Daniele et al. [6], Giannessi [12] and references therein.Very recently, implicit vector equilibrium problem (IVEP) in Hausdorff topological vec-tor spaces was introduced by Li et al. [15] and they established some existence resultson noncompact set by using KKM–Fan theorem. The IVEP includes the VEP, implicit

� To be presented as an invited talk in the “Fourth World Congress of Nonlinear Analysts” (WCNA-2004),to be held at Orlendo, Florida, USA during June 30 through July 7, 2004.

E-mail address: [email protected].

0362-546X/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2005.01.070

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mailto:[email protected]

e1824 A. Khaliq / Nonlinear Analysis 63 (2005) e1823–e1831

variational inequalities and implicit complementarity problems as special cases. Lin andPark [20] and Ding [8,9] introduced and studied the quasi-equilibrium problems in G-convexspaces and general topological spaces using fixed point approach. Recently, generalizationof quasi-equilibrium problems to vector-valued function has been studied by Ansari andYao [3], Ansari and Flores-Bazan [1], Khaliq and Krishan [17] and references therein.It is a unified model of several known problems, for instance, vector quasi-optimizationproblem, vector quasi-saddle point problem, vector quasi-variational inequality and vectorquasi-variational-like inequality problems, see [4,21,22].

The main purpose of this paper is to establish some existence results for solutions toimplicit vector quasi-equilibrium problem (IVQEP) under compact and noncompact settingsby using one person game theorems. As special cases we derive some existence results forsolutions to Variational inequalities. Our results generalize and improve the correspondingresults in the literature.

2. Preliminaries

Let X be a vector space and K ⊂ X. We shall denote by co(K), the convex hull of K. If Kis a subset of a topological space X, the interior of K in X is denoted by intX(K) and closureof K in X is denoted by clX(K) or simply int(K), and cl(K) if there is no ambiguity,respectively. Let X and Y be two sets. We shall denote by 2X the family of all subsetsof X and if F, G : X → 2Y be multifunctions, then the multifunction F ∩ G : X → 2Y

is defined by (F ∩ G)(x) = F(x) ∩ G(x) for each x ∈ X. Let X and Y be topologicalvector spaces and T : X → 2Y be a multifunction. The graph of T denoted by G(T ) is theset {(x, y) ∈ X × Y : x ∈ X, y ∈ T (x)} and the multifunction T : X → 2Y is defined byT (x)={y ∈ Y : (x, y) ∈ clX×YG(T )}. The set clX×YG(T ) is called adherence of the graphof T. The multifunction clT : X → 2Y is defined by (clT )(x) = clT (x) for each x ∈ X. Itcan be seen easily that clT (x) ⊂ T (x) for each x ∈ X. The inverse of T denoted by T −1 is amultifunction from R(T ), range of T, to X defined by x ∈ T −1(y) if and only if y ∈ T (x).Also, T is said to be upper semicontinuous on X if for each x ∈ X and each open set U in YcontainingT (x), there exists an open neighbourhoodV of x in X such thatT (y) ⊆ U , for eachy ∈ V . Let K be a nonempty convex subset of X and C : K → 2Y be a multifunction such thatfor each x ∈ K, C(x) is a closed, convex cone with int C(x) �= ∅, where int C(x) denotesthe interior of C(x). It is clear that the cone C(x) for each x ∈ K can define on Y a partialorder �Cx

by y�Cxz if and only if z−y ∈ C(x). We shall write y≺Cx z if z−y ∈ int C(x) in

the case int C(x) �= ∅. The multifunction T : K → 2Y is said to be Cx—convex if for eachx, y ∈ K and � ∈ [0, 1], T (�y + (1 − �)x)�Cx

�T (y) + (1 − �)T (x). Let f : K × K → Y

be a vector valued bifunction, A : K → 2K a multifunction and g : K → K a mapping.We consider the following implicit vector quasi-equilibrium problem:

Find x∗ ∈ K such that

x∗ ∈ clK A(x∗) and f (g(x∗), y) /∈ − intY C(x∗) for all y ∈ A(x∗). (IVQEP)

For A(x)=K for each x ∈ K , (IVQEP) is known as IVEP [15], which includesVEP [12,24],equilibrium problems [4,19], vector optimization problems, Nash equilibrium problem forvector valued functions and vector saddle point problems as special cases.

A. Khaliq / Nonlinear Analysis 63 (2005) e1823–e1831 e1825

If g is the identity mapping on K, then (IVQEP) reduces to the following vector quasi-equilibrium problem:


x∗ ∈ clK A(x∗) and f (x∗, y) /∈ − intY C(x∗), ∀y ∈ A(x∗). (VQEP)

This problem was considered by Khaliq and Krishan [17], which includes vector quasi-equilibrium problems, vector quasi-variational-like inequality problems and vector quasi-variational inequality problems as special cases; see, for example [3,14,16–18] andreferences therein.

If Y = R and C(x) = R+ for all x ∈ K , the (IVQEP) reduces to the following implicitquasi-equilibrium problem:


x∗ ∈ clK A(x∗) and f (g(x∗), y)�0, ∀y ∈ A(x∗). (IQEP)

This problem includes equilibrium problems considered in [4,8,9,20] as special cases.The following results which can be obtained easily from Theorem 2 in [10] and Theorem

2 in [11] as special cases will be used in proving the main results of this paper, see also Kimand Tan [13].

Lemma A. Let K be a nonempty compact convex subset of a Hausdorff topological vectorspace X. Suppose that A, clX A, P : K → 2K are multifunctions such that for all x ∈K, A(x) is a nonempty convex set, for all y ∈ K, A−1(y) is an open set in K,clX A is uppersemicontinuous, for all x ∈ K, x /∈ co P (x) and and for all y ∈ K, P −1(y) is open in K.Then there exists x∗ ∈ K such that x∗ ∈ clK A(x∗) and A(x∗) ∩ P(x∗) = ∅.

Lemma B. Let K be a nonempty convex subset of a locally convex Hausdorff topologicalvector space X and D be a nonempty compact subset of K. Suppose that A, P : K → 2D

and clXA : K → 2K are multifunctions such that for all x ∈ K, A(x) is a nonemptyconvex set, for all y ∈ D, A−1(y) is an open set in K, clXA is upper semicontinuous, forall x ∈ K, x /∈ co P (x) and for all y ∈ D, P −1(y) is open in K. Then there exists x∗ ∈ K

such that x∗ ∈ clKA(x∗) and A(x∗) ∩ P(x∗) = ∅.

3. Existence results

We first establish an existence result for solution to (IVQEP) in compact settings.

Theorem 3.1. Let K be a nonempty compact convex subset of a Hausdorff topologicalvector space X and Y be an ordered Hausdorff topological vector space. Let C : K → 2Y

and A : K → 2K be the multifunctions such that ∀x ∈ K, A(x) is nonempty convexand ∀y ∈ K, A−1(y) is open in K, clK A : K → 2K is upper semicontinuous and ∀x ∈K, C(x) is closed, convex and pointed cone in Y such that intY C(x) is nonempty. Letf : K × K → Y be a vector valued bifunction and g : K → K a mapping. Suppose that


the following assumptions hold:

(1) g is continuous,(2) f (g(x), x) /∈ − intY C(x), ∀x ∈ K ,(3) f is Cx—convex in the second argument and continuous in the first argument, and(4) the point to set mapping W : K → 2Y defined by W(x) = Y\(−intY C(x)) ∀x ∈ K ,

is upper semicontinuous on K.

Then there exists x∗ ∈ K such that

x∗ ∈ clK A(x∗) and f (g(x∗), y) /∈ − intY C(x∗) for all y ∈ A(x∗).

Proof. Define G : K → 2K by

G(x) = {y ∈ K : f (g(x), y) ∈ −intY C(x)}, ∀x ∈ K .

We first prove that x /∈ co G(x), ∀x ∈ K . Suppose that there exists xo ∈ K such thatxo ∈ co P (xo). This implies that xo can be expressed as

xo =∑i∈I

�iyi , with �i �0,∑i∈I

�i = 1, i = 1, . . . , n,

where {yi : i ∈ N} is a finite subset of K, I ⊂ N be arbitrary nonempty subset where N

denotes the set of natural numbers. This follows

f (g(xo), yi) ∈ −intY C(xo) for all i = 1, . . . , n.

Hence,∑i∈I

�if (g(xo), yi) ∈ −intY C(xo). (3.1)

From assumption (3), we have

f (g(xo), xo) = f

(g(xo),

∑i∈I

�iyi

)�Cxo

∑i∈I

�if (g(xo), yi).

Hence,∑i∈I

�if (g(xo), yi) − f (g(xo), xo) ∈ C(xo). (3.2)

From (3.1) and (3.2) and Lemma 1.1, we have

f (g(xo), xo) ∈ −intY C(xo).

which contradicts assumption (2).


Now we show that G−1(y) is open in K, which is equivalent to show that [G−1(y)]c =K\G−1(y) is closed. Indeed we have

G−1(y) = {x ∈ K : y ∈ G(x)}= {x ∈ K : f (g(x), y) ∈ −intY C(x)}.

[G−1(y)]c = {x ∈ K : f (g(x), y) /∈ − intY C(x)}.Let u ∈ [G−1(y)]c, the closure of [G−1(y)]c in K. We claim that u ∈ [G−1(y)]c. Indeed,let {x�}�∈� be a net in [G−1(y)]c such that x� → u. Then we have f (g(x�), y) /∈ −intY C(x�) ∀y ∈ K , that is, f (g(x�), y) ∈ W(x�)=Y\(−intY C(x)). Since f is continuousin the first argument and g is continuous, we have f (g(x�), y) → f (g(u), y). By uppersemicontinuity of W, it follows that f (g(u), y) ∈ W(u), that is f (g(u), y) /∈ − intY C(u)

and so [G−1(y)]c is closed. Thus all the hypotheses of Lemma A are satisfied. Hence thereexists x∗ ∈ K such that

x∗ ∈ clKA(x∗) and A(x∗) ∩ G(x∗) = ∅,

which implies that there exists x∗ ∈ K such that

x∗ ∈ clKA(x∗) and f (g(x∗), y) /∈ − intY C(x∗)

for all y ∈ A(x∗). �

Example 3.1. Let X=Y =R and K=[o, 1]. It is easy to see that A : K → 2K , g : K → K ,C : K → 2Y and f : K × K → Y defined as

A(x) ={ {0} if x = 0

[0, x) x ∈ (0, 1],g(x) = x, C(x) = R+ and f (x, y) = x − y, ∀x, y ∈ K satisfies all the assumptions ofTheorem 2.1 and there exists x∗ = 1 ∈ K such that

x∗ ∈ clKA(x∗) and f (g(x∗), y) /∈ − intY C(x∗)

for all y ∈ A(x∗).

For the noncompact case we need the following concept of escaping sequences introducedin Border [5].

Definition 3.1. Let X be a topological space and K a subset of X such that K =⋃∞n=1Kn,

where {Kn}∞n=1 is an increasing sequence of nonempty compact sets in the sense that Kn ⊆Kn+1 for all n ∈ N. A sequence {xn}∞n=1 in K is said to be escaping sequence from K(relative to {Kn}∞n=1) if for each n there is an M such that k�M, xk /∈ Kn.

Theorem 3.2. Let K be a nonempty subset of a Hausdorff topological vector space X andK =⋃∞

n=1Kn, where {Kn}∞n=1 is an increasing sequence of nonempty, compact and convexsubsets of K. Let Y, f, g, C, W and A be the same as in Theorem 3.1 and satisfy all the


conditions. In addition, suppose that for each sequence {xn}∞n=1 in K with xn ∈ Kn, n ∈ N

which is escaping from K relative to {Kn}∞n=1, there exists m ∈ N andym ∈ Km ∩ A(xm)

such that for each xm ∈ clK A(xm),

f (g(xm), ym) ∈ −intY C(xm).



Proof. Since for each n ∈ N, Kn is compact and convex set in X, applying Theorem 2.1for all n ∈ N, there exists xn ∈ Kn such that

xn ∈ clK A(xn) and f (g(xn), z) /∈ − intY C(xn) for all z ∈ A(xn). (3.3)

Suppose that the sequence {xn}∞n=1 is escaping from K relative to {Kn}∞n=1. By the givenassumption there exists m ∈ N and zm ∈ Km ∩ A(xm) such that for each xm ∈ clK A(xm),

f (g(xm), zm) ∈ −intY C(xm),

which contradicts (3.3). Hence {xn}∞n=1 is not an escaping sequence from K relative to{Kn}∞n=1. Therefore, there exists r ∈ N and there is some subsequence {xjn} of {xn}∞n=1which must lie entirely in Kr . Since Kr is compact, there is a subsequence {xin}in∈� of{xjn} in Kr and there exists x∗ ∈ Kr such that xin → x∗, where in → ∞. Since {Kn}∞n=1is an increasing sequence for all y ∈ K , there exists i0 ∈ � with io > r , such that y ∈ Ki0 ,for all in ∈ � and in > io, we have y ∈ Ki0 ⊆ Kin , such that

f (g(xin), y) /∈ − intY C(xin),

which implies that

f (g(xin), y) ∈ W(xin).

Also, since f is continuous in the first argument and g is continuous, we have

f (g(xin), y) → f (g(x∗), y).

By the upper semicontinuity of W, we have

f (g(x∗), y) ∈ W(x∗).

That is,

f (g(x∗), y) /∈ − intY C(x∗).

Since clK A : → 2K is upper semicontinuous with compact values, there exists x∗ ∈ K

such that

x∗ ∈ clK A(x∗) and f (g(x∗), y) /∈ − intY C(x∗)

for all y ∈ A(x∗). �


Theorem 3.3. Let K be a nonempty convex subset of a locally convex Hausdorff topologicalvector space X and D be a nonempty compact subset of K. Let Y be an ordered Hausdorfftopological vector space. Let f : K×K → Y be a vector valued bifunction and g : K → K

a mapping. Let C : K → 2Y be a multifunction such that for each x ∈ K, C(x) is closed,convex and pointed cone inY with intY C(x) �= ∅. LetA, clK A : K → 2D be multifunctionssuch that for each x ∈ K, A(x) is nonempty convex, for each y ∈ K, A−1(y) is open in Kand clK A is upper semicontinuous. Suppose that conditions (1)–(4) of Theorem 3.1 holds.Then there exists x∗ ∈ K such that


Proof. We consider the set

G(x) = {y ∈ D : f (g(x), y) ∈ −intY C(x)} for all x ∈ K .

Then by using the same argument, which we have used in proving Theorem 3.1, we havex /∈ co G(x) for each x ∈ K and G−1(y) is open for each y ∈ D. Thus all the conditions ofLemma B are satisfied. Hence there exists a solution to (IVQEP). �

Remark 3.1. Theorems 3.1–3.3 generalize and improve the corresponding results in[3,4,8,9,12,17,24].

4. Applications

In this section we establish some existence results for variational inequalities. We needthe following:

Lemma 4.1 (Ding [7]). Let X and Y be topological vector spaces and let L(X, Y ) beequipped with the uniform convergence topology �. Then the bilinear form 〈., .〉 : L(X, Y )×X → Y is continuous on (L(X, Y ), �) × X.

Theorem 4.1. Let K be a nonempty compact convex subset of a Hausdorff topologicalvector space X and Y be an ordered Hausdorff topological vector space. Let C : K → 2Y

and A : K → 2K be the multifunctions such that ∀x ∈ K, A(x) is nonempty convexand ∀y ∈ K, A−1(y) is open in K, clK A : K → 2K is upper semicontinuous and ∀x ∈K, C(x) is closed, convex and pointed cone in Y such that intY C(x) is nonempty. LetL(X, Y ) be the space of all continuous linear operators from X toY and 〈u, x〉, the evalutionof u ∈ L(X, Y ) at x ∈ X. Let T : K → L(X, Y ), g : K → K and � : K × K → X bemappings. Suppose that the following assumptions hold:

(1) � is Cx—convex in the first argument and g and T are continuous;(2) 〈T (g(x)), �(x, g(x))〉 /∈ − intY C(x), ∀x ∈ K;(3) the point to set mapping mappingW : K → 2Y defined byW(x)=Y\(−intY C(x)) ∀x ∈

K , is upper semicontinuous on K.



x∗ ∈ clK A(x∗) and 〈T (g(x∗)), �(y, g(x∗))〉 /∈ − intY C(x∗), (GVQVLIP)

f or all y ∈ A(x∗).

Proof. Taking f (g(x), y)=〈T (g(x)), �(y, g(x))〉 for all x, y ∈ K , and using Lemma 4.1,we see that all the assumptions of Theorem 3.1 are satisfied. Hence the conclusion followsfrom Theorem 3.1. �

Corollary 4.1. If in Theorem 4.1 we assume that g is the identity mapping on K,h : K → K ,�(y, x) = y − h(x) for all x, y ∈ K , and all the assumptions are satisfied, then there existsx∗ ∈ K such that

x∗ ∈ clK A(x∗) and 〈T (x∗), y − h(x∗)〉 /∈ − intY C(x∗), (GVQVIP)

for all y ∈ A(x∗).

Corollary 4.2. If in Theorem 4.1 we assume that Y = R, C(x) = R+ for each x ∈ K ,L(X, Y ) = X∗ and all the assumptions are satisfied, then there exists x∗ ∈ K such that

x∗ ∈ clK A(x∗) and 〈T (g(x∗)), �(y, g(x∗))〉�0, for all y ∈ A(x∗). (GQVIP)

Remark 4.1. Theorem 4.1 extends and generalizes the corresponding results in[14,13,18,23]. Also, by making use of Theorems 3.2 and 3.3, we can derive the existenceresults in noncompact settings.

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Implicit vector quasi-equilibrium problems with applications to variational inequalities