Semilocal convergence of a continuation method under ω-differentiability condition. (2016)

• Record Type: Journal Article
• Authors: Prashanth, M.; Gupta, D.K.; Motsa, S.S.
• Is Part Of: International journal of computing science and mathematics. Volume 7:Number 5(2016)
• Page Start: 395
• Page End: 409
• Publication Date: 2016
• Subjects: Lipschitz condition -- H&amp -- #246 -- lder conditions -- continuation method -- Fr&amp -- #233 -- chet derivative -- semilocal convergence -- differentiability -- nonlinear operator equations -- Banach spaces; Mathematics -- Periodicals; Computer science -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.285
• Languages: English
• ISSNs: 1752-5055

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 Fréchet 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 ω(0) ≥ 0. This condition generalises the Lipschitz and the Hölder continuity conditions on the first Fréchet derivative used for this purpose. Example can be given to show that the ω-continuity condition works even when the Lipschitz and the Hölder continuity conditions on the first Fréchet derivative fail. This also avoids the computation of second Fréchet 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.