Argyros, I.K.George, S.2020-03-312020-03-312013Asian-European Journal of Mathematics, 2013, Vol.6, 3, pp.-https://idr.nitk.ac.in/handle/123456789/11233We present a new semilocal convergence analysis of Newton's method on Riemannian manifolds with values in a cone in order to solve the inclusion problem. Using more precise majorizing sequences than in earlier studies such as [J. H. Wang, S. Huang and C. Li, Extended Newton's method for mappings on Riemannian manifolds with values in a cone, Taiwanese J. Math. 13(2B) (2009) 633-656] and the concept of L-average Lipschitz condition we provide: weaker sufficient convergence conditions; tighter error analysis on the distances involved and an at least as precise information on the solutions. These advantages are obtained using the same parameters and functions. Applications include the celebrated Newton-Kantorovich theorem. 2013 World Scientific Publishing Company.Article