A constructive fixed point theorem for min-max functions

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Abstract
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, F i (x 1 + h, .. . , x n + h) = Fi(x 1 , … , x n ) + h, monotonic , x y F(x) F(y), and nonexpansive in the l infinity 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) = x i + h for some h 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.

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