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廣義向量平衡態問題

中 華 民 國 94 年 10 月 12 日

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Generalized Vector Equilibrium Problems

Œi_rNSC 93-2115-M-110-012

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x¹ß{Õ;(Yungyen Chiang)

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E-mail

[email protected]

Abstract. Let L(X, Z) be the space of continuous linear mappings between topological vector spaces, where Z is pre-ordered by a closed convex cone C with nonempty interior IntC. In this project, we extend the (S)+ and (S)1+ conditions

to multivalued mappings from nonempty subsets of X into L(X, Z), and derive some existence results for generalized vector variational inequalities associated with multivalued mappings satisfying the (S)+ or (S)1+ condition.

Key words. Generalized vector varia-tional inequalities, the (S)+ and (S)1+

conditions, L-topology.

1. Introduction. Let K be a nonempty subset of X. The generalized vector vari-ational inequality associated with a map-ping T : K −→ 2L(X, Z), GVVI(T, K) for short, is the problem of finding (_bx, _by) ∈ K × L(X, Z) such that y ∈ T (_{b b}x) and h_by, u − _bxi ∈ (−IntC)c _{for all u ∈ K,}

where (−IntC)c _{is the complement of}

For any multivalued mapping T as given above, its graph is defined by G(T ) = {(x, y) : y ∈ T (x)}.

The motivation of the work is to prove vector versions of a result due to Cubiotti and Yao [4, Theorem 3.1], and a result obtained by Guo and Yao [5, Theorem 3.1]. To motivate our formulation for the (S)+ condition, we state Guo and Yao’s

result as follows :

Let K be a nonempty weakly compact convex subset of a real reflexive Banach space B, and let T : K −→ 2B∗ _{be a}

multivalued mapping. Assume that the following conditions are satisfied.

(i) T has nonempty closed and convex values.

(ii) T is bounded, i.e., T (K) is a bounded subset of B∗.

(iii) If V is a finite-dimensional subspace of B with K ∩ V is nonempty, then T is upper semicontinuous from the norm topology of K ∩ V to the weak topology of B∗.

(iv) If {xn}∞n=1 is a sequence in K

con-verging weakly to x ∈ K and if yn ∈

T (xn) satisfying lim sup n→∞

hyn, xn −

xi ≤ 0, then xn −→ x and {yn} has

a subsequence converging weakly to some y ∈ T (x).

Then there exists (_bx,_by) ∈ G(T ) such that h_by, u −_bxi ≥ 0 for all u ∈ K.

For any normed space X, let X_s∗ de-note the space X∗ equipped with the weak-star topology. For a given sequence {xn}∞n=1 in X, we write xn

w

−→ x ∈ X when {xn}∞n=1 converges weakly to x.

Note that, for a reflexive Banach space B, if y ∈ B∗ and {yn}∞n=1is a sequence in

B∗, then yn w

−→ y if and only if {yn}∞n=1

converges in B_s∗ to y. Now, the condition (iv) of [5, Theorem 3.1] can be restated as : If {(xn, yn)}∞n=1 is a sequence in G(T ) with xn w −→ x ∈ K and lim inf n→∞ hyn, x − xni ≥ 0 ,

then xn −→ x and {yn}∞n=1 has a

sub-sequence converging in B_s∗ to some y ∈ T (x).

Let K be a nonempty subset of a normed space X. A mapping T : K −→ 2X∗ _{is called to satisfy the}

sequen-tial (S)+-condition if for any sequence

{(xn, yn)}∞n=1in G(T ) with xn w −→ x ∈ K and lim inf n→∞ hyn, x − xni ≥ 0 , there is a subsequence of {(xn, yn)}∞n=1 converging in X × X_s∗ to some (x, y) ∈ G(T ).

2. Preliminaries. The (S)+-condition

subsets of a topological vector space X into L(X, Z) is formulated analogously to the sequential (S)+-condition given in

Section 1. While the (S)1₊ condition is formulated analogously to the sequential (S)1

+-condition given in [4, 6]. To state

these conditions, we need the L-topology defined in [3], and need the topology of bounded convergence and the topology of simple convergence on L(X, Z). Also, we need limit inferiors of nets in Z in-troduced in [2].

Throughout this section, let X de-note a topological vector space. The L-topology on X is the L-topology having the sets `−1(U ) as subbasis, where U is open in Z and ` ∈ L(X, Z). Let XL

de-note the space X equipped with the L-topology. Note that XL is a topological

vector space, and that if X is Hausdorff and locally convex then XL is Hausdorff

[3, Theorem 3.1].

A subset E of X will be called L-closed (respectively, L-open) if E is closed (re-spectively, open) in XL. Similarly, E

is called L-compact if it is compact in XL. When Z = IR, the notion of

L-compactness reduces to the notion of weak compactness.

For any given net {xα} in X, we shall

write xα−→ x ∈ X when {xα} converges

to x in the original topology on X. The net {xα} will be called L-convergent to x,

written by xα L

−→ x, if {xα} converges in

XL to x, i.e., h`, xαi −→ h`, xi in Z for

every ` ∈ L(X, Z).

Next, we recall the topology of bounded convergence and the topology of simple convergence on L(X, Z).

Let NZ be the family of

0-neighborhoods in Z. For any given family E of nonempty subsets of X, if E ∈ E and V ∈ NZ, we denote [E, V ] by

the set :

{f ∈ L(X, Z) : f (E) ⊂ V } . Let B(X) denote the family of all nonempty bounded subsets of X, and F (X) denote the set of all nonempty fi-nite subsets of X. If either E = F (X) or E = B(X), then the family

{[E, V ]E : E ∈ E and V ∈ NZ}

is the 0-neighborhood base in L(X, Z) for a unique translation-invariant topol-ogy TE. If E = F (X), then TE is the

topology of simple convergence (or the topology of pointwise convergence); the resulting t.v.s is denoted by Ls(X, Z).

When E = B(X), then TE is the topology

of bounded convergence, and the result-ing t.v.s is denoted by Lb(X, Z). Note

that Ls(X, IR) = Xs∗ is the weak-star

topology on X∗, and Lb(X, IR) = Xb∗ is

For any net {yα} in L(X, Z), if {yα}

converges in Ls(X, Z) to y ∈ L(X, Z),

then we shall write yα s

−→ y.

As in the scalar case, limit inferiors of nets in Z are defined by using vec-tor superiors and inferiors introduced in [1]. For a subset E of Z, the supe-rior of E with respect to C is written by Sup (E, C), and the inferior is writ-ten by Inf (E, C). As C is fixed, we simply write Sup (E, C) = Sup E and Inf (E, C) = Inf E.

For a given net {zα}α∈I in Z, let Aα =

{zβ : β  α} for every α ∈ I. The limit

inferior of {zα}α∈I is defined by

Liminf zα = Sup (

[

α∈I

Inf Aα) .

Now, let K be a nonempty subset of X. A multivalued mapping T from K into L(X, Z) is said to satisfy the (S)+

-condition if for any net {(xα, yα)} in

G(T ) with xα L

−→ x and

Liminf hyα, x − xαi ⊂ (−IntC)c

there is a subnet of {(xα, yα)} converging

in X × Ls(X, Z) to some point (x, y) ∈

G(T ).

While T is said to satisfy the (S)1 +

con-dition if for any net {(xα, yα)} in the

graph of T with xα L −→ x ∈ K, yα s −→ y ∈ L(X, Z) and Liminf hyα, x − xαi ⊂ (−IntC)c, {xα}

has a subnet {xλ} such that xλ −→ x in

K.

3. Existence Results. In this section, let X and Z be Hausdorff t.v.s., let K be a nonempty convex subset of X, and let T be a multivalued mapping from K into Y = Ls(X, Z) with nonempty

values.

Theorem 1. Assume that X is locally bounded and K is L-compact. Then GVVI(T, K) has a solution if

(i) T has convex values;

(ii) T : co(E) −→ 2Y _{is u.s.c for every}

E ∈ F (K);

(iii) T satisfies the (S)+-condition;

(iv) T (K) is bounded in Lb(X, Z) and is

contained in a compact subset of Y . Theorem 2. Assume that X and Z are locally convex with X barreled, and that K is L-compact. If T satisfies the conditions (i), (ii) and (iii) of Theorem 1 with T (K) contained in a compact subset of Y , then GVVI(T, K) has a solution.

Theorem 2 is a vector version of [5, Theorem 3.1].. While Theorem 3 given

below is a vector version of Cubiotti and Yao’s result [4, Theorem 3.1].

Theorem 3. Assume that X and Z are locally convex with X barreled, and that K is L-compact. Then GVVI(T, K) has a solution if

(i) T (x) is closed and convex for every x ∈ K;

(ii) T (K) is contained in a compact sub-set of Y ;

(iii) T is u.s.c and satisfies the (S)1 +

con-dition.

Finally, the following theorem general-ize [4, Theorem 3.1] to barreled normed spaces.

Theorem 4. Let X be a barreled normed space, K be weakly compact, and let T : K −→ 2Xs∗ be a mapping. Then GVI(T, K) has a solution if

(i) T (x) is nonempty, closed and convex for each x ∈ K;

(ii) T (K) is contained in a weakly com-pact subset of X_b∗;

(iii) T is u.s.c and satisfies the sequential (S)1

+ condition.

References

[1] Q.H. Ansari, X.Q. Yang and J.C. Yao, Existence and Duality of Im-plicit Vector Variational Problems, Numer. Funct. Anal. Optim., 22, pp. 815-829. 2001.

[2] O. Chadli, Y. Chiang and S. Huang, Topological pseudomonotonicity and vector equilibrium problems, J. Math. Anal. Appl., 270, pp. 435-450, 2002.

[3] Y. Chiang and J.C. Yao, Vector Vari-ational Inequalities and the (S)+

-condition, J. Optim. Theory Appl., 123, pp. 271-290, 2004.

[4] P. Cubiotti and J.C. Yao, Multi-valued (S)1

+ Operators and

Gener-alized Variational Inequalities, Com-put. Math. Appl., 29, pp. 49-56, 1995. [5] J.S. Guo and J.C. Yao, Variational Inequalities with Nonmonotone Op-erators, J. Optim. Theory Appl., 80, pp. 63-74, 1994.

[6] G. Isac and M.S. Gowda, Opera-tors of Class (S)1₊, Altman’s Condi-tion and the Complementarity Prob-lem, Journal of the Faculty of Sci-ence, University of Tokyo, Section IA, Mathematics 40, pp. 1-16, 1993.