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

• Autorzy: Boris A. Godunov , Petr P. Zabreĭko
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 39B12: Iteration theory, iterative and composite equations
• 65J15: Equations with nonlinear operators (do not use )

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1995
• Tom: 116
• Numer: 3
• Strony: 225-238

Daty

wydano

1995

otrzymano

1994-10-13

poprawiono

1995-05-09

Twórcy

autor

Boris A. Godunov

• Brest Engineering and Construction Institute, Moskovskaya, 267, 224017 Brest, Belorussia

autor

Petr P. Zabreĭko

• Faculty of Mechanics and Mathematics, Belorussian State University, PR. F. Skoriny, 4, 220050 Minsk, Belorussia

Bibliografia

• [1] J. Daneš, On the local spectral radius, Časopis Pěst. Mat. 112 (1987), 177-187.
• [2] N. Dunford and J. Schwartz, Linear Operators I, Interscience Publ., Leyden, 1963.
• [3] F. R. Gantmacher, The Theory of Matrices, Chelsea, New York, 1959.
• [4] B. A. Godunov, The behavior of successive approximations for nonlinear operators, Dokl. Akad. Nauk Ukrain. SSR 4 (1971), 294-297 (in Russian).
• [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] P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964.
• [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] L. V. Kantorovich and G. P. Akilov, Functional Analysis, Nauka, Moscow, 1977 (in Russian); English transl.: Pergamon Press, Oxford, 1982.
• [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] 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] 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).