Generalization of istratescu's theorem on contractive mappings in metric spaces

M. L. Katkov

Zapiski Nauchnykh Seminarov POMI, Investigations in topology. Part III, Tome 83 (1979), pp. 73-82

Résumé

In this paper we consider mappings $T\colon X\to X$ of metric spaces, satisfying the condition: $$ d(T_x,T_y)\leqslant\omega(\alpha_1d(x,y)+\alpha_2d(x,Tx)+\alpha_3d(y,Ty)+\alpha_4d(x,Ty)+\alpha_5d(y,Tx), $$ where $\omega$ is some right semicontinuous function. We prove that if $\omega$ is a nondecreasing function, $\omega(r)$ for $r>0$, $r-\omega(r)\to\infty$ as $r\to\infty$, $\sum^5_{i=1}\alpha_i(x,y)\leqslant1$, then the map $T$ has a fixed point $\xi$ and $\lim_{n\to\infty}T^nx=\xi$ for any point $x\in X$. Interesting examples are given.