A constructive fixed point theorem for min-max functions∗
نویسندگان
چکیده
Min-max functions, F : R → R, 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 `∞ norm—so-called topical functions—which have appeared recently in the work of several authors. Our main result characterises those min-max functions which have a (generalised) fixed point, where Fi(~x) = xi + 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.
منابع مشابه
Constructive Proof of Tychonoff’s Fixed Point Theorem for Sequentially Locally Non-Constant Functions
We present a constructive proof of Tychonoff’s fixed point theorem in a locally convex space for uniformly continuous and sequentially locally non-constant functions. Keywords—sequentially locally non-constant functions, Tychonoff’s fixed point theorem, constructive mathematics.
متن کاملA New Common Fixed Point Theorem for Suzuki Type Contractions via Generalized $Psi$-simulation Functions
In this paper, a new stratification of mappings, which is called $Psi$-simulation functions, is introduced to enhance the study of the Suzuki type weak-contractions. Some well-known results in weak-contractions fixed point theory are generalized by our researches. The methods have been appeared in proving the main results are new and different from the usual methods. Some suitable examples ar...
متن کاملBrouwer's fixed point theorem with sequentially at most one fixed point
We present a constructive proof of Brouwer’s fixed point theorem with sequentially at most one fixed point, and apply it to the mini-max theorem of zero-sum games.
متن کاملProof of Constructive Version of the Fan-Glicksberg Fixed Point Theorem Directly by Sperner’s Lemma and Approximate Nash Equilibrium with Continuous Strategies: A Constructive Analysis
It is often demonstrated that Brouwer’s fixed point theorem can not be constructively proved. Therefore, Kakutani’s fixed point theorem, the Fan-Glicksberg fixed point theorem and the existence of a pure strategy Nash equilibrium in a strategic game with continuous (infinite) strategies and quasi-concave payoff functions also can not be constructively proved. On the other hand, however, Sperner...
متن کاملConstructive Proof of Brouwer’s Fixed Point Theorem for Sequentially Locally Non-constant and Uniformly Sequentially Continuous Functions
We present a constructive proof of Brouwer’s fixed point theorem for sequentially locally non-constant and uniformly sequentially continuous functions based on the existence of approximate fixed points. And we will show that our Brouwer’s fixed point theorem implies Sperner’s lemma for a simplex. Since the existence of approximate fixed points is derived from Sperner’s lemma, our Brouwer’s fixe...
متن کامل