Semilocal convergence of Stirling's method for fixed points in Banach spaces

The aim of this paper is to discuss the semilocal convergence of Stirling's method used to find fixed points of nonlinear operator equations in Banach spaces. This convergence is achieved using recurrence relations under the assumption that the first Fréchet derivative of the involved operator satisfies the ω-continuity condition. The ω-continuity condition relaxes the Lipschitz/Hölder continuity condition on the first Frèchet derivative and is given by ||F′ (x) - F′ (y)|| ≤ ω || x - y|| ∀x, y ∈ Ω; where ω R+ → R+ is a continuous and non-decreasing function such that ω(0)≥ 0. This work has importance as it does not require the evaluation of second order Fréchet derivative and successfully works for examples for which Lipschitz/Hölder continuity condition on the first Fréchet derivative may fail. A priori error bounds along with the domains of existence and uniqueness for a fixed point are derived. Finally, two numerical examples involving integral equations of Fredholm types are worked out and the results obtained are compared with those obtained by Newton's method. It is found that our approach gives better existence and uniqueness domains for both the examples considered. However, a priori error bounds for our approach are better in one example but not so good in another.