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 , where is a complete metric space with a metric , satisfying the following condition
where , . It is proved that , the convergence of the Picard iteration to some for any , and the estimate
(note that can have more than one fixed point). In the second generalization, the author considers operators satisfying the following condition
where also , and is defined by the same formula. It is proved that has a unique fixed point , the Picard iteration converges to for any 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.