Semilocal Convergence Analysis for Two-Step Newton Method under Generalized Lipschitz Conditions in Banach Spaces

TL;DR

This paper develops a semilocal convergence analysis for a two-step Newton method in Banach spaces under generalized Lipschitz conditions, achieving Q-cubic convergence and applicable to solving nonsymmetric algebraic Riccati equations.

Contribution

It introduces a new convergence analysis for the two-step Newton method under generalized Lipschitz conditions, extending classical results and covering key special cases.

Findings

Achieves Q-cubic convergence under additional conditions.

Provides convergence results for Kantorovich, Smale, and Nesterov-Nemirovskii cases.

Applies the theory to approximate solutions of nonsymmetric algebraic Riccati equations.

Abstract

In the present paper, we consider the semilocal convergence problems of the two-step Newton method for solving nonlinear operator equation in Banach spaces. Under the assumption that the first derivative of the operator satisfies a generalized Lipschitz condition, a new semilocal convergence analysis for the two-step Newton method is presented. The Q-cubic convergence is obtained by an additional condition. This analysis also allows us to obtain three important spacial cases about the convergence results based on the premises of Kantorovich, Smale and Nesterov-Nemirovskii types. An application of our convergence results is to the approximation of minimal positive solution for a nonsymmetric algebraic Riccati equation arising from transport theory.