Semilocal convergence of Stirling's method under Hölder continuous first derivative in Banach spaces

Abstract

The aim of this paper is to establish the semilocal convergence of Stirling's method used to find fixed points in Banach spaces assuming the Hölder continuity condition on the first Fréchet derivative of nonlinear operators. This condition generalizes the Lipschtiz continuity condition used earlier for the convergence. Also, the Hölder continuity condition holds on some problems, where the Lipschiz continuity condition fails. The R-order of convergence and a priori error bounds are also derived. On comparison with Newton's method, larger domains of existence and uniqueness of fixed points are obtained. An integral equation of Hammerstein type of second kind is solved to show the efficiency of our convergence analysis.