Full characterizations of minimax inequality, fixed point theorem, saddle point theorem, and KKM principle in arbitrary topological spaces
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© 2016, Springer International Publishing. 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, γ-recursive transfer lower semicontinuity, which fully characterizes the existence of γ-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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