Constructive fixed point theorem for min-max functions

Min-max functions, F:Rⁿ → Rⁿ, arise in modelling the dynamic behaviour of discrete event systems. They form a dense subset of those functions which are homogeneous, F_i(x₁ + h,...,x_n + h) = F_i(x_i,...,x_n) + h, monotonic, qq ≤ qq → F(qq) ≤ F(qq), 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 F_i(qq) = x_i + h for some h qq R. We deduce several earlier fixed point results. The proof is inspired by Howard's 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.