Local convergence of Newton-like methods for generalized equations

摘要

We are concerned with the problem of approximating a locally unique solution of a generalized equation using a Newton-like method in a Banach space setting. Using some ideas introduced by us in [I.K. Argyros, A unifying local–semilocal convergence analysis and applications for Newton-like methods, J. Math. Anal. Appl. 298 (2004) 374–397; I.K. Argyros, Approximate solution of operator equations with applications, World Scientific Publ. Comp., New Jersey, USA, 2005] for nonlinear equations, we provide a local analysis leading to the super–linear convergence of the method. Our approach has the following advantages (under the same computational cost) over earlier work [M.H. Geoffroy, A. Piétrus, Local convergence of some iterative methods for solving generalized equations, J. Math. Anal. Appl. 290 (2004) 497–505]: finer error on the distances involved, and a larger radius of convergence leading to a wider choice of initial guesses and fewer computations to arrive at a desired error tolerance.