World Applied Programming, Vol (2), No (1), January 2012. 7-11

ISSN: 2222-2510

©2011 WAP journal. www.waprogramming.com

7

System of Generalized Vector Quasi-Variational-Like

Inequalities with Set-Valued Mappings

Shamshad Husain Sanjeev Gupta*

Department of Applied Mathematics, Department of Applied Mathematics,

Z. H. College of Engineering & Technology, Z. H. College of Engineering & Technology,

Aligarh Muslim University, Aligarh Muslim University

Aligarh-202002, India

s_husain68@yahoo.com

Aligarh-202002, India

guptasanmp@gmail.com

Abstract: In this paper, we introduce and study the system of generalized vector quasi-variational-like inequalities in

Hausdorff topological vector spaces. By means of fixed point theorem, we obtain existence theorem for the system of

generalized vector quasi-variational-like inequalities and also derive the existence result of solutions for the generalized

vector quasi-variational-like inequalities.

Key word: The system of generalized vector quasi-variational-like inequalities; Fixed point theorem; Open lower

section; Upper semicontinuous; Weak C-diagonal quasiconvexity.

I. INTRODUCTION

Throughout this paper, unless otherwise specified, assume that I is an index set. For each ,i I∈ let Zi be a locally convex

topological vector space (l.c.s., in short) and Ki be a nonempty convex subset of Hausdorff topological vector space (t.v.s., in

short) Ei. Let Yi be a subset of continuous function space ( ), L E Zi i from Ei into Zi, where ( ), L E Zi i is equipped with a σ -

topology. Let int A and co A denote the interior and convex hull of a set A, respectively. Let :C K Zi i⊸ be a set-valued

mapping such that int Ci (x) ≠ ∅ for each x ∈K. Denote that K Kii I= ∏ ∈ and E Eii I= ∏ ∈ .

For each ,i I∈ let : K K Ei i i iη × → be a vector-valued mapping, ( ) ( ): , , ,G L E Z L E Zi ii⊸ :T K Y

i i⊸ and

:D K Ki i⊸ be three set-valued mappings. Then, we introduce a system of generalized vector quasi-variational-like inequalities

(SGVQVLI, in short) which is to find ( , ) x t K Y∈ × such that ( )x D xi i∈ and ( )t T xi i∈ and

( ) ( ) ( ), , int , ,G t y x C x y D xi i i i i i i iη ⊆ − ∀ ∈/

where ,l x⟨ ⟩ denotes the evaluation of ( , )l L E Z∈ at x E∈ . By the corollary of Schaefer [14], ( , )L E Z becomes a l.c.s.. By

Ding [6], the bilinear map : ( , )L K Z K Z⟨., .⟩ × → is continuous. Present work is motivated and inspired by Peng [12, 13] on

the existence of solutions for vector quasi-variational-like inequalities in Hausdorff topological vector spaces.

The above system of generalized vector quasi-variational-like inequalities encompasses many models of system of variational

inequalities. For example, if Gi is an identity mapping, the following problems are the special cases of SGVQVLI:

(1) SGVQVLI reduces to the problem of finding x K∈ such that for each ,i I∈ ( )x D xi i∈ and

( ) ( ) ( ) ( ), : , , int ,y D x t T x t y x C xi i i ii i i i iη∀ ∈ ∃ ∈ ∉ −

which is introduced and studied by Peng [12].

(2) For each ,i I∈ ( ) ( ), , and ,y x y x C x Ci ii i i iη = − = where Ci is a convex, closed, pointed cone in Zi with

int C ≠ ∅, then the SGVQVLI reduces to the problem of finding x K∈ such that for each ,i I∈ ( )x D xi i∈ and

( ) ( ), : , int ,y D x t T x t y x Ci i ii i i i∀ ∈ ∃ ∈ − ∉ −

Shamshad Husain and Sanjeev Gupta, World Applied Programming, Vol (2), No (1), January 2012.

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which is introduced and studied by Allevi et al. [3].

(3) If I is a singleton set, SGVQVLI reduces to the problem of finding x K∈ such that ( )x D x∈ and

( ) ( ) ( ) ( ), : , , int ,y D x t T x t y x C xη∀ ∈ ∃ ∈ ∉ −

which is introduced and studied by Ding [5].

In addition, if ( )D x K= SGVQVLI reduces to the problem of finding x K∈ and

( ) ( ) ( ), : , , int ,y K t T x t y x C xη∀ ∈ ∃ ∈ ∉ −

which is studied by Ahmad [2].

If ( ),y x y xη = − then SGVQVLI reduces to the problem of finding x K∈ and

( ) ( ), : , int ,y K t T x t y x C x∀ ∈ ∃ ∈ − ∉ −

which is studied by Fang [7].

In this Paper, we introduce the system of generalized vector quasi-variational-like inequalities and by using the fixed point theorem, we establish an existence result of its solutions. As a special case, we also derive the existence result of generalized vector quasi-variational-like inequalities. Our results extend and improve some main results of Allevi et al. [3], Peng et al. [12, 13], Husain et al. [9, 10]. The technical instrument in our proof is similar to that employed by Hou et al. [8], Tian [16].

II. PRELIMINARIES

In this section, in order to prove the main result, we need the following definitions and lemmas.

Definition 2.1 [8] Let E and Z be two t.v.s. and K be a convex subset of t.v.s. E. Let :C K Z⊸ and : K K Zθ × ⊸ be two set-

valued mappings. Assume given any finite subset { }1 2 , ,...., nx x xΛ = in K, any

1

nx xi i

i

α= ∑=

with 0i

α ≥ for i = 1,2,...,n,

and 1.

1

n

ii

α =∑=

Then θ is said to be weak C-diagonally quasiconvex (WC-DQC, in short) in the second argument if for some

xi ∈ Λ ,

( ), int ( ).x x C xiθ ⊆ −/

Definition 2.2. [17] Let A and B be two topological spaces and :T A B⊸ be a set-valued mapping. Then,

(1) T is said to have open lower sections if the set T –-

(y) = {x∈ A : y ∈T(x)} is open in A for every y ∈ B.

(2) T said to be upper semicontinuous (u.s.c., in short) if for each x ∈ A and each open set C in B with T (x) ⊂ C, there exists an

open neighborhood O of x in A such that ( )T u C∈ for each u O∈ .

(3) T is said to be closed if for any net { xα} in A such that xα → x and any net {yα} in B such that yα → y and yα ∈ T (xα) for any

α, we have y ∈ T (x).

Definition 2.3. [4] Let A and B be two topological spaces. If :T A B⊸ is u.s.c. set-valued mapping with closed values, then T is

closed.

Lemma 2.4. [15] Let A and B be two topological spaces. Suppose that :T A B⊸ is u.s.c. mapping with compact values.

Suppose {xα} is a net in A such that xα → xo. If yα ∈ T (xα) for each α, then there are a yo ∈ T (xo) and a subnet {yβ} of {yα} such

that yβ → yo .

Lemma 2.5. [18] Let A and B be two topological spaces. If :T A B⊸ and :K A B⊸ are set-valued mappings having open

lower sections, then

(1) a set-valued mapping :J A B⊸ defined by, for each x ∈ A, J (x) = co T(x) has open lower sections;

Shamshad Husain and Sanjeev Gupta, World Applied Programming, Vol (2), No (1), January 2012.

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(2) a set-valued mapping : A Bθ ⊸ defined by, for each x ∈ A, θ (x) = T(x) ∩ K (x) has open lower sections.

For each i ∈ I, Ei a Hausdorff t.v.s.. Let {Ki} be a family of nonempty compact convex subsets with each Ki in Ei . Let

K Kii I= ∏ ∈ and E Eii I= ∏ ∈ . The following system of fixed-point theorem is needed in this paper.

Lemma 2.6. [1] For each ,i I∈ let :T K Ki i⊸ be a set-valued mapping. Assume that the following conditions hold.

(1) For each ,i I∈ Ti is convex set-valued mapping;

(2) ( ){int : }.K T x x Ki i i i−= ∈∪

Then there exists x K∈ such that ( ) ( )x T x T xii I∈ = ∏ ∈ , that is ( )x T xi i∈ , for each ,i I∈ where ix is the

projection of x onto Ki .

III. MAIN RESULTS

In this section, we present existence result of a solution for the SGVQVLI by using the fixed point theorem.

Theorem 3.1. For each ,i I∈ let Zi be a l.c.s., Ki a nonempty compact convex subset of Hausdorff t.v.s. Ei , Yi a nonempty

compact convex subset of ( ), L E Zi i , which is equipped with a σ -topology. For each ,i I∈ assume that the following

conditions are satisfied.

(1) :D K Ki i⊸ and :T K Yi i⊸ are two nonempty convex set-valued mappings and have open lower sections;

(2) For each ,t Yi i∈ and ,x coi i

∈ Λ the mapping ( ), ., :G t x K Zi i i i iη ⊸ is WC-DQC;

(3) For each ,y Ki i∈ the set ( ) ( ) ( ){ , : , , int }x t K Y G t y x C xi i i i i iη∈ × ⊆ − is open.

Then there exist ix ∈ Di ( )x and )(xTt ii ∈ such that

( ) ( ) ( ), , int ,G t y x C x y D xi i i i i i i iη ⊆ − ∀ ∈/ .

That is, SGVQVLI has a solution ( ), .x t K Y∈ ×

Proof. Define a set-valued mapping :P K Y Ki i× ⊸ by

Pi (x, t) = {yi ∈ Ki : ⟨Gi ti, ηi (yi , xi)⟩ ⊂ − int Ci (x)}, ∀ (x, t) ∈ K × Y. (3.1)

We first prove that xi ∉ co (Pi (x, t)) for all (x, t) ∈ K × Y. To see this, suppose, by way of contradiction, that there exist some i

∈ I and some point ( )tx, ∈ K × Y such that ix ∈ co ( )( )txPi , . Then, these exist finite points niii yyy ,....,,

21 in Ki and αj > 0 with

11

n

jj

α =∑=

such that

1

nx yi j i j

i

α= ∑=

and yi j∈ co ( )( ),P x ti for all j = 1,2,.... n, such that

( ) ( ), , int , 1, 2,...., .G t y x C x j ni i i i ij iη ⊆ − = (3.2)

Equations (3.1) and (3.2) contradict the hypothesis (2). Hence, xi ∉ co (Pi (x, t)).

By hypothesis (3), for each i I∈ and each ,y Ki i∈ we know that

( ) ( ) ( ) ( ){ }, : , , intQ y x t K Y G t y x C xi i i i i i iiη

−= ∈ × ⊆ − is open and so Pi has open lower sections.

For each i ∈ I, consider a set-valued mapping :Q K Y Ki i

× ⊸ defined by

( ) ( )( ) ( ) ( ), co , , , .Q x t P x t D x x t K Yi i i= ∩ ∀ ∈ × (3.3)

Shamshad Husain and Sanjeev Gupta, World Applied Programming, Vol (2), No (1), January 2012.

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Since Di has open lower sections by hypothesis (1), we may apply Lemma 2.5 to assert that the set-valued mapping Qi

has

also open lower sections. With the help of (3.3), we consider

( ) ( ){ }, : , .W x t K Y Q x t K Yii= ∈ × ≠ ∅ ⊂ × (3.4)

There are two cases to consider.

Case (I)- If Wi

= ∅ :

From (3.4), we have ( )( ) ( ) ( )co , , , .P x t D x x t K Yi i∩ = ∅ ∀ ∈ × This implies that, ( ) , ,x t K Y∀ ∈ ×

( , ) ( ) .P x t D xi i∩ = ∅

On the other hand, by condition (1), and the fact Ki is a compact convex subset of Ei, we can apply Lemma 2.6 to

assert the existence of a fixed point *

( *).x D xi i

∈

Since ( *)T xi ≠ ∅, picking *

( *)t T xi i

∈ , we have

( *, *) ( *) .P x t D xi i

∩ = ∅

From (3.4), we have ( *), ( *, *).y D x y P x ti i i i

∀ ∈ ∉ Hence, in this particular case, the assertion of the theorem holds.

Case (II)- If Wi

≠ ∅ :

Define a set-valued mapping :S K Y Ki i

× ⊸ by

(3.5)

From (3.5), ( , )S x ti is a convex set-valued mapping and for each u K∈ , ( ) ( ) ( ( ) )S u Q u D u Yi i i i− − −

= ∪ × is open. For each

,i I∈ consider the set-valued mapping :H K Y K Y× ×⊸ where ,H Hii I= ∏ ∈ defined by

( , ) ( ( , ), ( ( )).H x t S x t T xi i i

= (3.6)

By hypothesis (1) and (3.6), Hi satisfies all the conditions of Lemma 2.6. Therefore, there exists ( )*, *x t K Y∈ × such that

* *( , ( *, *).x t H x t

i i i) ∈ Suppose that ( )*, *x t Wi∈ , then

*co ( ( *, *)) ( *),x P x t D x

i i i∈ ∩

so that *co( ( *, *)) ,x P x ti i∈ This is a contradiction. Hence, ( )*, * .x t W

i∉ From (3.4), (3.5), and (3.6), we have

* *( , ( ( *), ( *)),x t D x T xi i i i) ∈ and ( *, *)Q x ti = ∅.

Thus * *

( *), ( *),x D x t T xi i i i

∈ ∈ co ( ( *, *)) ( *) .P x t D xi i∩ = ∅

This implies

( , ), ( , )( , )

( ), ( , ) \ .

Q x t x t Wi iS x t

i D x x t K Y Wi i

∈=

∈ ×

Shamshad Husain and Sanjeev Gupta, World Applied Programming, Vol (2), No (1), January 2012.

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( *, *) ( *) .P x t D xi i∩ = ∅

Consequently, the assertion of the theorem holds in this case. □

Remark 3.2. If I is a singleton set, by using Theorem 3.1, we obtain the following existence result of solutions for GVQVLI.

Corollary 3.3. Let Z be a l.c.s., K a nonempty compact convex subset of Hausdorff t.v.s. E, Y a nonempty compact convex

subset of L(E, Z), which is equipped with a σ -topology. Assume that the following conditions are satisfied.

(1) :D K K⊸ and :T K Y⊸ are two nonempty convex set-valued mappings and have open lower sections;

(2) For each t ∈ Y and x ∈ coΛ, the mapping ( ), ., :G t x Kη Ζ⊸ is WC-DQC;

(3) For each y ∈ K, the set ( ) ( ){( , ) : , , int }x t K Y Gt y x C xη∈ × ⊆ − is open.

Then there exist a point ( )x D x∈ and a point ( )t T x∈ such that

( ) ( ) ( ), , int , .G t y x C x y D xη ⊆ − ∀ ∈/

That is, GVQVLI has a solution ( )tx, ∈ K × Y.

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