WEIGHTED VARIATIONAL INEQUALITIES WITH SET-VALUED MAPPINGS

WEIGHTED VARIATIONAL INEQUALITIES WITH SET-VALUED MAPPINGS

MIRUNA BELDIMAN

We consider some systems of vector general variational inequalities and weighted variational inequalities, and find equivalence conditions between them. Then we establish for these classes of problems a few existence results under different types of generalized weighted monotonicity assumptions.

AMS 2000 Subject Classification: 47J20, 58E35.

Key words: weighted general variational inequalities, system of vector variational inequalities, generalized weighted pseudomonotonicity, hemicontinu- ity, convexity.

1. INTRODUCTION

Because of their applications in economics, game theory, mathematical physics, operations research and other areas, many classes of vector variational inequalities were intensively studied. For existence of solutions, resolution methods or equivalence with equilibrium and optimization problems see, for example, [11], [14], [15], [19], [17] and the references therein. For the study of variational inequalities systems in the single valued case, see [16] and [18].

Ansari, Schaible and Yao [4] introduced a system of vector equilibrium problems with vector valued bifunction defined over a product set, which in- cludes as particular cases systems of generalized vector variational inequalities and optimization problems.

The same authors [5] defined another systems of vector equilibrium prob- lems and obtained an existence theorem based on a known maximal element theorem. From this, they derived existence results for some systems of vector variational-like inequalities and systems of implicit vector variational inequal- ities.

Ansari and Khan [2] proved the existence of solution for a variational inequality over product sets, which is equivalent to a system of vector vari- ational inequalities, under generalized pseudo-monotonicity assumptions, in the meaning of Brezis [8]. They used a fixed point theorem of Chowdury and Tan [9].

REV. ROUMAINE MATH. PURES APPL.,52(2007),3, 315–327

Later, Allevi, Gnudi and Konnov [1] obtained existence results for a system of general vector variational inequalities and countable product of sets with set-valued mappings in real Banach spaces, using the classical Ky Fan lemma [13].

Ansari, Khan and Siddiqi [3] introduced weighted variational inequalities which involve set- valued mappings and established the relationship between systems of general vector variational inequalities and systems of weighted vari- ational inequalities, which can be written as duality products.

Following this direction, we extend their results for arbitrary spaces and general mappings between them. We define some systems of general vector variational inequalities and weighted variational inequalities and give equiva- lence conditions. Also, we obtain existence of solutions using a theorem proved in [7] for the single- valued case.

2. DEFINITIONS AND SOME PRELIMINARY RESULTS For each given m∈N, we denote byR^m₊ the nonnegative orthant ofR^m, that is,

R^m₊ ={u= (u₁, . . . , um)∈R^m :uj ≥0, forj= 1, . . . , m} and let

T₊^m=

u= (u₁, . . . , u_m)∈R^m₊ : m j=1

u_j = 1

be the simplex of R^m₊ . In what follows we will shorten “with respect to”

as “wrt”.

LetI ={1, . . . , n}be a finite index set. For eachi∈I let_ibe a positive integer, X_i a real topological vector space (not necessarily Hausdorff), K_i a nonempty convex subset ofX_i, and Y_i an arbitrary set. Put X =

i∈IX_i and K=

i∈IK_i.

For each x ∈ X, x_i ∈ X_i stands for ith coordinate of x and we write x= (xi)_i∈I. For eachi∈I letfi :K →Yi and Ψi:Yi×Ki×Ki →Rⁱ be two maps and Ψ = (Ψ_i)_i∈I. Assume that W = (W₁, . . . , W_n) ∈ ⁿ

i=1

R₊ⁱ\ {0}

is a given weight vector and consider the following problems, where “·” stands for the inner product onR₊ⁱ.

(Ψ-WVIP) Findx∈K such that

i∈I

Wi·Ψi(fi(x), xi;yi)≤0 for allyi ∈Ki,i∈I.

(Ψ-SWVI) Findx∈K such that,

W_i·Ψi(f_i(x), x_i;y_i)≤0 for ally_i ∈K_i,i∈I.

We denote by K^w (respectively K_s^w) the solution set of (Ψ-WVIP) (respectively, (Ψ-SWVI)).

Definition 2.1 ([7]). A family (f_i)_i∈I of functions is said to be (i) weighted monotone wrt(W,Ψ) if

i∈I

Wi·(Ψi(fi(y), xi;yi)−Ψi(fi(x), xi;yi))≤0

for allx, y∈K, andweighted strictly monotone wrt(W, ψ) if the inequality is strict for allx=y;

(ii)weighted pseudomonotone wrt (W,Ψ) if

i∈I

Wi·Ψi(fi(x), xi;yi)≤0⇒

i∈I

Wi·Ψi(fi(y), xi;yi)≤0

for allx, y∈K, andweighted strictly pseudomonotone wrt(W,Ψ) if the second inequality is strict for all x=y;

(iii)weighted maximal pseudomonotone wrt(W,Ψ) if it is weighted pseu- domonotone wrt (W,Ψ) and

i∈I

W_i·Ψ_i(f_i(z), x_i;z_i)≤0, ∀z∈(x, y]⇒

i∈I

W_i·Ψ_i(f_i(x), x_i;y_i)≤0 for allx, y∈K, where (x, y] =

i∈I(xi, yi], andweighted maximal strictly pseu- domonotone wrt(W,Ψ) if it is weighted strictly pseudomonotone wrt (W,Ψ) and (3.1) holds.

Definition 2.2 ([7]). A family (f_i)_i∈I of functions is said to be weighted hemicontinuous wrt (W,Ψ) if for all x, y ∈ K the mapping λ ∈ [0,1] →

i∈I

W_i·Ψ_i(f_i(x+λ(y−x)), x_i;y_i) is continuous.

Definition 2.3 ([7]). A family (f_i)_i∈I is said to be weighted B-pseudo- monotone wrt(W,Ψ) if for eachx∈Kand every net{x^α}_α∈ΓinKconverging to xwith

lim sup

α i∈I

W_i·Ψ_i(f_i(x^α), x^α_i;x_i) ≤0 we have

lim sup

α i∈I

W_i·Ψ_i(f_i(x^α), x^α_i;y_i) ≥

i∈I

W_i·Ψ_i(f_i(x), x_i;y_i), for ally∈K.

In what follows we shall use Theorem2.1 ([7]). Assume that

(i₁)the family (fi)_i∈I of functions is weighted maximal pseudomonotone wrt (W,Ψ);

(i₂) there exists a nonempty closed compact subset D of K and y∈D such that

i∈I

W_i·Ψ_i(f_i(x), x_i;y_i)>0 for all x∈K\D;

(i₃) the mapping y→

i Wi·Ψi(fi(x), xi;yi) is convex on K;

(i₄)

i∈IW_i·Ψ_i(f_i(x), x_i;x_i) = 0 for all x∈K;

(i₅)for x, y∈coAand every sequence {x^α}_α∈Γin K converging to xwe have lim

α∈Γ

i∈IWi·Ψi(fi(x^α), x^α_i;yi) =

i∈IWi·Ψi(fi(x), xi;yi).

Then there exists a solution x ∈ K of (Ψ-WVIP), hence a solution of (Ψ-SWVI). Furthermore, if W ∈

i∈IT₊ⁱ, then there exists a normalized solution x ∈ K of (Ψ-WVIP), hence a solution of (Ψ-SVVI)_w. Finally, if W ∈ ⁿ

i=1

intT₊ⁱ

,then x ∈K is a solution of (Ψ-SVVI).

Corollary2.1 [7]. Assume that conditions (i₂)–(i₅) from Theorem2.1 and (i₆) below hold:

(i₆) the family {f_i}i∈I of functions is weighted hemicontinuous and weighted pseudomonotone wrt (W,Ψ).

Then there exists a solution x ∈ K of (Ψ-WVIP),hence a solution of (Ψ-SWVI). If W ∈

i∈IT₊ⁱ then there exists a normalized solution x∈K of (Ψ-WVIP) hence a solution of (Ψ-SVVI)_w. Moreover, if W ∈ ⁿ

i=1

intT₊ⁱ then x∈K is a solution of(Ψ-WVIP),hence a solution of (Ψ-SVVI).

3. A CLASS OF WEIGHTED VARIATIONAL INEQUALITIES WHICH INVOLVE SET-VALUED MAPPINGS

In this section we introduce a class of generalized weighted variational inequalities over products of sets. It includes as particular cases the weighted variational inequalities for multivalued maps and systems of weighted varia- tional inequalities from [3] as well as the weighted Ψ-variational inequalities and Ψ-systems of vector variational inequalities from [7].

A few extensions of weighted monotonicity concepts are also considered, and under such assumptions some existence results for our generalized weighted variational problem and systems of generalized vector variational inequalities are obtained.

Fori∈I and j = 1,2, . . . , _i,_i ∈N, letF_ij :K →2^Yi be a multivalued map with nonempty values. Fori∈I set

Fi(x) =F_i1(x)× · · · ×F_ii(x), x∈K.

Also, put

F(x) ={Fi(x)}_i∈I.

Consider the following general systems of generalized vector variational inequalities.

(Ψ-SGVVI) Find x∈K such that

∃ui∈ Fi(x) with Ψi(ui, xi;yi)∈/R₊ⁱ\ {0}, ∀yi∈Ki, i∈I, and

(Ψ-SGVVI)_w Find x∈K such that

∃u_i ∈ Fi(x) with Ψ_i(u_i, x_i;y_i)∈/R₊ⁱ, ∀y_i∈K_i i∈I.

Also, consider the system:

(Ψ-SGVVI) Find x∈K such that

∃ui∈ Fi(x) with Ψi(ui, xi;yi)∈/R₊ⁱ, ∀yi∈Ki, i∈I.

Remark 3.1. Every solution of (Ψ-SGVVI) is a solution of (Ψ-SGVVI)_w and every solution of (Ψ-SGVVI)_w is a solution of (Ψ-SGVVI),but the con- verse implications are not true in general.

Thus (Ψ-SGVVI) is a weak formulation of (Ψ-SGVVI)_w. Remark 3.2. If for everyi∈I, Yi =X_i^∗ and

Ψ_i(u_i, x_i;y_i) =

u_i1, x_i−y_i, . . . ,u_ii, x_i−y_i ,

where ·,· denotes the continuous pairing between X_i^∗ and X_i, then (Ψ-SGVVI),(Ψ-SGVVI)_wand (Ψ-SGVVI)lead to (SGVVI),(SGVVI)_wfrom [3] and to the problem of systems of generalized vector variational inequalities introduced and studied by Ansari, Schaible and Yao [5],respectively.

Now, to solve (Ψ-SGVVI) and (Ψ-SGVVI)_w, we consider as in [3] a weighted variational inequality problem type. So, we introduce the following weighted generalized variational inequality problem over products of sets:

(Ψ-WGVIP) Find x∈K and u∈ F(x) such that

i∈I

W_i·Ψi(u, x_i;y_i)≤0, ∀y_i ∈K_i,i∈I.

We denote the solution of (Ψ-WGVIP) byK^wg.

Also consider the problem of systems of weighted generalized variational inequalities

(Ψ-SWGVI) Find x∈K and u∈ F(x) such that W_i·Ψ_i(u_i, x_i;y_i)≤0, ∀y_i∈K_i, i∈I.

Note that if W_i ∈ T₊ⁱ for each i∈ I, then the solution of (Ψ-WGVIP) or (Ψ-SWGVI) is said to be normalized. Denote by K_s^wg the solution set of (Ψ-SWGVI) and byKsn^wg the normalized solution set of (Ψ-SWGVI).

The following definitions can be seen as an extension of relative B-pseudo- monotonicity introduced in [2], weighted B-pseudomonotonicity for a family of functions given in [3] and of Definition 2.3.

Definition 3.1. A family{Fi_j} i∈I

j=1,_i is said to be

(i) weighted generalized pseudomonotone wrt (W,Ψ) if for all x, y ∈ K andu∈ F(x), v∈ F(y) we have

i∈I

W_i·Ψ_i(u_i, y_i;x_i)≤0⇒

i∈I

W_i·Ψ_i(v_i, y_i;x_i)≤0;

(ii) weighted generalized maximal pseudomonotone wrt (W,Ψ) if it is weighted generalized pseudomonotone wrt (W,Ψ) and for all x, y ∈ K and u∈ F(x) there exists δ₀∈(0,1] such that

i∈I

W_i·Ψ_i(s_i, z_i;x_i)≤0,∀s_i∈ Fi(z) ,i∈I,z∈(x, y]

⇒

i∈I

W_i·Ψ_i(u_i, y_i;x_i)≤0, wherez=x+t(y−x),0< t≤δ₀;

(iii)weightedu-hemicontinuous wrt(W,Ψ) if for allx, y∈Kthe mapping λ∈[0,1]→

i∈IWi·Ψi(si(x+λ(y−x)), yi;xi) is upper semicontinuous at 0, wheresi(x+λ(y−x))∈ Fi(x+λ(y−x)), i∈I.

Remark 3.3. In the case considered in Remark 3.2,we obtain the defini- tions of generalized weighted pseudomonotonicity wrtW,generalized weighted maximal pseudomonotonicity wrt W and weighted u-hemicontinuity respec- tively (see [3]).

Proposition 3.1. Assume that

•the family {F_ij} i∈I

j=1,_i is hemicontinuous wrt (W,Ψ)and weighted gen- eralized pseudomonotone wrt (W,Ψ) ;

• there exist δ > 0 and τ > 0 such that for any i ∈ I, x, y ∈ K and t∈(0, δ) we have

Ψ_i(u_i, x_i+t(y_i−x_i) ;x_i) =t^τΨ_i(u_i, y_i;x_i). Then the family{Fi_j} ^i∈I

j=1,i is weighted generalized maximal pseudomono- tone wrt (W,Ψ).

Proof. Letx, y∈K such that for anyu ∈ F(x) we have

i∈I

W_i·Ψi(u_i, y_i;x_i)>0.

By the weighted u-hemicontinuity of the family {Fi_j} i∈I

j=1,_i,there exists δ,0< δ ≤1,such that

i∈I

Wi·Ψi(si(x+t(y−x)), yi;xi)>0

for all s(x+t(y−x)) ∈ F(x+t(y−x)) and t ∈ (0, δ). This inequality together with (j₂) yields

i∈I

W_i·Ψ_i(s_i(x+t(y−x)), x_i+t(y_i−x_i) ;x_i)>0 for any t∈(0,min{δ, δ}). The proof is complete.

Corollary 3.1 ([3]). If the family {F_ij} i∈I

j=1,_i of multivalued maps F_ij : K → 2^Xi∗ is u-hemicontinuous and weighted generalized pseudomono- tone wrt W, then it is weighted generalized maximal pseudomonotone wrt the same weight vector W.

In the multivalued case we need some results about continuous selections.

Definition 3.2 ([20]). Consider two topological vector spaces S and Z, a subsetU of S, a multivalued map G:U → 2^Z\ {Φ} and g :U → Z. We say thatg is a selection of Gon U if g(x) ∈G(x) for allu ∈U. The selection g is said to be a continuous selection of G on U if it is continuous on U and a selection ofGon U.

For some results on the existence of continuous selections see [10,20]. It is easy to prove the following results.

Lemma3.1. Assume that

•for each i∈I and j= 1,2, . . . , i, fi_j is a selection of Fi_j on K;

•the family {F_ij} i∈I

j=1,_i is weighted generalized maximal pseudomonotone wrt (W,Ψ).

Then the family {fi_j} _i∈I

j=1,_i

is weighted maximal pseudomonotone wrt (W, Ψ).

Lemma 3.2. Assume that f_ij is a selection of F_ij on K for each i∈ I and j = 1,2, . . . , _i. Let x ∈ K and let u be given by u_ij ∈ F_ij(x) for each i∈I and j= 1,2, . . . , _i. Then

•x∈K^w ⇒(x, u)∈K^wg;

•x∈K_s^w ⇒(x, u)∈Ks^wg.

Lemma3.3. Under the assumptions of Lemma 3.1,

•if W ∈ ⁿ

i=1T₊ⁱ and x∈K_s^w,then (x, u) is a solution of (Ψ-SGVVI)_w;

•if W ∈ ⁿ

i=1

intT₊ⁱ

andx∈K_s^w,then(x, u)is a solution of (Ψ-SGVVI).

As for the existence of a solution of (Ψ-SGVVI),we have the following results.

Theorem3.1. Assume that (j₁) the family {F_ij} i∈I

j=1,_i is weighted generalized maximal pseudomono- tone wrt (W,Ψ) ;

(j₂) for each i∈I, j = 1, . . . , i there exists a selection (not necessarily continuous)f_ijof F_ij onKfor which assumptions (i₁)−(i₃)from Theorem2.1 are satisfied;

(j₃) there exists a nonempty closed compact subset D of K and y∈D such that

i∈I

W_i·Ψ_i(u_i, x_i;y_i)>0

for all u_i∈ Fi(x), i∈I,andx∈K\D.

Under these assumptions (k₁) K^wg=∅ and K_s^wg=∅;

(k₂) if W ∈

i∈IT₊ⁱ then the solution set of (Ψ-SGVVI)_w is nonempty;

(k₃)if W ∈

i∈I

intT₊ⁱ

then the solution set of (Ψ-SGVVI)is nonempty.

Proof. By (j₂),for i∈ I and j = 1, . . . , _i there is a function f_ij such that f_ij(x) ∈ F_ij(x) for all x ∈ K. Using Lemma 3.1 we get that

f_ij is weighted maximal pseudomonotone wrt (W,Ψ). Now, we see that the as-i,j

sumptions of Theorem 2.1 are satisfied for{f_i}_i,so that K^w =K_s^w =∅. Let x ∈ K_s^w. Put ui_j =fi_j(x) for i ∈I and j = 1, . . . , i. Hence ui = f_i(x) ∈ F_i(x) and Lemma 3.2 implies that (x, u) ∈ K^wg and (x, u) ∈ Ks^wg, whereu= (ui)_i∈I.

For (k₂) and (k₃) we see that (x, u) is a normalized solution of (Ψ-SWGVI).

Now,using Lemma 3.3 we obtain that (x, u) is a solution of (Ψ-SGVVI)_w,in case (k₂), and a solution of (Ψ-SGVVIP) in case (k₃). Thus, the proof is complete.

Using Lemma 3.1 we have

Corollary 3.2. Assume that (j₂) and (j₃) from Theorem 3.1 hold as well as

(j₄) the family {Fi_j} i∈I

j=1,_i is u-hemicontinuous and weighted generalized pseudomonotone wrt (W,Ψ).

Then the conclusions of Theorem 3.1 hold.

Corollary3.3 ([3]). Let the family Fi_j

i∈I

j=1,_i of multivalued maps be weighted generalized maximal pseudomonotone wrt W. Asume that

• for each i ∈ I and each j = 1, . . . , _i there exists a selection (not necessarily continuous) f_ij of F_ij on K;

•there exists a nonempty closed compact subset Dof Kand y∈Dsuch that

i∈IW_i· u_i, x_i−y_i>0 for allx∈K\D,andu_i ∈ Fi(x),i∈I.

Then the solution set K^wg of (WGVIP) is nonempty and so is Ks^wg. Furthermore, if W ∈

i∈IT₊ⁱ (respectively W ∈

i∈I

intT₊ⁱ

) then the solution set of (SGVVI)_w (respectively (SGVVI))is nonempty.

Theorem3.2. Assume (i₃)from Theorem 2.1 and that (j₅) the family

F_ij

i∈I

j=1,_i is weighted generalized pseudomonotone wrt (W,Ψ) ;

(j₆) for each i ∈ I and j = 1, . . . , _i there exists a continuous selection f_ij on K for which assumptions (i₃)−(i₅) from Theorem 2.1 are satisfied.

Then the conclusions of Theorem 3.1 hold.

Proof. According to (j₆) we have a family of continuous functions f_ij

i,j

such that f_ij(x) ∈ F_ij(x) for all x ∈ K. By (i₄) and Lemma 3.1, the fami- ly

fi_j

i,j of continuous functions is weighted pseudomonotone wrt (W,Ψ).

Hence this family is weighted pseudomonotone and weighted hemicontinuous wrt (W,Ψ). Now, we can use Corollary 2.1. Therefore, there existsx ∈K^w, hencex∈K_s^w.

If we put u_ij = f_ij(x) ∈ F_ij(x) for i ∈ I, j = 1, . . . , _i, we can use Lemma 3.2 and then get that (x, u) ∈ K^wg and (x, u) ∈ K_s^wg, where u= (u_i)_i∈I.

For the last part of this theorem we use Lemma 3.3. The proof is com- plete.

Now, by Proposition 3.1 and Lemma 3.1 we obtain

Corollary 3.4. Assume (i₁) from Corollary 3.2 and (i₂), (i₃) from Theorem3.2 Then (k₁)−(k₃) from Theorem 3.2 hold.

Corollary3.5 ([3]).Let the family F_ij

i∈I

j=1,_i of multivalued maps be weighted generalized pseudomonotone wrt W. Assume that

•for each i∈I and each j= 1, . . . , i there exists a continuous selection f_ij of F_ij on K;

•there exists a nonempty closed compact subset Dof Kand y∈Dsuch that

i∈IW_i· u_i, x_i−y_i>0,for all x∈K\D and u_i ∈ Fi(x), i∈I. Then the solution set K^wg of (WGVIP) is nonempty and so is K_s^wg. Furthermore,if W ∈

i∈IT₊ⁱ(respectively,W ∈

i∈I

intT₊ⁱ

),then the solution set of (SGVVI)_w (respectively, (SGVVI)) is nonempty.

Corollary3.6 ([3]).Let the family F_ij

i∈I

j=1,_i of multivalued maps be weightedu-hemicontinuous and weighted generalized pseudomonotone wrt W. Assume that

• for each i ∈ I and each j = 1, . . . , _i there exists a selection (not necessarily continuous) fi_j of Fi_j on K;

•there exists a nonempty closed compact subset Dof K and y∈Dsuch that

i∈IWi· ui, xi−yi>0 for all x∈K\D andui ∈ Fi(x),i∈I.

Then the solution set K^wg of (WGVIP) is nonempty and so is Ks^wg. Furthermore, if W ∈

i∈IT₊ⁱ (respectively W ∈

i∈I

intT₊ⁱ

) then the solution set of (SGVVI)_w (respectively (SGVVI)) is nonempty.

Corollary3.7 ([3]). Let the family F_ij

i∈I

j=1,_i of multivalued maps be weighted generalized pseudomonotone wrt W. Assume that

•for each i∈I and each j= 1, . . . , _i there exists a continuous selection fi_j of Fi_j on K;

•there exists a nonempty closed compact subset Dof K and y∈Dsuch that

i∈IW_i· u_i, x_i−y_i>0 for all u_i ∈ Fi(x),i∈I,x∈K\D.

Then, the solution set K^wg of (WGVIP) is nonempty and so is Ks^wg. Furthermore,if W ∈

i∈IT₊ⁱ then the normalized solution set of (SGVVI)_w is nonempty.

To conclude, we consider the case of multivalued maps of B-pseudo- monotone type.

Definition 3.3. A family Fi_j

i∈I

j=1,_i is said to beweighted generalized B- pseudomonotone wrt (W,Ψ) if for each x ∈ K and every net {x^α}_α∈Γ in K converging tox with

lim inf

α∈Γ

i∈I

Wi·Ψi(u^α_i, x^α_i;xi)≥0, ∀u^α_i ∈Fi(x^α)

we have

i∈I

W_i·Ψ_i(u_i, x_i;y_i)≥lim sup

α∈Γ

i∈I

W_i·Ψ_i(u^α_i, x^α_i;y_i) ,

for allui∈Fi(x), u^α_i ∈Fi(x^α),i∈I, andy ∈K.

Remark 3.4. If we take Ψi as in Remark 3.2 we obtain the definition of weighted generalizedB-pseudomonotonicity wrtW from [3].

We easily get

Lemma3.4. Assume that

• the family Fi_j

i∈I

j=1,_i is weighted generalized B-pseudomonotone wrt (W,Ψ) ;

• for each i ∈ I and j = 1, . . . , _i, f_ij : K → Y_i is a selection of F_ij : K→2^Yi on K.

Then the family f_ij

i,j is weighted B-pseudomonotone wrt (W,Ψ).

Now, from Lemma 3.4 and Theorem 3.2 we obtain Theorem3.3. Assume (i₃) from Theorem 3.1 and that

• the family F_ij

i∈I

j=1,_i is weighted generalized B-pseudomonotone wrt (W,Ψ) ;

•for each i∈I, j = 1, . . . , i and all A∈ F(K)there exists a continuous selection f_ij of F_ij on coA which satisfies the assumptions (i₃)–(i₅) from Theorem 2.1.

Then (k₁)–(k₃) from Theorem 3.1 are true.

Corollary3.8 ([3]). Let the family Fi_j

i∈I

j=1,_i of multivalued maps be weighted generalized B-pseudomonotone wrt W. Assume that

• for each i ∈ I, j = 1, . . . , _i and for all A ∈ F(K) there exists a continuous selection fi_j of Fi_j on coA;

•there exists a nonempty closed compact subset Dof Kand y∈Dsuch that

i∈IWi· ui, xi−yi>0,for all ui∈ Fi(x), i∈I,and x∈K\D.

Then the solution set K^wg of (WGVIP) is nonempty and so is Ks^wg. Furthermore,if W ∈

i∈IT₊ⁱ (respectively, W ∈

i∈I

intT₊ⁱ

),then the solution set of (SGVVI)_w (respectively, (SGVVI)) is nonempty.

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Received 18 July 2006 University of Bucharest

Faculty of Mathematics and Computer Science Str. Academiei 14

010014 Bucharest, Romania