Majorization fixed point principle and application to nonlinear integral equations

TL;DR

This paper introduces a modified fixed point principle that extends the Kantorovich method to nondifferentiable operators, providing exact estimates and applications to nonlinear integral equations.

Contribution

It presents a new fixed point principle for nondifferentiable operators, with precise estimates and applications to nonlinear integral equations.

Findings

Exact estimates of fixed point domain radii

New a priori and a posteriori error bounds

Applications to various nonlinear integral operators

Abstract

The successive approximations method allows us to solve problems concerning existence and uniqueness of fixed points of wide classes of operators. The classical result in this field, such as Banach -- Caccioppoli principle together with some its modification and generalisations, is applicable to operators satisfying Lipschitz condition with a small coefficient or, in other words, to operators with the compression property. However, the successive approximations method works well for other classes of operators that are not compressions. In particular, the well known Kantorovich fixed point principle for differentiable operators deals with operators that, in general, are not compression; moreover, this principle covers some cases when Banach -- Caccioppoli principle is nonapplicable. In the article it is presented some modification of Kantorovich fixed point principle that covers nondifferentiable operators. We describe the exact (unimprovable) estimates of the internal and external radius of the domain of existence of a unique fixed point of the operator under consideration and, in addition, we present new apriori and aposteriori error estimates for successive approximations to the corresponding fixed point. Some applications of the new fixed point principle to nonlinear integral operators of different types are given as well.