Article: Semilocal convergence of Stirling's method for fixed points in Banach spaces Journal: International Journal of Mathematics in Operational Research (IJMOR) 2016 Vol.9 No.2 pp.243 - 257 Abstract: 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. Inderscience Publishers - linking academia, business and industry through research

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

Authors: D.K. Gupta; S.K. Parhi; Sukhjit Singh

Addresses: Department of Mathematics, Indian Institute of Technology, Kharagpur 721 302, West Bengal, India ' Department of Mathematics, International Institute of Information Technology, Bhubaneswar 751003, Odisha, India ' Department of Mathematics, Indian Institute of Technology, Kharagpur 721 302, West Bengal, India

Abstract: 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.

Keywords: nonlinear operator equations; Stirling's method; omega-condition; Fréchet derivative; semi-local convergence; fixed points; Banach spaces.

DOI: 10.1504/IJMOR.2016.078003

International Journal of Mathematics in Operational Research, 2016 Vol.9 No.2, pp.243 - 257

Received: 22 Oct 2014
Accepted: 15 Nov 2014
Published online: 31 Jul 2016 *

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

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