Advertisement

Ukrainian Mathematical Journal

, Volume 62, Issue 10, pp 1635–1648 | Cite as

Functions of shift operator and their applications to difference equations

  • A. V. Chaikovs’kyi
Article

We study the representation for functions of shift operator acting upon bounded sequences of elements of a Banach space. An estimate is obtained for the bounded solution of a linear difference equation in the Banach space. For two types of differential equations in Banach spaces, we present sufficient conditions for their bounded solutions to be limits of bounded solutions of the corresponding difference equations and establish estimates for the rate of convergence.

Keywords

Banach Space Difference Equation Unit Circle Shift Operator Bounded Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. E. Slyusarchuk, “Bounded and almost periodic solutions of difference equations in a Banach space,” in: Analytic Methods for Investigation of Solutions of Nonlinear Differential Equations [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1975), pp. 147–156.Google Scholar
  2. 2.
    A. Ya. Dorogovtsev, Periodic and Stationary Modes of Infinite-Dimensional Deterministic and Stochastic Dynamical Systems [in Russian], Vyshcha Shkola, Kiev (1992).Google Scholar
  3. 3.
    A. G. Baskakov and A. I. Pastukhov, “Spectral analysis of the operator of weighted shift with unbounded operator coefficients,” Sib. Mat. Zh., 42, No. 6, 1231–1243 (2001).MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    M. F. Gorodnii, “Bounded and periodic solutions of one difference equation and its stochastic analog in a Banach space,” Ukr. Mat. Zh., 43, No. 1, 41–46 (1991).MathSciNetCrossRefGoogle Scholar
  5. 5.
    V. V. Smagin, “Estimates in strong norms for the error of the projection-iterative method for the approximate solution of an abstract parabolic equation,” Mat. Zametki, 62, Issue 6, 898–909 (1997).MathSciNetGoogle Scholar
  6. 6.
    S. I. Piskarev, “Approximation of positive C0-semigroups of operators,” Differents. Uravn., 27, No. 7, 1245–1250 (1991).MathSciNetzbMATHGoogle Scholar
  7. 7.
    V. L. Makarov and I. P. Havrylyuk, “Exponentially convergent methods of parallel discretization for evolution equations of the first order,” Dopov. Nats. Akad. Nauk Ukr., No. 3, 24–28 (2002).Google Scholar
  8. 8.
    M. F. Gorodnii and A. V. Chaikovs’kyi, “On approximation of a bounded solution of a differential equation with unbounded operator coefficient,” Dopov. Nats. Akad. Nauk Ukr., No. 6, 10–14 (2002).Google Scholar
  9. 9.
    N. Dunford and J. T. Schwartz, Linear Operators, General Theory [Russian translation], Inostrannaya Literatura, Moscow (1962).Google Scholar
  10. 10.
    B. V. Shabat, Introduction to Complex Analysis [in Russian], Nauka, Moscow (1969).Google Scholar
  11. 11.
    A. Zygmund, Trigonometric Series, Vol. 1, Cambridge University Press, Cambridge (1959).zbMATHGoogle Scholar
  12. 12.
    A. G. Baskakov, “Abstract harmonic analysis and asymptotic estimates for elements of inverse matrices,” Mat. Zametki, 52, Issue 2, 17–26 (1992).MathSciNetGoogle Scholar
  13. 13.
    A. V. Chaikovs’kyi, “On the existence and uniqueness of bounded solutions of differential equations with shifts in a Banach space,” Dopov. Nats. Akad. Nauk Ukr., No. 8, 33–37 (2000).Google Scholar
  14. 14.
    A. V. Chaikovs’kyi, “On the approximation of bounded solutions of differential equations with shift of argument in a Banach space by solutions of difference equations,” Visn. Kyiv. Univ., Mat. Mekh., Issue 8, 125–129 (2002).Google Scholar
  15. 15.
    V. K. Dzyadyk, Introduction to the Theory of Uniform Approximation of Functions by Polynomials [in Russian], Nauka, Moscow (1977).Google Scholar
  16. 16.
    Yu. L. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in a Banach Space [in Russian], Nauka, Moscow (1970).Google Scholar
  17. 17.
    T. Kato, Perturbation Theory for Linear Operators [Russian translation], Mir, Moscow (1972).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • A. V. Chaikovs’kyi
    • 1
  1. 1.Shevchenko Kyiv National UniversityKyivUkraine

Personalised recommendations