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Full characterizations of minimax inequality, fixed point theorem, saddle point theorem, and KKM principle in arbitrary topological spaces

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Abstract

This paper provides necessary and sufficient conditions for the existence of solutions for some important problems from optimization and non-linear analysis by replacing two typical conditions—continuity and quasiconcavity with a unique condition, weakening topological vector spaces to arbitrary topological spaces that may be discrete, continuum, non-compact or non-convex. We establish a single condition, \(\gamma \)-recursive transfer lower semicontinuity, which fully characterizes the existence of \(\gamma \)-equilibrium of minimax inequality without imposing any restrictions on topological space. The result is then used to provide full characterizations of fixed point theorem, saddle point theorem, and KKM principle.

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Notes

  1. The FS is for Fan [2] and Sonnenschein [36].

  2. The SS is for Shafer and Sonnenschein [37].

  3. When \(\gamma =\sup _{y \in X} \phi (y, y)=+\infty \), any point in X is clearly a \(\gamma \)-equilibrium with \(\gamma =+\infty \).

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Authors and Affiliations

  1. Department of Economics, Texas A&M University, College Station, TX, 77843, USA

    Guoqiang Tian

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Correspondence to Guoqiang Tian.

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I shall thank an anonymous referee for helpful comments and suggestions. Financial support from the National Natural Science Foundation of China (NSFC-71371117) and the Key Laboratory of Mathematical Economics (SUFE) at Ministry of Education of China is gratefully acknowledged.

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Tian, G. Full characterizations of minimax inequality, fixed point theorem, saddle point theorem, and KKM principle in arbitrary topological spaces. J. Fixed Point Theory Appl. 19, 1679–1693 (2017). https://doi.org/10.1007/s11784-016-0314-z

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