On Existence of Solutions for η – Generalized Implicit Vector Variational- Like Inequalities

On Existence of Solutions for – Generalized Implicit Vector Variational- Like Inequalities

1

TIRTH RAM

Abstract. In this work, we intend to introduce and study a class of generalized im- plicit vector variational- like inequalities and a class of generalized implicit strong vec- tor variational -like inequalities in the setting of Hausdorff topological vector spaces.An equivalence result concerned with two classes of generalized implicit vector variational- like inequalities is proved under the suitable conditions.By using F KKM theorem, some new existence results of solutions for the generalized implicit vector variational-like inequalities and generalized strong implicit vector variational-like inequalities are obtained under some suitable conditions.

Key Words : Generalized parametric quasi-variational inclusions, sensitivity analy- sis, resolvent operator, Hausdorff metric.

1. Introduction

   Vector variational inequality was first introduced and studied by Giannessi [1] in the setting of finite dimensional Eucledean spaces. since then,the theory with applications for vector variational like inequalities, vector problems, vector equilibrium problems, and vector optimization problems, have been studied and generalized by many authors(see, e.g.,[2-14] and refernces therein). Recently Yu et al [15] considered a more general form of weak vector variational inequalities and proved some new results on the existence of solutions of the new class of weak vector variational inequalities in the setting of Haus- dorff topological vector spaces and Ahmed and Khan [16] introduced and considered weak vector variational like inequalities with generally convex mapping and gave some ex- istence results.

   On the other hand, Fang and Huang [17] studied some existence results of solutions for a class of strong vector variational inequalities in Banach spaces which give a positive answer to an open problem proposed by Chen and Hou [18].

   In 2008, Lee et al.[19] introduced a new class of strong vector variational type inequalities in Banach spaces. They obtained the existence theorems of solutions for the inequalities without monotonicity in Banach spaces by using Broweder Fixed point Theorem. Moti- vated and inspired by the work mentioned above , in this paper we introduce and study

   a class of generalized implicit vector variational- like inequalities and a class of

   generalized strong implicit vector variational -like inequalities in the setting of Haudorff toplogical vector spaces. We first show an equivalence theorem concerned with the two

   classes of generalized implicit vector variational -like inequalities under suitable con-

   ditions. By ussing FKKM theorem, we prove some new existence results of solutions for

   12000 Mathematics Subject Classification , 47J20,47J25, 49J40

   1

   2 T. RAM

   the generalized implicit vector variational like inequalities and generalized strong implicit vector variational-like inequalities under some suitable conditions.The results pre- sented in this paper improve and generalize some known results due to Ahmed and khan [16], Lee et al. [19], and Yu et al. [15].

2. Preliminaries

   Let X and Y be two real Hausdorff topological vector spaces, K X a nonempty, closed, and convex subset, and C Y a closed, convex, and pointed cone with apex at the origin. Recall that the Hausdorff topological vector space Y is said to be an ordered Hausdorff toplogical vector space by (Y, C) if ordering relations are defined in Y as follows:

   x, y Y, x y y x C,

   x, y Y, x i y y x / C,

   If the intC /= , then the weak ordering relations in Y is defined as follows:

   x, y Y, x < y y x intC,

   x, y Y, x y y x / intC,

   Let L(X, Y ) be the space of all continuous linear maps from X toY andT : X L(X, Y ). We denote the value of l L(X, Y ) on x Xby(l, x). throughout this paper, we assume that C(x) : x K is a family of closed, convex, and pointed cones of Y such that intC /= for allx K, is a mapping from K × K into Y .

   In this paper, we consider the following two kinds of vector variational inequalities:- Generalized Implicit Vector Variational-Like Inequality (in short, GIVVLI): for each z K and (0, 1], findx K such that

   (T (x + (1 )z), (y, x)) + f (y, g(x)) / intC(x), y K,

   Generalized Strong Implicit Vector Variational-Like Inequality(in short, GSIVVLI): for each z K and (0, 1], findx K such that

   (x + (1 )z), (y, x)) + f (y, g(x)) / C(x) \ {0}, y K,

   n

   Definition 2.1 Let T : K L(X, Y ) and : K × K K be two mappings and

   C = xK C(x) /= . T is said to be monotone in C if and only if

   (2.1)

   (T (x) T (y), (x, y)) C, x, y K.

   Definition 2.2 Let T : K L(X, Y ) and : K × K K be two mappings. We say that T is hemicontinuous if, for given any x, y, z K and (0, 1], the mapping t I (T ((x + (1 t)(y x)) + (1 )z), (x, y)) is continuous at 0+.

   3

   Definition 2.3 A multivalued mapping Let A : X XY is said to be upper semicontin- uous on X if, for all x X and for each open set G in Y with A(x) G, there exist an open neighbourhood O(x) of x X such that A(xI ) G for all xI O(x).

   Lemma 2.4([21]).Let (Y, C) be an ordered topological vector space with a closed, pointed

   convex cone C with intC(x)

   . Then for any y, z Y, we have

   1. y z intC and y / intC imply z / intC;

   2. y z C and y / intC imply z / intC;

   3. y z intC and y / intC imply z / intC;

   4. y z C and y / intC imply z / intC.

   Lemma 2.5([22]).Let M be a nonempty closed , and convex subset of a Hausdorff topological space, and G : M 2M is a multivalued map.Suppose that for any finite

   i=1

   x1, x2, · · · xn ln G(xi) (i.e F is a KKM mapping) G(x) is closed for each x M

   and compact for some x M, where we have conv deotes the convex hull operator.Then

   nxM G(x) .

   Lemma 2.6([23]).Let X Hausdorff topological linear space, A1, A2, · · · An be nonempty,

   i=1

   closed compact and convex subsets of X. Then conv(ln Ai) is compact.

   Lemma 2.7([24]).Let X and Y be two topological spaces.If A : X 2Y is upper semi- continuous with closed values, then A is closed.

Theorem 3.1. Let K be nonempty set, closed and convex subset of Hausdorff topological vector space X , and (Y, C(x)) an ordered topological vector space with intC(x) /= for all x K. Let g : K K,and let : K × K X and f : K × K X be affine mappings such that (x, x) = f (x, g(x)) = 0 for each x K. Let T : K L(X, Y ) be an

hemicontinuos mapping. If C = nxK C(x) . and T is monotone in C, then for

each z K, (0, 1], the following statements are equivalent

1. find x0 K such that (Tz(x0), (y, x0)) + f (y, g(x0)) / intC(x0), for all y K;

2. find x0 K such that (Tz(y), (y, x0)) + f (y, g(x0)) / intC(x0), for all y K;

where T(z) is defined by Tz(x) = T (x + (1 )z for all x K.

Proof. Suppose that (i) holds. We can find x0 K, such that

(3.1)

(Tz(x0), (y, x0)) + f (y, g(x0)) / intC(x0), y K

Since T is monotone, for each x, y K, we have

(3.2)(T (y + (1 )z) T (x + (1 )z), (y + (1 )z, x + (1 )z)) C

4 T. RAM

On the other hand, we know is affine and (x, x) = 0. It follws that

(Tz(y) Tz(x), (y, x))

1

= (T (y + (1 )z) T (x

(3.3)

+ (1 )z), (y + (1 )z, x + (1 )z)) C

Hence Tz is also monotone. That is

(3.4)

(Tz(x0), (y, x0)) (Tz(y), (y, x0)) Cy K.

n

since C = xK C(x), for all y K

(3.(5T)z(x0), (y, x0) + f (y, g(x0))) (Tz(y), (y, x0)) f (y, g(x0)) C C(x0). By Lemma 2.4,

(3.6)

(Tz(x0), (y, x0)) + f (y, g(x0)) / intC(x0), y K,

and so x0 is a solution of (ii). Conversly suppose that (ii) holds. Then there exists x0 K

such that

(3.7)

(Tz(y), (y, x0) + f (y, g(x0))) / intC(x0), y K.

For each y K, t (0, 1), we let yt = ty + (1 t)x0. Obviously, yt K

(3.8)

(Tz(yt), (yt, x0) + f (yt, g(x0))) / intC(x0),

Since f and are affine and (x0, x0) = f (x0, g(x0)) = 0, we have

(3.9)

(T ((ty + (1 t)x0) + (1 )z), t(y, x0)) + tf (y, g(x0)) / intC(x0),

That is (3.10)

(T ((x0 + t(y x0)) + (1 )z), (y, x0)) + f (y, g(x0)) / intC(x0).

Considering the hemicontinuity of T and letting t 0+, we have

(3.11)

(Tz(x0), (y, x0) + f (y, g(x0))) / intC(x0), y K.

This completes the proof.

Remark If C(x) = C and f (y, g(x)) = 0 for all x, y K, then Theorem 3.1 is reduced to Lemma 5 of [17].

Let K be a closed convex subset of a toppological linear space X, and {C(x) : x K} a family of closed, convex and a pointed cones of a topological space Y such that intC(x) for all x K. Throught this paper, we define set- valued mapping C : K L(X, Y ) as follows:

(3.12)

C = Y \ {intC(x)}, x K.

5

n

Theorem 3.2. Let K be nonempty, closed and convex subset of Hausdorff topological vector space X , and (Y, C(x)) an ordered topological vector space with intC(x) /= for all x K. Let g : K K,and let : K × K X and f : K × K X be affine mappings such that (x, x) = f (x, g(x)) = 0 for each x K. Let T : K L(X, Y ) be an hemicontinuos mapping. Assume the following conditions are satisfied

1. I f C = xK C(x) =/ . and T is monotone in C,

2. C : K 2Y is upper semicontinuous set- valued mapping.

Then for each z K, (0, 1], there exists x0 K such that

(3.13)

(T (x0 + (1 )z), (y, x0)) + f (y, g(x0)) / intC(x0), for all y K.

Proof. For each y K, we denote Tz(x) = T (x + (1 )z), and define

F1(y) = {x K : (Tz(x), (y, x)) + f (y, g(x))},

F2(y) = {x K : (Tz(y), (y, x)) + f (y, g(x))}

Then F1(y) and F2(y) are nonempty since y F1(y) and y F2(y). The proof is divided into the following three steps.

1. First, we prove the following conclusion: F1 is a KKM mapping. Indeed, assume that

   F1 is not a KKM mapping; then there exist u1, u2, · · · um K, t1 0, t2 0, · · · tm 0

   i=1

   i=1

   with m ti = 1 and w = m tiui such that

   I

   m

   (3.14)

   That is, (3.15)

   w / F1(ui), i = 1, 2, · · · m

   i=1

   i = 1, 2, · · · m, (Tz(w), (ui, w)) + f (ui, g(w)) intC(w).

   since and f are affine, we have

   m m

   (Tz(w), (ui, w)) + f (w, g(w)) = (Tz(w), (i=1tiui, w)) + f (i=1tiui, g(w))

   =

   (3.16)

   m i=1

   ti((Tz(w), (ui, w)) + f (ui, g(w)) intC(w).

   On the other hand,we know (w, w) = f (w, g(w)) = 0 then we have

   0 = (Tz(w), (w, w)) + f (w, g(w)) intC(w).

   It is impossible and so F1 = K 2K is a KKM mapping.

2. Further, we prove that

   (3.17)

   n F1(y) = n F2(y)

   yK

   yK

   yK

   yK

   Infact, if x F1(y), then(Tz(x), (y, x)) + f (y, g(x))) / intC(x) from the proof of

   theorem 3.1, we know that Tz is monotone in C(z). it follows that

   (3.18)

   (Tz(y) Tz(x), (y, x)) C,

   6 T. RAM

   and so

   (3.19)(Tz(x), (y, x)) + f (y, g(x)) (Tz(y), (y, x)) f (y, g(x)) C C(x) By Lemma 2.4, we have

   (3.20)

   (Tz(y), (y, x)) + f (y, g(x)) / intC(x)

   and so x F2(y) for each y K. That is, F1(y) F2(y) and so

   (3.21)

   n F1(y) n F2(y)

   yK

   yK

   Conversely suppose that x nyK F2(y). Then

   (3.22)

   (Tz(y), (y, x)) + f (y, g(x)) / intC(x)y K.

   it follows from Theorem 3.1 that

   That is, x nyK F1(y). and so

   (3.23)

   (Tz(x), (y, x)) + f (y, g(x)) / intC(x)y K.

   (3.24)

   n F2(y) n F1(y),

   which implies that

   yK

   yK

   (3.25)

   n F1(y) = n F2(y)

   yK

   yK

   yK

   yK

   n

3. We prove that yK F2(y) /= Indeed, since F1 is a KKM mapping, we know that, for any finite set {y1, y2, · · · yn} K, one has

   n n

   (3.26)

   conv{y1, y2, · · · yn} I F1(yi) I F2(yi))

   i=1

   This shows that F2 is also a KKM mapping.

   i=1

   Now we prove that F2(y) is closed for all y K. Assume that there exists a net {x}

   F2(y) with x x K. Then

   (3.27)

   (Tz(y), (y, x)) + f (y, g(x)) / intC(x)

   Using the definition of C, we have

   (3.28)

   (Tz(y), (y, x)) + f (y, g(x)) / intC(x)

   Since and f are continuous, it follows that

   (3.29)

   (Tz(y), (y, x)) + f (y, g(x)) (Tz(y), (y, x)) + f (y, g(x))

   Since C is upper semicontinuous mapping with close values, by Lemma 2.7, we know that

   C is closed, and so

   (3.30)

   This implies that (3.31)

   (Tz(y), (y, x)) + f (y, g(x)) C(x).

   (Tz(y), (y, x)) + f (y, g(x)) / intC(x)

   7

   n

   we know that F2(y) is compact. By Lemma 2.5, we have yK F2(y) /= , and it follows

   , and so F2(y) is closed.Considering the compactness of K and closedness of F2(y) K

   that nyK F1(y) , that is, for each z K and (0, 1],there exists x0 K such that

   (3.32) (T (x0 + (1 )z), (y, x0)) + f (y, g(x0)) / intC(x0)y K.

   , Thus, GIV V LI is solvable. This complete the proof.

   Remark If C(x) = C and f (y, g(x)) = 0 for all x, y K in theorem 3.3., the condition

   1. holds and condition (i)is equivalent to the monotonicity of T . Thus, it is easy to

      see that Theorem 3.3 is a generalization of [17, Theorem 6].

      In the above theorem, K is compact. In the following theorem, under some suitable conditions, we prove a new existence result of solutions for GIV V LI without the conditions of compactness of K.

      n

      Theorem 3.3. Let K be nonempty, closed and convex subset of Hausdorff topological vector space X , and (Y, C(x)) an ordered topological vector space with intC(x) /= for all x K. Let g : K K,and let : K × K X and f : K × K X be affine mappings such that (x, x) = f (x, g(x)) = 0 for each x K. Let T : K L(X, Y ) be an hemicontinuos mapping. Assume the following conditions are satisfied

      1. If C = xK C(x) /= . and T is monotone in C,

      2. C : K 2Y is upper semicontinuous set- valued mapping.

      3. there exists a nonempty compact and convex subset D of K and for each z

   K, (0, 1], x K \ D, there exist y0 D such that

   (3.33)

   (T (y0 + (1 )z), (y0, x)) + f (y0, g(x)) intC(y0).

   Then for each z K, (0, 1], there exists x0 D such that

   (3.34)

   (T (x0 + (1 )z), (y, x0)) + f (y, g(x0)) / intC(x0), for all y K.

   Proof. By Theorem 3.1, we know that the solution set of the problem (ii) in Theorem 3.1 is equivalent to the solution set of the following variational inequality: find x K,such that

   (3.35)

   (T (y + (1 )z), (y, x)) + f (y, g(x)) / intC(x), y K.

   For each z K and (0, 1], we denote Tz(x) = T (x + (1 )z). Let G : K 2D be defined as follows:

   (3.36)

   G(y) = {x D : (Tz(y), (y, x)) + f (y, g(x)) / intC(x), y K.}

   Obviously , for each y K,

   (3.37)

   G(y) = {x D : (Tz(y), (y, x)) + f (y, g(x) / intC(x)} D.

   Using the proof of Theorem 3.3, we obtain that G(y) is a closed subset of D. Considering the compactness of D and closedness of G(y), we know that G(y) is compact.

   i=1

   Now we prove that for any finite set {y1, y2, · · · yn} K, one has nn G(yi) /= Let

   8 T. RAM

   i=1

   Yn = ln {yi}. Since Y is a real Hausdorrf topological vector space, for each yi

   {y1, y2, · · · yn}, {yi} is compact and convex. Let N = conv(D Yn). By Lemma 2.6,

   we know that N is a compact and convex subset of K.

   Let F1, F2 : N 2N be defined as follows:

   (3.38)

   (3.39)

   F1(y) = {x N : (Tz(x), (y, x)) + f (y, g(x)) / intC(x), y N ; }

   F2(y) = {x N : (Tz(y), (y, x)) + f (y, g(x)) / intC(x), y N.}

   Using the proof of Theorem 3.3, we obtain

   (3.40)

   n F1(y) = n F2(y) ,

   y N

   and so there exists y0 nyN F2(y).

   yN

   Next we prove that y0 D. In fact, if y K \ D, then the assumption implies that there

   exists u Dsuch that have

   (3.41)

   (T (u + (1 )z), (u, y0)) + f (u, g(y0)) intC(u),

   Which contradicts y0 F2(u) and so y0 D.

   Since {y1, y2, · · · yn} N and G(yi) = F2(yi) D for each yi {y1, y2, · · · yn} it follows

   i=1

   i=1

   that y0 nn G(yi). Thus for any finite set {y1, y2, · · · yn} K , we have nn G(yi) /= .

   n

   Considering the compactness of G(y) for each y K, we know that there exists x0 D

   such that x0 yK G(y). Therefore, the solution set of GIV V LI is nonempty.

   This completes the proof.

   In the following, we prove the solavability of GSIV V LI under some suitable ondi- tions by using F KKM theorem.

   Theorem 3.4. Let Xbe a Hausdorff topological linear space, K X a nonempty, closed, and convex set, and (Y, C(x)) an ordered Hausdorff topological vector space with intC(x) =/ for all x K. Assume that for each y K, x (x, y) and x f (g(x)) are affine, (x, y) + (y, x) = 0, and f (g(x), y) + f (y, g(x)) = 0 for all x K, where g : K K. Let T : K L(X, Y ) be mapping such that

   1. for each z, y K, (0, 1], the set {x K : (T (x + (1 )z), (y, x)) +

      f (y, g(x)) C(x) \ {0}} is open in K;

   2. there exists a nonempty compact and convex subset D of K and for each z

K, (0, 1], x K \ D, there exists u D such that

(3.42)

(T (x + (1 )z), (y, x)) + f (y, g(x)) C(x) \ {0}

Then for each z K, (0, 1], there exists x0 K such that

(3.43)

(T (x0 + (1 )z), (y, x0)) + f (y, g(x)) / C(x0) \ {0}y K.

Proof. For each z K and (0, 1], we denote Tz(x) = T (x+(1)z). Let G : K 2D be defined as follows:

(3.44)

G(y) = {x K : (Tz(x), (y, x)) + f (y, g(x)) / C(x) \ {0}}y K.

9

Obviously, for each y K,

(3.45)

G(y) = {x D : (Tz(x), (y, x)) + f (y, g(x)) / C(x) \ {0}} D.

Since G(y) is closed subset of D, considering the compactness of D and closedness of G(y) is compact.

Now we prove that for any finite set {y1, y2, · · · yn} K, one has nn G(yi) .

Let Yn =

i=1{yi}. Since Y is real Hausdorff topological vector space, for each yi

ln i=1

{y1, y2, · · · yn}, {yi} is compact and convex. Let N = conv(D Yn). By Lemma2.6, we know that N is compact and convex subset of K. Let F : N to2N be defined as folows:

(3.46)

F (y) = {x N : (Tz(x), (y, x)) + f (y, g(x)) / C(x) \ {0}}, y N.

We claim that F is KKM mapping.Indeed, assume that F is not a KKM mapping. Then

i=1

i=1

there exist u1, u2, · · · um K, t1 0, t2 0, · · · tm 0 with m ti = 1 and w = m tiui

such that

(3.47)

That is, (3.48)

m

I

w / F (ui), i = 1, 2, · · · m

i=1

i = 1, 2, · · · m (Tz(w), (ui, w)) + f (ui, g(w)) C(w) \ {0}.

Since and f are affine, we have

m m

(Tz(w), (w, w)) + f (w, g(w)) = (Tz(w), ( tiui, w)) + f ( tiui, g(w))

i=1

m

i=1

(3.49)

= ti(Tz(w), ui, w)) + f (ui, g(w)) C(w) \ {0}.

i=1

On the other hand, we know that (w, w) = f (w, g(w)) = 0, and so

(3.50)

0 = (Tz(w), (w, w)) + f (w, g(w)) C(w) \ {0}.

n

which is impossible. Therefore, F : N 2N is a KKM mapping. Since F (y) is a closed subset of N , it follows that F (y) is compact. By Lemma 2.5, we have

(3.51)

F (y) /= .

yN

n

Thus, there exists y0 yN F (y). Next we prove that y0 D. In fact, if y N \ D,

then the condition (ii) implies that there exists u D such that

(3.52)

(T (y0 + (1 )z), (u, y0)) + f (u, g(y0)) C(y0) \ {0},

which contradicts y0 F (u) and so y0 D. Since {y1, y2, · · · yn} N and G(yi) =

F (yi) D for each yi {y1, y2, · · · yn}. Thus, for any finite set {y1, y2, · · · yn} K, we

i=1

have n G(yi) /= . Considering the compactness of G(y) for each y K, it is easy

10 T. RAM

to know that there exists x0 D such that x0 yKG(y) =/ . Therefore, for each

z K, (0, 1], there exists x0 K such that

(3.53)

(T (x0 + (1 )z), (y, x0)) + f (y, g(x0)) C(x0) \ {0}, y K.

Thus, GSIVVI is solvable. This completes the proof. Remark 3.8. If K is compact,

C(x) = C, g = I and = 1, then Theorem 3.7 is reduced to Theorem 2.1 in [20].

References

1. F.Giannessi, Theorems of alternative, quadratic programs and complimentarity problems ,in Vari- ational Inequalities and Complimentarity Problems, R.W. Cottle, F. Giannessi, and J.L. Lions,Eds., 151-186, John Wiley and Sons, New York, NY, USA, 1980.

2. G.Y. Chen, X.X. Huang, and X.Q. Yang, Vector optimization: Set valued and Variational Analysis, vol.541 of Lecture Notes in Economics and Mathematical System, Springer Berlin, Germany, 2005.

3. Y.P. Fang and N.J. Huang, Feasibility and Solvability of vector Variational inequalities with moving cones in Banach spaces, Nonlinear Analysis, Theory, Method and Application vol. 70(5),pp.2024- 2034, 2009.

4. F.Giannessi, Ed.,Vector Variational Inequalities and Vector Equilibrium, vol. 38 of Nonconvex Opti- mization and its Applications, Academic Publishers, Dordrecht, the Netherlands, 2000.

5. S.M.Guu, N.J.Huang and J.Li, Scalarization approaches for set valued vector optimization prob- lems and vector variational inequalities, Journal of Mathematical Analysis and Applications, vol.356,no.2,pp 564-576,2009.

6. N.J.Huang and C.J. Gao, Some generalized vector Variational inequalities and complimentarity prob- lems for multi-valued mapping, Applied Mathematics Letters,vol 16(7),pp.1003-1010, 2003

7. N.J.Huang,X.J.Long, and H.B.Thompson,Generalized F- variational inequalities and vector F- com- plimentarity problems for point to set valued mapping, mathematical and computer modelling, vol.48,no.5-6,pp.908-917,2008.

8. N.J.huang, X.J.Long, and C.W.zhao,Well posedness of for vector quasi-equilibrium problems with applications, Journal of Industrial and Management optimization, vol.5,no.2,pp.341-349,2009.

9. N.J.Huang, A.M.Rubinov, and X.Q.Yang, vector optimization problems with nonconvex preferences,

   Journal of global optimization, vol.40,no.4.pp.765-777,2008.

10. N.J.Huang,X.Q.Wang and W.K.Chan,Vector Complimentarity problems with a variable ordering re- lation, European Journal of Operational Research , vol.176,no.1,pp.15-26,2007.

11. G.Isac, V.A. Bulavsky, and V.V.Kalashnikov,Complimentarity, Equilibrium, efficiency and Econom- ics, vol.63 of nonconvex Optimization and its Applications,Kluwer Academic Publishers,Dordrecht, The Netherlands,2002.

12. J.Li, N.J.Huang, and J.K.Kim, On Implicit Vector equilibrium problems, Journal of Mathematical analysis and applications,vol.283,no.2,pp.501-512,2003.

13. X.J.Long, N.J.Huang, and K.L.Teo, Existence and stability of solutions for generalized strong vector quasi-equilibrium problems,Mathematical and Computer Modelling, Vol.47,no.3-4,pp.445-451,2008.

14. R.Y.Zhong, N.J.Huang, and M.M.wong,Connectedness and Path-connectedness of solution sets to symmetric vector equilibrium problems, Taiwanese Journalm of Mathematics, vol.13,no.2,pp.821- 836,2009.

15. M.Yu. S.Y. Wang, W.T.Fu, and W.S.Xiao, On the existence and connectedness of solutions sets of vector Variational inequalities, Mathematical Methods Operations Research, vol. 54,(2),pp.201- 15,2001.

16. R. Ahmed and Z. Khan, Vector Variational- like Inequalities with generally convex mappings, The Australian and New Zealand Industrial and Applied Mathematics Journal, vol. 49, pp. E 33-E46, 2007

17. Y.P. Fang and N.J.Huang, Strong vector Variational inequalities in Banach Spaces, Applied Math- ematics Letters,vol 19(4),pp.362-368, 2006

    11

18. ] G.Y. Chen and S.H. Hou, Existence of solutions for vector Variational inequalities in Vector Varia- tional Inequalities and Vector Equilibria, F. Giannessi Eds.vol. 38 of Academic Publishers, Dordrecht, the Netherlands, 2000.

19. B.S.Lee, M.F.Khan, and Salahuddin,Generalized vector variational type inequalities Computer and Mathematics with Applications,vol.55(6)pp.1164- 1169,2008.

20. G.Y.Chen, Existence of solutions for a vector variational inequality: an extension of the Hartmann Stampachhia theorem, Journal of Optimization Theory and Applications,vol.74(3)pp.445-456,1992.

21. K.Fan,some properties of convex sets relared to fixed point theorems, Mathematische Annalen,vol.266,no.4,pp.519-537,1984.

22. A.E. Taylor, An Introduction to Functional Analysis, John Wiley and Sons, New York, NY,USA,1963.

23. J.P. Aubin and I.Ekland, Applied Nonlinear Analysis, Pure and Applied Mathematics, Wiley Inter- science, New York, NY,USA,1984.

24. ] M.K.Ahmed and Salahudin, Existence of solutions for generalized implicit vector Variational like inequalities, Nonlinear analysis:Theorey Methods and Applications, vol.67,no.2,pp.430-441,2007.

25. ] T.Jabarootian and J.Zafarani,Generalized vector Variational- like inequalities, Journal of optimiza- tion Theory and Applications, vol.136,no.1,pp.15-30,2008.

Department of mathematics, University of Jammmu, Jammu 180 006, india

E-mail address: tir1ram2@yahoo.com