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The successive approximations method allows us to solve problems concerningexistence and uniqueness of fixed points of wide classes of operators. Theclassical result in this field, such as Banach -- Caccioppoli principletogether with some its modification and generalisations, is applicable tooperators satisfying Lipschitz condition with a small coefficient or, in otherwords, to operators with the compression property. However, the successiveapproximations method works well for other classes of operators that are notcompressions. In particular, the well known Kantorovich fixed point principlefor differentiable operators deals with operators that, in general, are notcompression; moreover, this principle covers some cases when Banach --Caccioppoli principle is nonapplicable. In the article it is presented somemodification of Kantorovich fixed point principle that covers nondifferentiableoperators. We describe the exact (unimprovable) estimates of the internal andexternal radius of the domain of existence of a unique fixed point of theoperator under consideration and, in addition, we present new apriori andaposteriori error estimates for successive approximations to the correspondingfixed point. Some applications of the new fixed point principle to nonlinearintegral operators of different types are given as well.

Majorization fixed point principle and application to nonlinear integral equations