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Proceedings of the American Mathematical Society

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An asymptotic bound for the iterates of certain real functions near a contractive fixed point

Author: Lawrence J. Wallen
Journal: Proc. Amer. Math. Soc. 109 (1990), 395-398
MSC: Primary 26A18
MathSciNet review: 1010806
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Abstract: If $ x = 0$ is a contractive fixed point for the function $ F$ , then under certain conditions, the iterates $ {F_k}(a)$ are asymptotically equal to the numbers $ {\xi _k}$ defined by $ k = \int_{{\xi _k}}^a {\frac{{du}}{{u - F(u)}}} $. Using somewhat different hypotheses, we give a more precise bound on $ {F_k}(a)/{\xi _k}$ .

References [Enhancements On Off] (What's this?)

  • [1] N. G. de Bruijn, Asymptotic methods in analysis, Wiley, New York, 1961.
  • [2] Vladimir Drobot and Lawrence J. Wallen, Asymptotic iteration, Math. Mag. 64 (1991), no. 3, 176–180. MR 1110745, 10.2307/2691299
  • [3] A. M. Ostrowski, Solution of equations in Euclidean and Banach spaces, Academic Press [A Subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Third edition of Solution of equations and systems of equations; Pure and Applied Mathematics, Vol. 9. MR 0359306

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Article copyright: © Copyright 1990 American Mathematical Society