Extended High Convergence Compositions for Solving Nonlinear Equations in Banach Space

Abstract

The local as well as the semi-local convergence analysis is provided for two compositions for solving Banach space valued operator nonlinear equations. These compositions are defined on the real line. They were shown to be efficient and of convergence order six. But, the convergence in the local convergence case utilized assumptions reaching the seventh derivative not on the composition. Moreover, no computable error estimates on the distances or uniqueness of the solution regions were provided, limiting the applicability of these compositions. The new convergence analysis is using conditions only on the operators on these compositions. Moreover, computable error estimates and uniqueness results are developed based on [Formula: see text]-continuity and in the more general setting of Banach space valued operators. Numerical applications are presented to validate the theoretical aspects.