Local convergence for a family of third order methods in Banach spaces

Abstract

We present a local convergence analysis of a family of third order methods for approximating a locally unique solution of nonlinear equations in a Banach space setting. Recently, the semilocal convergence analysis of this method was studied by Chun, Stanic ˘ a and Neta in [10]. ˘ These authors extended earlier results by Kou, Li [17] and others [8, ?, 11, 13, 14]. The convergence analysis is based on hypotheses up to the second Frechet derivative of the operator involved. This work further extends the ´ results of [10] and provides computable convergence ball and computable error bounds under hypotheses only up to the first Frechet derivative.