*(English)*Zbl 1199.54209

This article deals with two generalizations of the classical Banach-Caccioppoli fixed point principle for contractions in complete metric spaces. In the first of them, the author considers operators $T:X\to X$, where $(X,d)$ is a complete metric space with a metric $d$, satisfying the following condition

where $0\in (0,1)$, $L\ge 0$. It is proved that $\text{Fix}\left(T\right)\ne \varnothing $, the convergence of the Picard iteration ${x}_{n+1}=T{x}_{n}$ to some ${x}_{*}\in \text{F}ix\left(T\right)$ for any ${x}_{0}\in X$, and the estimate

(note that $T$ can have more than one fixed point). In the second generalization, the author considers operators $T:X\to X$ satisfying the following condition

where also $0\in (0,1)$, $L\ge 0$ and $M(x,y)$ is defined by the same formula. It is proved that $T$ has a unique fixed point ${x}_{*}$, the Picard iteration ${x}_{n+1}=T{x}_{n}$ converges to ${x}_{*}$ for any ${x}_{0}\in X$ and the estimate (1) holds, and, moreover, the following estimate

also holds. In the end of the article, the author gives some illustrative examples and some remarks about relations between new generalizations of the Banach-Caccioppoli fixed point principle and early versions of such generalizations.

##### MSC:

54H25 | Fixed-point and coincidence theorems in topological spaces |