Fixed points for discontinuous quasi-monotone maps in $\textbf {R}^ n$

Abstract:

Let ${K_n}$ be the unit cube in ${R^n}$ and $f = ({f_1},{f_2}, \ldots ,{f_n}):{K_n} \to {R^n}$. It is known that $f$ has maximal and minimal fixed points in ${K^n}$ if $f:{K_n} \to {K_n}$ and $f$ is monotone increasing. In this paper, a weaker condition, namely quasi-monotonicity, is considered and it is proved that the above mentioned conclusion is still true if $f$ is either quasi-monotone and \[ \lim \inf \limits _{t \to 0} \frac {{[{f_i}(x + t{e_i}) - {f_i}(x)]}}{t} \ne - \infty ,\] or $- f$ is quasi-monotone and \[ \lim \sup \limits _{t \to 0} \frac {{[{f_i}(x + t{e_i}) - {f_i}(x)]}}{t} \ne + \infty \].