Fon-Che Liu

On a form of KKM principle and SupInfSup inequalities of von Neumann and of Ky Fan type

Abstract A version of KKM principle is considered together with its application to SupInfSup inequalities of von Neumann and of Ky Fan type. These inequalities which are respectively generalizations of von Neumann minimax theorem and minimax inequality of Ky Fan provide new light in obtaining either new results or new proofs for known results.

On the localization of rectangular partial sums for multiple Fourier series

The question of the localization for rectangular partial sums of the multiple Fourier series for functions of Sobolev spaces is settled.

Equilibrium value and measure of systems of functions

If λ and μ satisfy (1), then λ is called an equilibrium value and μ an equilibrium measure of the systems {fi}, {gi} in this order. If μ is supported by a point x0, then x0 is called an equilibrium point of the systems {fi}, {gi}. Let S be the standard (n − 1)-simplex in R, i.e. S is the set of all those points x = (x1, . . . , xn) of R with all xi ≥ 0 and ∑n i=1 xi = 1. In [3] Ky Fan has shown the existence and uniqueness of an equilibrium value and an equilibrium point of systems {f1, . . . , fn}, {g1, . . . , gn} of continuous functions defined on S under the following conditions: