Semilocal Convergence Analysis for Two-Step Newton Method under Generalized Lipschitz Conditions in Banach Spaces
This work provides a theoretical convergence framework for the two-step Newton method, but it is an incremental extension of existing convergence analyses to generalized Lipschitz conditions.
The paper presents a semilocal convergence analysis for the two-step Newton method under generalized Lipschitz conditions in Banach spaces, achieving Q-cubic convergence. The results are applied to approximate the minimal positive solution of a nonsymmetric algebraic Riccati equation from transport theory.
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.