(Communicated by William J. Davis). ABSTRACT. In this note, we prove a generalization of a generalized quasi- variational inequality of Shih and Tan in locally ...
PROCEEDINGS of the AMERICAN MATHEMATICAL Volume
103, Number
2, June
SOCIETY 1988
SHORTER NOTES The purpose of this department is to publish very short papers of unusually polished character, for which there is no other outlet.
REMARK ON A GENERALIZED QUASI-VARIATIONAL INEQUALITY WON KYU KIM (Communicated
by William J. Davis)
ABSTRACT. In this note, we prove a generalization of a generalized quasivariational inequality of Shih and Tan in locally convex Hausdorff topological vector spaces.
In a recent paper [1], Shih and Tan proved some generalized quasi-variational inequalities in locally convex Hausdorff topological vector spaces (Theorems 1, 2 and 3 in [1]) or in normed spaces (Theorem 4 in [1]). In this note, we prove the following generalization of their Theorem 4 to locally convex Hausdorff topological vector spaces as follows: THEOREM. Let E be a locally convex Hausdorff topological vector space, and X be a nonempty compact convex subset of E. Let S: X —»2X be continuous such that for each x G X, S(x) is a nonempty closed convex subset of X, and letT: X —»2E be upper semicontinuous from the relative topology of X to the strong topology of E* such that for each x G X, T(x) is a nonempty compact convex subset of E*. Then there exists a point y G X such that
(l)yGS(y), (2) infu€T(y) Re(u, y —x) < 0 for all x G S(y). PROOF. By Theorem 3 of [1], it suffices to show that the set ^2 := S 2/ € X| sup inf Re(u, y - x) > 0 > { x€S(y) u€T(y) J is open in X. Let yo GÜ2- Then there exists xo G S (yo) such that
t= Received
by the editors
inf
ueT(y0)
Re(u,yo —xo) > 0.
May 1, 1987 and, in revised form, November
1980 Mathematics Subject Classification (1985 Revision). Primary
20, 1987.
49A29, 49A40; Secondary
52A07. Key words and phrases. Locally convex Hausdorff topological vector space, variational inequality, compact convex set, strong topology, weak'-topology. ©1988
American
0002-9939/88
667
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generalized
Mathematical
quasi-
Society
$1.00 + $.25 per page
W. K. KIM
668
Let Uo := {p G E*\ Re(p, yo —xq) > t/2}. Then t/0 is a weak*-open neighborhood of T(yo). Let W := {p G E*\ supx yeX \(p, x —y)\ < t/6}. Then W is a strongly open neighborhood of 0 € E*, so that U\ — T(y0) + W is a strongly open neighborhood of T(yo). Now let U := mîueT(yo) Re(u, y —yo) are continuous, there exist an open neighborhood V of xq such that infu€r(¡/0) Re(i¿, yo — x) > t/2 for all x G V, and an open
neighborhood V2 of yo such that | infu6r(yo) Re(u,y —yo)\ < t/6 for all y G V2. Since xq G S (yo) n V and S is lower semicontinuous
at yo, there exists an open
neighborhood V3 of j/0 such that S(y) Í) V ^ 0 for all y G V3. Now let V :— (2/0- zo + V) n (Vi n V2 n V3); then F is an open neighborhood of yo. For each y € V, there exists x G V such that y = j/o - xq + x G V\ n V2 PI V3. Since
y G Vi n V2n V3, there exists ii G S (y) n V, and T(y) C Í7 C T(y0) + IV- It follows that inf
u€T{y)
Re(u,y
— xi) >
>
inf
Re(u,y
inf
Re(u,y-xi)--
inf
Re(u,y -y0)+
ueT{y0)
u€T(y0)
>
— x\) + inf Re(u,y uÇW
b
u€T(y0)
>-6
t
— x\)
inf
Re(u, y0 - xi) - -
u€T{y0) t
+ 2-6
t
D
t
= 6>°-
Therefore we have supxeS(yj infueT(y) Re(u, y - x) > 0, so that t/gE2. that £2 is open in X, and it completes the proof.
This shows
REFERENCES 1. M. H. Shih and K. K. Tan, Generalized quasi-variational
inequalities
in locally convex topological
vector spaces, J. Math. Anal. Appl. 108 (1985), 333-343.
Department of Mathematics Cheongju 360-763, Korea
Education,
Chungbuk
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National
University,
