2024/04/08 by Jiehua Mai, Enhui Shi, Mai, Jiehua +5
Computer Science · Mathematics · #54H20 #55M20 #55M25 #Advanced Optimization Algorithms Research #Algebraic Topology (math.AT) #Dynamical Systems (math.DS) #FOS: Mathematics #Fixed Point Theorems Analysis #Geometric Topology (math.GT) #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2404.05248
openalex publication_date 2024/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the existence of fixed points for continuous maps f from an n-ball X in \mathbb Rn to \mathbb Rn with n≥ 1. We show that f has a fixed point if, for some absolute retract Y⊂∂ X, f(Y)⊂ X and ∂ X-Y is an (f, X)-blockading set. For n≥ 2, let D be an n-ball in X and Y be an (n-1)-ball in ∂ X. Relying on the result just mentioned, we show the existence of a fixed point of f, if D and Y are well placed and behave well under f, and \rm deg(fD)=-\rm deg(f∂ Y), where fD=f|D: D → ℝn and f∂ Y=f|∂ Y: ∂ Y → ∂ Y. The degree \rm deg(fD) of fD is explicitly defined and some elementary properties of which are investigated. These results extend the Brouwer fixed point theorem.