Advertisement

Nonlinear Oscillations

, Volume 9, Issue 1, pp 1–12 | Cite as

Bounded solutions of linear differential equations in a banach space

  • A. A. Boichuk
  • A. A. Pokutnii
Article

Abstract

For a linear inhomogeneous differential equation in a Banach space, we find a criterion for the existence of solutions that are bounded on the entire real axis under the assumption that the homogeneous equation admits an exponential dichotomy on the semiaxes. This result is a generalization of the Palmer lemma to the case of infinite-dimensional spaces. We consider examples of countable systems of ordinary differential equations that have bounded solutions.

Keywords

Banach Space Linear Differential Equation Singular Integral Operator Solvability Condition 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.
    Yu. M. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in a Banach Space [in Russian], Nauka, Moscow (1970).Google Scholar
  2. 2.
    K. J. Palmer, “Exponential dichotomies and transversal homoclinic points,” J. Different. Equat., 55, 225–256 (1984).MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    I. Ts. Gokhberg and N. Ya. Krupnik, Introduction to the Theory of One-Dimensional Singular Integral Operators [in Russian], Shtiintsa, Kishinev (1973).Google Scholar
  4. 4.
    A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, VSP, Utrecht (2004).MATHGoogle Scholar
  5. 5.
    V. S. Korolyuk and A. F. Turbin, Mathematical Foundations of Phase Lumping of Complex Systems [in Russian], Naukova Dumka, Kiev (1978).Google Scholar
  6. 6.
    V. A. Trenogin, Functional Analysis [in Russian], Nauka, Moscow (1980).Google Scholar
  7. 7.
    A. A. Boichuk, “Solutions of weakly nonlinear differential equations bounded on the whole line,” Nonlin. Oscillations, 2, No. 1, 3–10 (1999).MathSciNetGoogle Scholar
  8. 8.
    A. M. Samoilenko and Yu. V. Teplinskii, Countable Systems of Differential Equations [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1993).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • A. A. Boichuk
    • 1
  • A. A. Pokutnii
    • 2
  1. 1.Institute of MathematicsUkrainian Academy of SciencesKiev
  2. 2.Shevchenko Kiev National UniversityKiev

Personalised recommendations