Abstract
Given a nonlinear mapping \(G:D\subseteq \mathbb{R}^n\rightarrow \mathbb{R}^n\) differentiable at a fixed point \(x^\ast\), the Ostrowski theorem offers the sharp sufficient condition \[\rho(G′(x^\ast))<1\] for \(x^\ast\) to be an attraction point, where \(\rho\) denotes the spectral radius. However, no estimate for the size of an attraction ball is known.
We show in this note that such an estimate may be readily obtained in terms of \(\|G^\prime(x^\ast)\|<1\) (with \(\|\cdot \|\) an arbitrary given norm) and of the Hölder (in particular Lipschitz) continuity constant of \(G’\).
An elementary example shows that this estimate may be sharp. Our assumptions do not necessarily require \(G\) to be of contractive type on the whole estimated ball.
Authors
Emil Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)
Keywords
Cite this paper as:
E. Cătinaş, Estimating the radius of an attraction ball, Appl. Math. Lett., 22 (2009) no. 5, pp. 712-714.
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0893-9659
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Google Scholar citations
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