Stud. Univ. Babe¸ s-Bolyai Math. 66(2021), No. 4, 757–768 DOI: 10.24193/subbmath.2021.4.12 Extended local convergence for Newton-type solver under weak conditions Ioannis K. Argyros, Santhosh George and Kedarnath Senapati Abstract. We present the local convergence of a Newton-type solver for equations involving Banach space valued operators. The eighth order of convergence was shown earlier in the special case of the k-dimensional Euclidean space, using hypotheses up to the eighth derivative although these derivatives do not appear in the method. We show convergence using only the first derivative. This way we extend the applicability of the methods. Numerical examples are used to show the convergence conditions. Finally, the basins of attraction of the method, on some test problems are presented. Mathematics Subject Classification (2010): 65F08, 37F50, 65N12. Keywords: Banach space, Newton-type, local convergence, Fr´ echet derivative. 1. Introduction Let Ω ⊂B 1 be nonempty, open, and B 1 , B 2 be Banach spaces. B(B 1 , B 2 )= {G : B 1 −→ B 2 be bounded and linear}, T (x,d)= {y ∈B 1 : ‖y − x‖ <d; d> 0} and ¯ T (x,d)= {y ∈B 1 : ‖y − x‖≤ d; d> 0}. One of the greatest challenges in Computational Mathematics is to find a solution x ∗ of the equation F (x)=0, (1.1) where F :Ω −→ B 2 is Fr´ echet differentiable operator. Notice that a plethora of applications from Mathematics, Science and Engineering are reduced to a form as (1.1) by utilizing Mathematical modeling [1-19]. The solution x ∗ is sought in closed form, but this can be achieved only in some cases. Hence, researchers develop iterative methods, generating a sequence approximating x ∗ under certain initial conditions.