Taiwanese Journal of Mathematics

ON THE SOLUTION EXISTENCE OF GENERALIZED VECTOR QUASI-EQUILIBRIUM PROBLEMS WITH DISCONTINUOUS MULTIFUNCTIONS

B. T. Kien, N. Q. Huy, and N. C. Wong

Full-text: Open access

Abstract

In this paper we deal with the following generalized vector quasi-equilibrium problem: given a closed convex set $K$ in a normed space $X$, a subset $D$ in a Hausdorff topological vector space $Y$, and a closed convex cone $C$ in $R^n$. Let $\Gamma: K\to 2^K$, $\Phi : K\rightarrow 2^{D}$ be two multifunctions and $f : K\times D\times K\to R^n$ be a single-valued mapping. Find a point $(\hat x, \hat y)\in K\times D$ such that \begin{gather} (\hat x, \hat y)\in \Gamma(\hat x)\times\Phi(\hat x),\,\, {\rm and}\,\, \{f(\hat x, \hat y, z): z\in\Gamma(\hat x)\}\cap (-{\rm Int }C)=\emptyset. \notag \end{gather} We prove some existence theorems for the problem in which $\Phi$ can be discontinuous and $K$ can be unbounded.

Article information

Source
Taiwanese J. Math., Volume 13, Number 2B (2009), 757-775.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405401

Digital Object Identifier
doi:10.11650/twjm/1500405401

Mathematical Reviews number (MathSciNet)
MR2510830

Zentralblatt MATH identifier
1176.49008

Subjects
Primary: 49J40: Variational methods including variational inequalities [See also 47J20] 49J45: Methods involving semicontinuity and convergence; relaxation 49J53: Set-valued and variational analysis [See also 28B20, 47H04, 54C60, 58C06] 46N10: Applications in optimization, convex analysis, mathematical programming, economics 91B50: General equilibrium theory

Keywords
solution existence generalized vector quasi-equilibrium problem implicit generalized quasivariational inequality lower semicontinuity upper semicontinuity Hausdorff lower semicontinuity $C$-convex $C$-lower semicontinuity $C$-upper semicontinuity

Citation

Kien, B. T.; Huy, N. Q.; Wong, N. C. ON THE SOLUTION EXISTENCE OF GENERALIZED VECTOR QUASI-EQUILIBRIUM PROBLEMS WITH DISCONTINUOUS MULTIFUNCTIONS. Taiwanese J. Math. 13 (2009), no. 2B, 757--775. doi:10.11650/twjm/1500405401. https://projecteuclid.org/euclid.twjm/1500405401


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