On fixed point theorems of contractive type

Abstract:

Let $G$ be a continuous map of a nonempty compact metric space $(X,d)$ into itself, such that for some positive integer $m$, the iterated map ${G^m}$ satisfying \[ d({G^m}(x),{G^m}(y)) < \max \left \{ {d(x,y),d(x,{G^m}(x)),d(y,{G^m}(y)),d(x,{G^m}(y)),d(y,{G^m}(x))} \right \} \] for all $x$, $y \in X$ with $x \ne y$. It is shown that (i) $G$ has a unique fixed point ${x^ * } \in X$; (ii) the sequence of iterates $\left \{ {{G^k}(x)} \right \}$ converges to ${x^ * }$ for any $x \in X$; (iii) given $\lambda$, $0 < \lambda < 1$, there exists a metric ${d_\lambda }$, topologically equivalent to $d$, such that ${d_\lambda }(G(x)$, $G(y)) \leqslant \lambda {d_\lambda }(x,y)$ for all $x$, $y \in X$.