A constructive fixed point theorem for min-max functions

Min-max functions, F: R^n- R^n, arise in modelling the dynamic behaviour of discrete event systems. They form a dense subset of those functions which are homogeneous, Fi(X1 + h, . . ., Xn + h) = Fi(X1, ..., Xn) +h, monotonic, x <= y - F(x)<= F(y), and nonexpansive in the l norm-so-called topical functions which have appeared recently in the work of several authors. Our main result characterizes those min-max functions which have a (generalized) fixed point) where Fi(x) = xi+h for some h E R. We deduce several earlier fixed point results. The proof is inspired by Howardʹs policy improvement scheme in optimal control and yields an algorithm for finding a fixed point, which appears efficient in an important special case. An extended introduction sets the context for this paper in recent work on the dynamics of topical functions.