Convergence of a third order method for fixed points in Banach spaces

Abstract

In this paper, the semilocal convergence of a third order Stirling-like method used to find fixed points of nonlinear operator equations in Banach spaces is established under the assumption that the first Fréchet derivative of the involved operator satisfies ω-continuity condition. It turns out that this convergence condition is weaker than the Lipschitz and the Hölder continuity conditions on first Fréchet derivative of the involved operator. The importance of our work lies in the fact that numerical examples can be given to show that our approach is successful even in cases where Lipschitz and Hölder continuity conditions on first Fréchet derivative fail. It also avoids the evaluation of second order Fréchet derivative which is difficult to compute at times. A priori error bounds along with the domains of existence and uniqueness of a fixed point are derived. The R-order of the method is shown to be equal to (2p + 1) for p ∈ (0,1]. Finally, two numerical examples involving nonlinear integral equations are worked out to show the efficacy of our approach.