Some extensions of the Brouwer fixed point theorem

Abstract

We study the existence of fixed points for continuous maps $f$ from an $n$-ball $X$ in ${\mathbb{R}}^n$ to ${\mathbb{R}}^n$ with $n\ge 1$. We show that $f$ has a fixed point if, for some absolute retract $Y\subset \partial X$, $f(Y)\subset X$ and $\partial X-Y$ is an $(f,X)$-blockading set. For $n\ge 2$, let $D$ be an $n$-ball in $X$ and $Y$ be an $(n-1)$-ball in $\partial X$. 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 $\mathrm{deg}(f_D)=-\mathrm{deg}(f_{\partial Y})$, where $f_D=f|D:D\to {\mathbb{R}}^n$ and $f_{\partial Y}=f|\partial Y:\partial Y\to \partial Y$. The degree $\mathrm{deg}(f_D)$ of $f_D$ is explicitly defined and some elementary properties of which are investigated. These results extend the Brouwer fixed point theorem.