A fixed point theorem for monotone functions

Abstract

The main result of this work states that if $f:R_{+}^m\to R_{+}^m$ is increasing and continuous and the set $S=\{x\ge 0:f\left(x\right)\le x\}$ is bounded and contains some $x^{\prime }>0$ then there is a non-zero fixed point of $f$, i.e. $f\left(x\right)=x\ne 0$. If $f:R^m\to R^m$ is increasing and continuous and the set $\{x:f\left(x\right)\le x\}$ is bounded and contains $x^{″}$ and $x^{\prime }$, $x^{″}<x^{\prime }$ there are multiple fixed points.