A class of iterative methods for solving nonlinear operator equations

TL;DR

This paper introduces a general principle for analyzing gradient-like iterative methods to solve nonlinear operator equations in Hilbert and Banach spaces, providing convergence conditions, rate estimates, and error bounds.

Contribution

It formulates a unifying principle that simplifies convergence analysis and error estimation for classical iterative methods on complex spaces.

Findings

Established convergence conditions for various methods.

Provided effective a priori and aposteriori error estimates.

Applied the principle to classical methods like minimal residuals and steepest descent.

Abstract

The article deals with gradient-like iterative methods for solving nonlinear operator equations on Hilbert and Banach spaces. The authors formulate a general principle of studying such methods. This principle allows to formulate simple conditions of convergence of the method under consideration, to estimate the rate of this convergence and to give effective a priori and aposteriori error estimates in terms of a scalar function that is constructed on the base of estimates for properties of invertibility and smoothness of linearizations of the left-hand side of the equations under study. The principle is applicable for analysis of such classical methods as method of minimal residuals, method of steepest descent, method of minimal errors and others. The main results are obtained for operator equations on Hilbert spaces and Banach spaces with a special property, that is called Bynum property.