Minimax Inequalities and Related Theorems for Arbitrary Topological Spaces: A Full Characterization

This paper provides a complete solution to the problem of minimax inequality for an arbitrary topological space that may be discrete, continuum, non-compact or non-convex. We establish a single condition, γ-recursive transfer lower semicontinuity, which is necessary and sufficient for the existence of equilibrium of minimax inequality without imposing any kind of convexity nor any restriction on topological space. Since minimax inequality provides the foundation for many of the modern essential results in diverse areas of mathematical sciences, the results can be used to significantly generalize or fully characterize many other important existence results such as KKM lemma, variational inequality, saddle point, fixed-point and coincidence theorems, etc. As illustrations, we show how they can be employed to fully characterize fixed point theory, saddle point theory, and the FKKM theory. The method of proof adopted to obtain our main results is also new and elementary — neither fixed-point-theorem nor KKM-theorem approach is used. 2010 Mathematical subject classification: 49K35, 90C26, 55M20 and 91A10.