Semilocal convergence of a continuation method under ω-differentiability condition

Abstract

The aim of this paper is to study the semilocal convergence of a continuation method combining the Chebyshev's method and the convex acceleration of Newton's method for solving nonlinear operator equations in Banach spaces. This is carried out by deriving a family of recurrence relations based on two parameters under the assumption that the first Frechet derivative satisfies the ω-continuity condition 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 ω0 ≥ 0. This condition generalises the Lipschitz and the Holder continuity conditions on the first Frechet derivative used for this purpose. Example can be given to show that the ω-continuity condition works even when the Lipschitz and the Holder continuity conditions on the first Frechet derivative fail. This also avoids the computation of second Frechet derivative which is either difficult to compute or unbounded at times. An existence and uniqueness theorem is established along with a priori error bounds. Two numerical examples are worked out to demonstrate the efficacy of our approach.