Berge's maximum theorem to vector-valued functions with some applications

Volume 10, Issue 4, pp 1861--1872 http://dx.doi.org/10.22436/jnsa.010.04.46

Publication Date: April 20, 2017 Submission Date: February 14, 2017

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Authors

Qiu Xiaoling - School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China. Peng Dingtao - School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China. Yu Jian - School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China.

Abstract

In this paper, we introduce pseudocontinuity for Berge’s maximum theorem for vector-valued functions which is weaker than semicontinuity. We prove the Berge’s maximum theorem for vector-valued functions with pseudocontinuity and obtain the set-valued mapping of the solutions is upper semicontinuous with nonempty and compact values. As applications, we derive some existence results for weakly Pareto-Nash equilibrium for multiobjective games and generalized multiobjective games both with pseudocontinuous vector-valued payoffs. Moreover, we obtain the existence of essential components of the set of weakly Pareto-Nash equilibrium for these discontinuous games in the uniform topological space of best-reply correspondences. Some examples are given to investigate our results.

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Keywords

• Maximum theorem
• vector-valued functions
• set-valued mapping
• pseudocontinuity
• weakly Pareto-Nash equilibrium
• essential components.

MSC

•  47H04
•  90C31
•  91A10

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