Geometric characteristics for convergence and asymptotics of successive approximations of equations with smooth operators

Boris Godunov; Petr Zabreĭko

Studia Mathematica (1995)

  • Volume: 116, Issue: 3, page 225-238
  • ISSN: 0039-3223

Abstract

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We discuss the problem of characterizing the possible asymptotic behaviour of the iterates of a sufficiently smooth nonlinear operator acting in a Banach space in small neighbourhoods of a fixed point. It turns out that under natural conditions, for the most part of initial approximations these iterates tend to "lie down" along a finite-dimensional subspace generated by the leading (peripherical) eigensubspaces of the Fréchet derivative at the fixed point and moreover the asymptotic behaviour of "projections" of the iterates on this subspace is determined by the arithmetic properties of the leading eigenvalues.

How to cite

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Godunov, Boris, and Zabreĭko, Petr. "Geometric characteristics for convergence and asymptotics of successive approximations of equations with smooth operators." Studia Mathematica 116.3 (1995): 225-238. <http://eudml.org/doc/216230>.

@article{Godunov1995,
abstract = {We discuss the problem of characterizing the possible asymptotic behaviour of the iterates of a sufficiently smooth nonlinear operator acting in a Banach space in small neighbourhoods of a fixed point. It turns out that under natural conditions, for the most part of initial approximations these iterates tend to "lie down" along a finite-dimensional subspace generated by the leading (peripherical) eigensubspaces of the Fréchet derivative at the fixed point and moreover the asymptotic behaviour of "projections" of the iterates on this subspace is determined by the arithmetic properties of the leading eigenvalues.},
author = {Godunov, Boris, Zabreĭko, Petr},
journal = {Studia Mathematica},
keywords = {peripherical eigensubspaces; asymptotic behaviour; iterates of a sufficiently smooth nonlinear operator acting in a Banach space; fixed point; Fréchet derivative at the fixed point},
language = {eng},
number = {3},
pages = {225-238},
title = {Geometric characteristics for convergence and asymptotics of successive approximations of equations with smooth operators},
url = {http://eudml.org/doc/216230},
volume = {116},
year = {1995},
}

TY - JOUR
AU - Godunov, Boris
AU - Zabreĭko, Petr
TI - Geometric characteristics for convergence and asymptotics of successive approximations of equations with smooth operators
JO - Studia Mathematica
PY - 1995
VL - 116
IS - 3
SP - 225
EP - 238
AB - We discuss the problem of characterizing the possible asymptotic behaviour of the iterates of a sufficiently smooth nonlinear operator acting in a Banach space in small neighbourhoods of a fixed point. It turns out that under natural conditions, for the most part of initial approximations these iterates tend to "lie down" along a finite-dimensional subspace generated by the leading (peripherical) eigensubspaces of the Fréchet derivative at the fixed point and moreover the asymptotic behaviour of "projections" of the iterates on this subspace is determined by the arithmetic properties of the leading eigenvalues.
LA - eng
KW - peripherical eigensubspaces; asymptotic behaviour; iterates of a sufficiently smooth nonlinear operator acting in a Banach space; fixed point; Fréchet derivative at the fixed point
UR - http://eudml.org/doc/216230
ER -

References

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  1. [1] J. Daneš, On the local spectral radius, Časopis Pěst. Mat. 112 (1987), 177-187. Zbl0645.47002
  2. [2] N. Dunford and J. Schwartz, Linear Operators I, Interscience Publ., Leyden, 1963. Zbl0128.34803
  3. [3] F. R. Gantmacher, The Theory of Matrices, Chelsea, New York, 1959. Zbl0085.01001
  4. [4] B. A. Godunov, The behavior of successive approximations for nonlinear operators, Dokl. Akad. Nauk Ukrain. SSR 4 (1971), 294-297 (in Russian). Zbl0215.21301
  5. [5] B. A. Godunov, Convergence acceleration in the method of successive approximations, in: Operator Methods for Differential Equations, Voronezh, 1979, 18-25 (in Russian). 
  6. [6] P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964. Zbl0125.32102
  7. [7] N. M. Isakov, On the behavior of continuous operators near a fixed point in the critical case, in: Qualitative and Approximate Methods for Investigation of Operator Equations, Yaroslavl', 1978, 74-89 (in Russian). 
  8. [8] L. V. Kantorovich and G. P. Akilov, Functional Analysis, Nauka, Moscow, 1977 (in Russian); English transl.: Pergamon Press, Oxford, 1982. Zbl0127.06102
  9. [9] M. A. Krasnosel'skiĭ, G. M. Vaĭnikko, P. P. Zabreĭko, Ya. B. Rutitskiĭ and V. Ya. Stetsenko, Approximate Solution of Operator Equations, Nauka, Moscow, 1969 (in Russian); English transl.: Noordhoff, Groningen, 1972. 
  10. [10] M. A. Krasnosel'skiĭ, E. A. Lifshits and A. V. Sobolev, Positive Linear Systems, Nauka, Moscow, 1985 (in Russian); English transl.: Heldermann, Berlin, 1989. 
  11. [11] P. P. Zabreĭko and N. M. Isakov, Reduction principle in the method of successive approximations and invariant manifolds, Sibirsk. Mat. Zh. 20 (1979), 539-547 (in Russian). 

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