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 Df 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 γ-condition and an extension of the theory of Smale's approximate zeros.

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Generalized vector variational inequality, Variable ordering relation, Cone mapping, KKM-Fan theorem, Brouwer fixed point theorem, Monotonicity, Complete continuity, Primary 49J30, 47H10, 47H17,