EXTENDED NEWTON’S METHOD FOR MAPPINGS ON RIEMANNIAN MANIFOLDS WITH VALUES IN A CONE

Abstract

Robinson's generalized Newton's method for nonlinear functions with values in a cone is extended to mappings on Riemannian manifolds with values in a cone. When ${\cal D}f$ satisfies the $L$-average Lipschitz condition, we use the majorizing function technique to establish the semi-local quadratic convergence of the sequences generated by the extended Newton's method. As applications, we also obtain Kantorovich's type theorem, Smale's type theorem under the $\gamma$-condition and an extension of the theory of Smale's approximate zeros.