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volume 4, issue 1, article 6, 2003.

Received 09 October, 2002;

accepted 14 November, 2002.

Communicated by:Th.M. Rassias

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Journal of Inequalities in Pure and Applied Mathematics

B-PSEUDOMONOTONE VARIATIONAL INEQUALITIES OVER PRODUCT OF SETS

QAMRUL HASSAN ANSARI AND ZUBAIR KHAN

Department of Mathematics Aligarh Muslim University Aligarh 202 002, India.

EMail:qhansari@sancharnet.in

c

2000Victoria University ISSN (electronic): 1443-5756 107-02

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Relatively B-Pseudomonotone Variational Inequalities Over

Product of Sets

Qamrul Hasan Ansari and Zubair Khan

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J. Ineq. Pure and Appl. Math. 4(1) Art. 6, 2003

Abstract

In this paper, we consider variational inequality problem over the product of sets which is equivalent to the problem of system of variational inequalities. These two problems are studied for single valued maps as well as for multivalued maps. New concept of pseudomonotonicity in the sense of Brézis is introduced to prove the existence of a solution of our problems. As an application of our results, the existence of a coincidence point of two families of operators is also established.

2000 Mathematics Subject Classification:49Jxx, 47Hxx

Key words: Variational inequalities, System of variational inequalities, Relatively B- pseudomonotone maps, Coincidence point.

This research was partially done during the visit of first author to the Abdus Salam ICTP, Trieste, Italy, and also supported by the Major Research Project of University Grants Commission, No. F. 8-1/2002 (SR-1), New Delhi, Govt. of India.

1. Introduction

It is mentioned by J.P. Aubin in his book [3] that the Nash equilibrium problem [12, 13] for differentiable functions can be formulated in the form of a variational inequality problem over product of sets (for short, VIPPS). Not only the Nash equilibrium problem but also various equilibrium-type problems, like, traffic equilibrium, spatial equilibrium, and general equilibrium programming problems, from operations research, economics, game theory, mathematical physics and other areas, can also be uniformly modelled as a (VIPPS), see for example [8,11,14] and the references therein. Pang [14] decomposed the orig- inal variational inequality problem defined on the product of sets into a system of variational inequalities (for short, SVI), which is easy to solve, to establish some solution methods for (VIPPS). Later, it was found that these two problems, (VIPPS) and (SVI), are equivalent. In the recent past (VIPPS) or (SVI) has been considered and studied by many authors, see for example [1,7,8,9,10,11] and references therein. Konnov [9] extended the concept of (pseudo) monotonicity and established the existence results for a solution of (VIPPS). Recently, Ansari and Yao [2] introduced the system of generalized implicit variational inequalities and proved the existence of its solution. They derived the existence results for a solution of system of generalized variational inequalities and used their results as tools to establish the existence of a solution of system of optimization problems, which include the Nash equilibrium problem as a special case, for nondifferentiable functions.

In this paper, we study (VIPPS) or (SVI) without using arguments from generalized monotonicity as considered in [9]. For this, we introduce the concept of relatively B-pseudomonotonicity which extends in a natural way the well-

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known pseudomonotonicity in the sense of Brézis [4] (see also [5]). By using a fixed point theorem of Chowdhury and Tan [6], we establish the existence results for a solution of (VIPPS) or (SVI) under relatively B-pseudomonotonicity.

As an application of our results, we prove the existence of a coincidence point for two families of nonlinear operators.

We also consider the generalized variational inequality problem over product of sets (for short, GVIPPS) and system of generalized variational inequalities (for short, SGVI). We adopt the technique of Yang and Yao [16] to establish the existence of a solution of (GVIPPS) or (SGVI) by using the existence results for a solution of (VIPPS) or (SVIP).

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2. Formulations and Preliminaries

LetI be a finite index set, that is, I ={1,2, . . . , n}. For eachi ∈ I, letXi be a topological vector space with its dualX_i^∗,K_i a nonempty and convex subset of X_i, K = Q

i∈IK_i, X = Q

i∈IX_i, andX^∗ = Q

i∈IX_i^∗. We denote by h·,·i the pairing betweenX_i^∗ and Xi. For each i ∈ I, when Xi is a normed space, its norm is denoted byk·k_iand the product norm on Xwill be denoted byk·k.

For eachx∈ X, we writex= (x_i)i∈I, wherex_i ∈X_i, that is, for eachx ∈X, xi ∈Xi denotes theith component ofx. For eachi ∈I, letfi :K →X_i^∗ be a nonlinear map. We consider the following variational inequality problem over product of sets (for short, VIPPS): Findx¯∈K such that

(2.1) X

i∈I

hf_i(¯x), y_i−x¯_ii ≥0, for all y_i ∈K_i, i∈I.

Of course, if we define the mappingf :K →X^∗by

(2.2) f(x) = f_i(x)

i∈I,

then (VIPPS) can be equivalently re-written as the usual variational inequality problem of findingx¯∈K such that

hf(¯x), y−xi ≥¯ 0, for all y∈K.

Very recently, Konnov [9] proved some existence results for a solution of (VIPPS) under relatively pseudomonotonicity or strongly relative pseudomonotonicity assumptions in the setting of Banach spaces.

We also consider the following problem of system of variational inequalities which has been studied by many authors because of its applications in various

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equilibrium-type problems from operations research, economics, game theory, mathematical physics and other areas, see for example, [1, 8, 10, 11, 14] and references therein:

(SVIP) Find x¯∈X such that for each i∈I,hf_i(¯x), y_i −x¯_ii ≥0,

for all y_i ∈K_i. It is easy to see that these two problems, (VIPPS) and (SVIP), are equivalent.

Indeed, that (SVIP) implies (VIPPS) is obvious. The reverse implication holds if we lety_j = ¯x_j for allj 6=i.

For each i ∈ I, when f_i and f are multivalued maps, then (VIPPS) and (SVIP) are called generalized variational inequality problem over product sets and system of generalized variational inequalities, respectively. More precisely, for eachi ∈ I, letF_i : K → 2^X^∗ⁱ be a multivalued map with nonempty values so that if we set

(2.3) F = (F_i :i∈I),

thenF : K → 2^X^∗ is a multivalued map with nonempty values. We shall also consider the following problems:

(GVIPPS) Findx¯∈Kandu¯∈F(¯x)such that X

i∈I

h¯ui, yi−x¯ii ≥0, for ally_i ∈K_i, i∈I, whereu_iis theith component ofu.

(SGVIP) Findx¯∈K and ¯u∈F(¯x) such that h¯u_i, y_i−x¯_ii ≥0,

for ally_i ∈K_i, i∈I,

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where u_i is the ith component ofu. As in the single valued case, it is easy to see that (GVIPPS) and (SGVIP) are equivalent.

In [2], Ansari and Yao used (SGVIP) as a tool to prove the existence of a solution of Nash equilibrium problem for nondifferentiable functions.

For every nonempty setA, we denote by2^A(respectively,F(A)) the family of all subsets (respectively, finite subsets) ofA. IfA is a nonempty subset of a vector space, thencoAdenotes the convex hull ofA.

The following result of Chowdhury and Tan [6] will be used to establish the main result of this paper.

Theorem 2.1. LetK be a nonempty and convex subset of a topological vector space (not necessarily Hausdorff) X and T : K → 2^K a multivalued map.

Assume that the following conditions hold:

(i) For allx∈K,T(x)is convex.

(ii) For eachA∈ F(K)and for ally∈coA,T⁻¹(y)∩coAis open incoA.

(iii) For each A ∈ F(K) and all x, y ∈ coA and every net {x_α}α∈Γ in K converging toxsuch thatty+ (1−t)x /∈T(xα)for allα∈Γand for all t∈[0,1], we havey /∈T(x).

(iv) There exist a nonempty, closed and compact subsetDofKand an element

˜

y∈Dsuch thaty˜∈T(x)for allx∈K\D.

(v) For allx∈D,T(x)is nonempty.

Then there existsxˆ∈Ksuch thatxˆ∈T(ˆx).

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3. Existence Results for (VIPPS) and (SVIP)

Definition 3.1. The mapf :K →X^∗, defined by (2.2), is said to be relatively B-pseudomonotone (respectively, relatively demimonotone) if for each x ∈ K and every net{x^α}α∈ΓinK converging tox(respectively, weakly tox) with

lim infα

"

X

i∈I

hfi(x^α), xi−x^α_ii

#

≥0

we have X

i∈I

hf_i(x), y_i−x_ii ≥lim sup_α

"

X

i∈I

hf_i(x^α), y_i−x^α_ii

#

for ally ∈K.

Of course, if I is a singleton set, then the above definition reduces to the definition of a pseudomonotone map, introduced by Brézis [4] (see also [5] and [3, p. 410]).

Now we are ready to establish the main result of this paper on the existence of a solution of (VIPPS) under relatively B-pseudomonotonicity assumption.

Theorem 3.1. For each i ∈ I, let K_i be a nonempty and convex subset of a real topological vector space (not necessarily, Hausdorff) X_i. Let f, defined by (2.2), be relatively B-pseudomonotone such that for eachA ∈ F(K), x 7→

P

i∈Ihf_i(x), y_i−x_iiis upper semicontinuous oncoA. Assume that there exists a nonempty, closed and compact subset D of K and an element y˜ ∈ D such that for allx∈K\D,P

i∈Ihf_i(x),y˜_i−x_ii<0. Then (VIPPS) has a solution.

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Proof. For eachx∈K, define a multivalued mapT :K →2^Kby T(x) =

(

y∈K :X

i∈I

hf_i(x), y_i−x_ii<0 )

.

Then for allx∈K,T(x)is convex. LetA∈ F(K), then for ally∈coA, [T⁻¹(y)]^c∩coA=

(

x∈coA:X

i∈I

hf_i(x), y_i−x_ii ≥0 )

is closed incoAby upper semicontinuity of the mapx7→ P

i∈Ihf_i(x), y_i−x_ii oncoA. HenceT⁻¹(y)∩coAis open incoA.

Suppose thatx, y ∈coAand{x^α}α∈Γis a net inKconverging toxsuch that X

i∈I

hfi(x^α),(tyi+ (1−t)xi)−x^α_ii ≥0, for all α∈Γ and all t ∈[0,1].

Fort= 0, we have X

i∈I

hf_i(x^α), x_i−x^α_ii ≥0, for all α∈Γ, and therefore

lim inf_α

"

X

i∈I

hf_i(x^α), x_i −x^α_ii

#

≥0.

By the relatively B-pseudomonotonicity off, we have

(3.1) X

i∈I

hf_i(x), y_i−x_ii ≥lim sup_α

"

X

i∈I

# .

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Fort= 1, we have X

i∈I

hf_i(x^α), y_i−x^α_ii ≥0, for all α∈Γ,

and therefore,

(3.2) lim infα∈Γ

"

X

i∈I

#

≥0.

From (3.1) and (3.2), we obtain X

i∈I

hf_i(x), y_i−x_ii ≥0,

and thusy /∈T(x).

Assume that for all x ∈ D, T(x) is nonempty. Then all the conditions of Theorem2.1 are satisfied. Hence there existsxˆ ∈ K such thatxˆ ∈ T(ˆx), that is,

0 =X

i∈I

hf_i(ˆx),xˆ_i−xˆ_ii<0,

a contradiction. Thus there existsx¯∈K such thatT(¯x) = ∅, that is, X

i∈I

hf_i(¯x), y_i−x¯_ii ≥0, for all y_i ∈K_i i∈I.

Hencex¯is a solution of (VIPPS).

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Corollary 3.2. For eachi∈ I, letK_i be a nonempty, closed and convex subset of a real reflexive Banach spaceXi. Letf, defined by (2.2), be relatively demi- monotone such that for each A ∈ F(K), x 7→ P

i∈Ihf_i(x), y_i −x_ii is upper semicontinuous oncoA. Assume that there existsy˜∈K such that

(3.3) lim

||x||→∞, x∈K

X

i∈I

hf_i(x),y˜_i−x_ii<0.

Then (VIPPS) has a solution.

Proof. Letα= lim||x||→∞, x∈KP

i∈Ihf_i(x),y˜_i−x_ii<0. Then by (3.3),α <0.

Let r > 0be such that ||˜y|| ≤ r andP

i∈Ihf_i(x),y˜_i −x_ii < ^α₂ for allx ∈ K with ||x|| > r. For eachi ∈ I, let K_i^r = {x_i ∈ K_i : ||x_i||_i ≤ r}, and we denote byK^r =Q

i∈IK_i^r. ThenK^r is a nonempty and weakly compact subset of K. Note that for anyx ∈ K \K^r, P

i∈Ihf_i(x),y˜_i−x_ii < ^α₂ < 0, and the conclusion follows from Theorem3.1.

As an application of Corollary 3.2, we establish the existence of a coincidence point for two families of nonlinear operators.

Corollary 3.3. For eachi∈I, letX_ibe a real reflexive Banach space. Letf, g : X → X_i^∗ be defined asf(x) = (f_i(x))i∈I andg(x) = (g_i(x))i∈I, respectively, for all x ∈ X, where for each i ∈ I, g_i : X → X_i^∗ is a nonlinear operator.

Assume that(f −g)is relatively demimonotone and for eachA∈ F(X),x7→

P

i∈Ihf_i(x), y_i −x_ii is upper semicontinuous on coA. Further, assume that there existsy˜∈X such that

lim

||x||→∞, x∈X

X

i∈I

h(f_i−g_i)(x),y˜_i−x_ii<0.

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Then there existsx¯∈X such thatf_i(¯x) =g_i(¯x)for eachi∈I.

Proof. From the Corollary3.2, there existsx¯∈Xsuch that for eachi∈I, hfi(¯x), yi−x¯ii ≥ hgi(¯x), yi−x¯ii, for all yi ∈Xi.

Therefore we have,f_i(¯x) =g_i(¯x)for eachi∈I.

Finally, we give another application of Corollary3.2in the setting of Hilbert spaces.

Corollary 3.4. For eachi ∈ I, let(Xi,h·,·i)be a real Hilbert space andKia nonempty, closed and convex subset ofX_i. Letf, defined by (2.2), be relatively demimonotone such that for each A ∈ F(K), x 7→ P

i∈Ihf_i(x), y_i −x_ii is lower semicontinuous oncoA. Assume that there existsy˜∈K such that

lim

||x||→∞, x∈K

X

i∈I

hx_i−f_i(x),y˜_i−x_ii<0.

Then there existsx¯∈K such that for eachi∈I,f_i(¯x) = ¯x_i.

Proof. For each i ∈ I, define a nonlinear operator S_i : K → X_i by S_i(x) = xi −fi(x)for allx ∈ K. Then obviously, for each i ∈ I, Si satisfies all the conditions of Corollary3.2. Hence there existsx¯∈K such that for eachi∈I, hS_i(¯x), y_i −x¯_ii ≥ 0for all y_i ∈ K_i. For eachi ∈ I, let y_i = f_i(¯x), we have

||¯xi−fi(¯x)|| ≤0. Therefore, for eachi∈I,fi(¯x) = ¯xi.

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4. Existence Results for (GVIPPS) and (SGVIP)

In this section, we adopt the technique of Yang and Yao [16] to derive the existence results for a solution of (GVIPPS) and (SGVIP) by using the results of Section3.

Definition 4.1. The multivalued mapF :K →2^X^∗, defined by (2.3), is said to be relatively B-pseudomonotone (respectively, relatively demimonotone) if for eachx∈ K and every net{x^α}α∈ΓinK converging tox(respectively, weakly tox) with

lim inf_α

"

X

i∈I

hu^α_i, x_i−x^α_ii

#

≥0, for all u^α_i ∈F_i(x^α)

we have, for allu_i ∈F_i(x)

X

i∈I

hui, yi−xii ≥lim sup_α

"

X

i∈I

hu^α_i, yi−x^α_ii

#

for allu^α_i ∈Fi(x^α)andy∈K.

Definition 4.2. Let Z be topological vector space and U a subset of Z. Let G : U → 2^Z^∗ be a multivalued map andg : U → Z^∗ a single valued map. g is called a selection ofGonU ifg(x) ∈G(x)for allx ∈U. Furthermore, the function g is called a continuous selection of G onU if it is continuous on U and a selection ofGonU.

For further details on continuous selections of multivalued maps, we refer to [15].

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It follows from (2.2) and (2.3) that if f : K → X^∗, defined by (2.2), is a selection of F :K →2^X^∗, defined by (2.3), then for each i∈I, fi :K →X_i^∗ is a selection ofF_i :K →2^Xⁱ^∗onK.

Lemma 4.1. Iff :K → X^∗, defined by (2.2), is a selection ofF : K → 2^X^∗, defined by (2.3), onK, then every solution of (VIPPS) is a solution of (GVIPPS).

Proof. Assume thatx¯∈Kis a solution of (VIPPS). Then X

i∈I

hfi(¯x), yi−x¯ii ≥0, for allyi ∈Ki, i∈I.

Letu¯_i =f_i(¯x), so thatu¯=f(¯x). Sincefis a selection ofF, we haveu¯∈F(¯x) such that

X

i∈I

h¯u_i, y_i−x¯_ii ≥0, for ally_i ∈K_i, i ∈I.

Hence(¯x,u)¯ is a solution of (GVIPPS).

Lemma 4.2. Let f : K → X^∗, defined by (2.2), be a selection of a mul- tivalued map F : K → 2^X^∗, defined by (2.3), on K. If F is relatively B- pseudomonotone (respectively, relatively demimonotone), then f is also rela- tively B-pseudomonotone (respectively, relatively demimonotone).

Proof. Let u_i = f_i(x)and u^α_i = f_i(x^α)for all x ∈ K, so thatu = f(x) and u^α =f(x^α). Then the result follows from the definitions.

In the remaining part of the paper, we shall assume that the pairingh·,·iis continuous.

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Theorem 4.3. For eachi∈I, letK_ibe a nonempty and convex subset of a real topological vector space (not necessarily, Hausdorff)Xi. Assume that

(i) F : K → 2^X^∗, defined by (2.3), is a relatively B-pseudomonotone multi- valued map;

(ii) there exists a continuous selectionf :K → X^∗, defined by (2.2), ofF on K;

(iii) there exist a nonempty, closed and compact subsetDofKand an element

˜

y∈Dsuch that for allx∈K\D,P

i∈Ihf_i(x),y˜_i −x_ii<0.

Then (GVIPPS) has a solution.

Proof. From condition (ii), there exists a continuous functions f : K → X^∗ such that f(x) ∈ F(x), for all x ∈ K. Since the pairing h·,·iis continuous, we have the map x 7→ P

i∈Ihf_i(x), y_i −x_ii is continuous on K. By Lemma 4.2,f is relatively B-pseudomonotone. Therefore by Theorem 3.1, there exists a solutionx¯∈ K of (VIPPS). For eachi ∈I, letu¯_i =f_i(¯x)∈ F_i(¯x). Then by Lemma4.1,(¯x,u)¯ is a solution of (GVIPPS).

Corollary 4.4. For each i ∈ I, letK_i be a nonempty and convex subset of a real reflexive Banach spaceX_i. Assume that

(i) F :K →2^X^∗, defined by (2.3), is a relatively demimonotone multivalued map;

(ii) there exists a continuous selectionf :K → X^∗, defined by (2.2), ofF on K;

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(iii) there existsy˜∈Ksuch that

||x||→∞, x∈Klim X

i∈I

hfi(x),y˜i−xii<0.

Then (GVIPPS) has a solution.

Proof. It follows by using the argument of Theorem4.3and Corollary3.2.

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