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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Applications of pseudo-monotone operators
with some kind of upper semicontinuity
in generalized quasi-variational inequalities
on non-compact sets


Authors: Mohammad S. R. Chowdhury and Kok-Keong Tan
Journal: Proc. Amer. Math. Soc. 126 (1998), 2957-2968
MSC (1991): Primary 47H04, 47H05, 47H09, 47H10; Secondary 49J35, 49J40, 54C60
MathSciNet review: 1459115
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Abstract: Let $E$ be a topological vector space and $X$ be a non-empty subset of $E$. Let $S:X\rightarrow 2^{X}$ and $T:X\rightarrow 2^{E^{*}}$ be two maps. Then the generalized quasi-variational inequality (GQVI) problem is to find a point $\hat y\in S(\hat y)$ and a point $\hat w\in T(\hat y)$ such that $Re\langle \hat w,\hat y-x\rangle \leq 0$ for all $x\in S(\hat y)$. We shall use Chowdhury and Tan's 1996 generalized version of Ky Fan's minimax inequality as a tool to obtain some general theorems on solutions of the GQVI on a paracompact set $X$ in a Hausdorff locally convex space where the set-valued operator $T$ is either strongly pseudo-monotone or pseudo-monotone and is upper semicontinuous from $co(A)$ to the weak$^{*}$-topology on $E^{*}$ for each non-empty finite subset $A$ of $X$.


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Additional Information

Mohammad S. R. Chowdhury
Affiliation: Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5
Email: mohammad@mscs.dal.ca

Kok-Keong Tan
Affiliation: Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5
Email: kktan@mscs.dal.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04436-0
PII: S 0002-9939(98)04436-0
Keywords: Generalized quasi-variational inequality, locally convex space, partition of unity, paracompact sets, lower semi-continuous, upper semi-continuous, strongly pseudo-monotone, pseudo-monotone and monotone operators
Received by editor(s): May 15, 1996
Received by editor(s) in revised form: March 7, 1997
Additional Notes: The work of the second author was partially supported by NSERC of Canada under grant A-8096.
Communicated by: Dale Alspach
Article copyright: © Copyright 1998 American Mathematical Society