Stud. Univ. Babe¸ s-Bolyai Math. 62(2017), No. 1, 127–135 DOI: 10.24193/subbmath.2017.0010 Ball convergence of a stable fourth-order family for solving nonlinear systems under weak conditions Ioannis K. Argyros, Munish Kansal and Vinay Kanwar Abstract. We present a local convergence analysis of fourth-order methods in order to approximate a locally unique solution of a nonlinear equation in Banach space setting. Earlier studies have shown convergence using Taylor expansions and hypotheses reaching up to the fifth derivative although only the first derivative appears in these methods. We only show convergence using hypotheses on the first derivative. We also provide computable: error bounds, radii of convergence as well as uniqueness of the solution with results based on Lipschitz constants not given in earlier studies. The computational order of convergence is also used to determine the order of convergence. Finally, numerical examples are also provided to show that our results apply to solve equations in cases where earlier studies cannot apply. Mathematics Subject Classification (2010): 65D10, 65D99. Keywords: Local convergence, nonlinear equation, Lipschitz condition, Fr´ echet derivative. 1. Introduction Let B 1 , B 2 be Banach spaces and D be a convex subset of B 1 . Let also L(B 1 ,B 2 ) denote the space of bounded linear operators from B 1 into B 2 . In the present paper, we deal with the problem of approximating a locally unique solution x ∗ of the equation F (x)=0, (1.1) where F : D ⊆ B 1 → B 2 is a Fr´ echet-differentiable operator. Numerous problems can be written in the form of (1.1) using Mathematical Modelling [3, 5, 8, 9, 12, 13, 18, 19, 22, 26, 28, 29, 30]. Analytical methods for solving such problems are almost non-existent and therefore, it is only possible to obtain approximate solutions by relying on numerical methods based on iterative procedure