Convergence of a Stirling-like method for fixed points in Banach spaces

Abstract

We have very few literature on fixed point iterative methods for solving nonlinear equations. We consider the Stirling method given by Rall (1969). Based on Stirling method, in this paper, we propose a third order Stirling-like method for finding fixed point for nonlinear equations in Banach spaces. We study the local and semilocal convergence of this method for finding the fixed points of nonlinear equations in Banach spaces. The convergence is established under the assumption that the first order Fréchet derivative satisfies the Lipschitz continuity condition. The existence and uniqueness theorem that establishes the convergence balls of these methods is obtained. We consider the numerical examples for local and semilocal convergence case and calculate the existence and uniqueness region of convergence balls even if we fail to apply the results in Rall (1975), Argyros (1995) and Parhi and Gupta (2010)[11,12] due to the F is not contraction on Ω .