J. Korean Math. Soc. 52 (2015), No. 1, pp. 23–41 http://dx.doi.org/10.4134/JKMS.2015.52.1.023 EXPANDING THE CONVERGENCE DOMAIN FOR CHUN-STANICA-NETA FAMILY OF THIRD ORDER METHODS IN BANACH SPACES Ioannis Konstantinos Argyros, Santhosh George, and ´ Angel Alberto Magre˜ n´ an Abstract. We present a semilocal convergence analysis of a third order method for approximating a locally unique solution of an equation in a Banach space setting. Recently, this method was studied by Chun, Stanica and Neta. These authors extended earlier results by Kou, Li and others. Our convergence analysis extends the applicability of these methods under less computational cost and weaker convergence criteria. Numerical examples are also presented to show that the earlier results cannot apply to solve these equations. 1. Introduction In this study we are concerned with the problem of approximating a locally unique solution x ∗ of the equation (1.1) F (x)=0, where F is a Fr´ echet-differentiable operator defined on a convex subset D of a Banach space X with values in a Banach space Y. Many problems in computational mathematics and other disciplines can be brought in a form like (1.1) using mathematical modelling [2, 4, 12, 15, 17, 20, 21]. The solutions of these equations can rarely be found in closed form. That is why most solution methods for these equations are usually iterative. In particular the practice of Numerical Functional Analysis for finding such solutions is essentially connected to Newton-like methods [1, 2, 4, 11, 12, 15, 16, 17, 19, 20, 21]. The study about convergence of iterative procedures is normally centered on two types: semi-local and local convergence analysis. The semi- local convergence matter is, based on the information around an initial point, to give criteria ensuring the convergence of the iterative procedures. While the local analysis is based on the information around a solution, to find estimates Received January 30, 2014; Revised April 4, 2014. 2010 Mathematics Subject Classification. 65H10, 65G99, 65K10, 47H17, 49M15. Key words and phrases. family of third order method, Newton-like methods, Banach space, semilocal convergence, majorizing sequences, recurrent relations, recurrent functions. c 2015 Korean Mathematical Society 23