Annales Univ. Sci. Budapest., Sect. Comp. 47 (2018) 127–139 ON THE LOCAL CONVERGENCE OF WEIGHTED-NEWTON METHODS UNDER WEAK CONDITIONS IN BANACH SPACES Ioannis K. Argyros (Lawton, USA) Janak Raj Sharma and Deepak Kumar (Longowal, India) Communicated by Ferenc Schipp (Received December 15, 2017; accepted February 25, 2018) Abstract. In this paper, we consider the weighted-Newton methods de- veloped in [18] and study their local convergence in Banach space. In the earlier study the Taylor expansion of higher order derivatives is used which may not exist or may be very expensive or impossible to compute. However, the hypotheses of present analysis are based on the first Fr´ echet- derivative only, thereby the applicability of methods is expanded. New analysis also provides radius of convergence, error bounds and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches that use Taylor expansions of higher order derivatives. Order of convergence of the methods is calculated by using computational order of convergence or approximate computational order of convergence with- out using higher order derivatives. Numerical tests are performed on some problems of different nature that confirm the theoretical results. 1. Introduction Let E 1 , E 2 be Banach spaces and D ⊆ E 1 be closed and convex. In this study, we locate a solution α of the nonlinear equation (1.1) F (x)=0, Key words and phrases : Weighted-Newton methods, local convergence, nonlinear systems, Banach space, Fr´ echet-derivative. 2010 Mathematics Subject Classification : 49M15, 41A25, 65H10.