Nash points, Ky Fan inequality and equilibria of abstract economies in Max-Plus and B-convexity

B-convexity was introduced in [W. Briec, C. Horvath, B-convexity, Optimization 53 (2004) 103–127]. Separation and Hahn–Banach like theorems can be found in [G. Adilov, A.M. Rubinov, B-convex sets and functions, Numer. Funct. Anal. Optim. 27 (2006) 237–257] and [W. Briec, C.D. Horvath, A. Rubinov, Separation in B-convexity, Pacific J. Optim. 1 (2005) 13–30]. We show here that all the basic results related to fixed point theorems are available in B-convexity. Ky Fan inequality, existence of Nash equilibria and existence of equilibria for abstract economies are established in the framework of B-convexity. Monotone analysis, or analysis on Maslov semimodules [V.N. Kolokoltsov, V.P. Maslov, Idempotent Analysis and Its Applications, Math. Appl., vol. 401, Kluwer Academic, 1997; V.P. Litvinov, V.P. Maslov, G.B. Shpitz, Idempotent functional analysis: An algebraic approach, Math. Notes 69 (2001) 696–729; V.P. Maslov, S.N. Samborski (Eds.), Idempotent Analysis, Advances in Soviet Mathematics, Amer. Math. Soc., Providence, RI, 1992], is the natural framework for these results. From this point of view Max-Plus convexity and B-convexity are isomorphic Maslov semimodules structures over isomorphic semirings. Therefore all the results of this paper hold in the context of Max-Plus convexity.