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Journal of Mathematical Sciences

, Volume 233, Issue 6, pp 853–874 | Cite as

On Stability of Perturbed Semigroups in Partially Ordered Banach Spaces

  • M. I. Kamenskii
  • I. M. Gudoshnikov
Article
  • 6 Downloads

Abstract

We prove necessary and sufficient stability conditions for perturbed semigroups of linear operators in Banach spaces with cones and consider examples using these conditions. In particular, we consider an example where the boundary-value problem is perturbed by a linear operator with a delayed independent variable and establish stability conditions for such a perturbed semigroup.

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References

  1. 1.
    R. Cooke, Infinite Matrices and Sequence Spaces, McMillan and Co. Ltd., London (1950).zbMATHGoogle Scholar
  2. 2.
    Yu. L. Daletskij and S. G. Krejn, Stability of Solutions of Differential Equations in Banach Space [in Russian], Nauka, Moscow (1970).zbMATHGoogle Scholar
  3. 3.
    K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York (2000).zbMATHGoogle Scholar
  4. 4.
    E. Hille and R. Phillips, Functional Analysis and Semigroups, Am. Math. Soc., Providence (1957).zbMATHGoogle Scholar
  5. 5.
    L. V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1977).zbMATHGoogle Scholar
  6. 6.
    T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin–Heidelberg–New York (1966).CrossRefzbMATHGoogle Scholar
  7. 7.
    A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis [in Russian], Fizmatlit, Moscow (2004).Google Scholar
  8. 8.
    M. A. Krasnosel’skij, Positive Solutions of Operator Equations [in Russian], Gos. Izdat. Fiz.-Mat. Lit., Moscow (1962).Google Scholar
  9. 9.
    M. A. Krasnosel’skij, P. P. Zabreyko, E. I. Pustyl’nik, and P. E. Sobolevskij, Integral Operators is Spaces of Summable Functions [in Russian], Nauka, Moscow (1966).Google Scholar
  10. 10.
    S. G. Krejn, Linear Equations in Banach Space [in Russian], Nauka, Moscow (1971).zbMATHGoogle Scholar
  11. 11.
    M. G. Kreyn and M. A. Rutman, “Linear operators preserving an invariant cone in a Banach space,” Usp. Mat. Nauk, 3, No. 1(23), 3–95 (1948).Google Scholar
  12. 12.
    M. A. Naymark, Linear Differential Operators [in Russian], Nauka, Moscow (1969).Google Scholar
  13. 13.
    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York (1983).CrossRefzbMATHGoogle Scholar
  14. 14.
    P. N. Shivakumar and K. C. Sivakumar, “A review of infinite matrices and their applications,” Linear Algebra Appl., 430, 976–998 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    P. N. Shivakumar, J. J. Williams, and N. Rudraiah, “Eigenvalues for infinite matrices,” Linear Algebra Appl., 96, 35–63 (1987).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Voronezh State UniversityVoronezhRussia

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