# Application of Upper Hemi-Continuous Operators on Generalized Bi-quasi-variational Inequalities in Locally Convex Topological Vector Spaces

Application of Upper Hemi-Continuous Operators on Generalized Bi-quasi-variational Inequalities... Let $$E$$ and $$F$$ be Hausdorff topological vector spaces over the field $$\Phi$$ , let $$\left\langle , \right\rangle :F \times E \to \Phi$$ be a bilinear functional, and let $$X$$ be a non-empty subset of $$E$$ . Given a set-valued map $$S:X \to 2^X$$ and two set-valued maps $$M,T:X \to 2^F$$ , the generalized bi-quasi-variational inequality (GBQVI) problem is to find a point $$\hat y \in X$$ and a point $$\hat w \in T(\hat y)$$ such that $$\hat y \in S(\hat y)$$ and $$\operatorname{Re} \left\langle {f - \hat w,\hat y - x} \right\rangle \leqslant 0$$ for all $$x \in S(\hat y)$$ and for all $$f \in M(\hat y)$$ or to find a point $$\hat y \in X,$$ a point $$\hat w \in T(\hat y)$$ and a point $$\hat f \in M(\hat y)$$ such that $$\hat y \in S(\hat y)$$ and $$\operatorname{Re} \left\langle {\hat f - \hat w,\hat y - x} \right\rangle \leqslant 0$$ for all $$x \in S(\hat y)$$ . The generalized bi-quasi-variational inequality was introduced first by Shih and Tan [8] in 1989. In this paper we shall obtain some existence theorems of generalized bi-quasi-variational inequalities as application of upper hemi-continuous operators [4] in locally convex topological vector spaces on compact sets. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Application of Upper Hemi-Continuous Operators on Generalized Bi-quasi-variational Inequalities in Locally Convex Topological Vector Spaces

, Volume 3 (4) – Oct 22, 2004
12 pages

/lp/springer_journal/application-of-upper-hemi-continuous-operators-on-generalized-bi-quasi-9zd9FV8CSG
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1009849400516
Publisher site
See Article on Publisher Site

### Abstract

Let $$E$$ and $$F$$ be Hausdorff topological vector spaces over the field $$\Phi$$ , let $$\left\langle , \right\rangle :F \times E \to \Phi$$ be a bilinear functional, and let $$X$$ be a non-empty subset of $$E$$ . Given a set-valued map $$S:X \to 2^X$$ and two set-valued maps $$M,T:X \to 2^F$$ , the generalized bi-quasi-variational inequality (GBQVI) problem is to find a point $$\hat y \in X$$ and a point $$\hat w \in T(\hat y)$$ such that $$\hat y \in S(\hat y)$$ and $$\operatorname{Re} \left\langle {f - \hat w,\hat y - x} \right\rangle \leqslant 0$$ for all $$x \in S(\hat y)$$ and for all $$f \in M(\hat y)$$ or to find a point $$\hat y \in X,$$ a point $$\hat w \in T(\hat y)$$ and a point $$\hat f \in M(\hat y)$$ such that $$\hat y \in S(\hat y)$$ and $$\operatorname{Re} \left\langle {\hat f - \hat w,\hat y - x} \right\rangle \leqslant 0$$ for all $$x \in S(\hat y)$$ . The generalized bi-quasi-variational inequality was introduced first by Shih and Tan [8] in 1989. In this paper we shall obtain some existence theorems of generalized bi-quasi-variational inequalities as application of upper hemi-continuous operators [4] in locally convex topological vector spaces on compact sets.

### Journal

PositivitySpringer Journals

Published: Oct 22, 2004

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