FAN'S INEQUALITY IN THE CONTEXT OF MP-CONVEXITY

Abstract

Several analogues of Fan’s inequality are proved in the context of Mp-convexity.As a consequence, a Nash equilibrium theorem is obtained. 1. Introduction In what follows we are interested in a class of functions having a nice behavior under the action of means. The weighted Mp-mean is de ned for pairs of positive numbers x; y by the formula Mp(x; y; 1 ; ) = 8> >: ((1 )x + y); if p 2 Rnf0g x y ; if p = 0 minfx; yg; if p = 1 maxfx; yg; if p =1; where 2 [0; 1]: If p is an odd number, we can extend Mp to pairs of real numbers. Let E be a linear topological space and assume that C is a nonempty compact and convex subset of E. De nition 1. We say that a function f : C ! R is Mp-concave if f((1 )x+ y) Mp(f(x); f(y); 1 ; ) for all x; y 2 C and 2 (0; 1): Thus the M1-concave functions are the usual concave functions, while the M1concave functions, are precisely the quasi-concave functions. A celebrated result due to Ky Fan asserts that any function f : C C ! R+ which is quasi-concave in the rst variable and lower semicontinuous in the second variable veri es the inequality (1.1) min y2C sup x2C f(x; y) sup z2C f(z; z): The aim of this paper is to prove a complementary result, precisely: Theorem 1. Suppose that f : C C ! R+ is a function which is Mp-concave and lower-semicontinuous in each variable. Then min y2C sup x2C M p (f(x; y); f(y; x); 1 ; ) sup z2C f(z; z); for all 2 (0; 1) and p 2 R. 2000 Mathematics Subject Classi cation. Primary 05C38, 15A15; Secondary 05A15, 15A18. Key words and phrases. Mpconvexity, min-max inequality, Nash equilibrium. Published in vol.: Applied Analysis and Di¤ erential Equations. Proc. ICAADE 2006, pp. 267-274, World Scienti c, Singapore, 2007 (Editors, Ovidiu Carja and Ioan I. Vrabie). ISBN 978-981-270-594-5, ISSN 981-270-594-5. Revised, February 12, 2009. 1 2 CONSTANTIN P. NICULESCU AND IONEL ROVENTA Our technics yields also the following fact: Theorem 2. Let C be a nonempty compact and convex subset of E; and let f : C C ! R+ be a function which is Mp-concave in the rst variable and lowersemicontinuous in the second variable. Then min y2C max x2C M p (f(y; y); f(x; x); 1 ; ) sup z2C f(z; z) for all 2 (0; 1) and p 2 R. For p an odd number f is allowed to take negative values. 2. Proof of the main result We actually prove a much more general result: Theorem 3. Assume f : C C ! R+ is a function which is Mp-concave and lower-semicontinuous in each variable and let g : C ! C be a continuous onto function. Then min y2C sup x2C M p (f(x; y); f(y; x); 1 ; ) sup z2C M p (f(z; g(z)); f(g(z); z); 1 ; ); for all 2 (0; 1) and p 2 R. Theorem 1 represents the particular case where g is the identity of C: The proof of Theorem 3 is based on the KKM-Theorem, whose statement is recalled here for the convenience of the reader: Theorem 4. (Knaster-Kuratowski-Mazurkievicz). Suppose that for every point x in a nonempty set X E there is an associated closed subset M(x) X such that