Skip to main content
Log in

Exponentially Dichotomous Difference Equations with Piecewise Constant Operator Coefficients

  • Published:
Ukrainian Mathematical Journal Aims and scope

We establish necessary and sufficient conditions for the exponential dichotomy of the solutions of linear difference equations with piecewise constant operator coefficients.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. V. E. Slyusarchuk, “Exponential dichotomy for solutions of discrete systems,” Ukr. Mat. Zh., 35, No. 1, 109–115 (1983); English translation: Ukr. Math. J., 35, No. 1, 98–103 (1983).

  2. A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1968).

    Google Scholar 

  3. V. E. Slyusarchuk, “Difference equations in function spaces,” Addition II to the Monograph by D. I. Martynyuk, Lectures on the Qualitative Theory of Difference Equations [in Russian], Naukova Dumka, Kiev (1972), pp. 197–224.

    Google Scholar 

  4. V. E. Slyusarchuk, “Bounded and almost periodic solutions of difference equations in a Banach space,” in: Analytic Methods for the Investigation of Solutions of Nonlinear Differential Equations [in Russian], Institute of Mathematics, Academy of Sciences of Ukr. SSR, Kiev (1975), pp. 147–156.

    Google Scholar 

  5. V. E. Slyusarchuk, “Bounded and almost periodic solutions of implicit difference equations in Banach spaces,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 6, 503–509 (1975).

    Google Scholar 

  6. I. M. Gelfand, D. A. Raikov, and G. E. Shilov, Commutative Normed Rings [in Russian], Fizmatgiz, Moscow (1960).

    Google Scholar 

  7. N. Dunford and J. T. Schwartz, Linear Operators. Part 1: General Theory, Interscience, New York (1958).

    MATH  Google Scholar 

  8. S. V. Coffman and J. J. Schaffer, “Dichotomies for linear difference equations,” Math. Ann., 172, 139–166 (1967).

    Article  MathSciNet  Google Scholar 

  9. A. Halanay and D. Wexler, Teoria Calitativ˘a a Sistemelor cu Impulsuri, Editura Academiei Republicii Socialiste România, Bucure¸sti (1968).

    Google Scholar 

  10. A. N. Sharkovskii, Yu. L. Maistrenko, and E. Yu. Romanenko, Difference Equations and Their Applications [in Russian], Naukova Dumka, Kiev (1986).

    MATH  Google Scholar 

  11. A. Ya. Dorogovtsev, Periodic and Stationary Modes of Infinite-Dimensional Deterministic and Stochastic Dynamical Systems [in Russian], Vyshcha Shkola, Kiev (1992).

    Google Scholar 

  12. V. Yu. Slyusarchuk, Invertibility of Nonlinear Difference Operators [in Ukrainian], National University of Water Management and Utilization of Natural Resources, Rivne (2006).

    Google Scholar 

  13. V. Yu. Slyusarchuk, Implicit Nondifferentiable Functions in the Theory of Operators [in Ukrainian], National University of Water Management and Utilization of Natural Resources, Rivne (2008).

    Google Scholar 

  14. V. E. Slyusarchuk, “Representation of the bounded solutions of discrete linear systems,” Ukr. Mat. Zh., 39, No. 2, 210–215 (1987); English translation: Ukr. Math. J., 39, No. 2, 176–180 (1987).

  15. V. E. Slyusarchuk, “Representations of bounded solutions of linear discrete equations,” Nelin. Kolyv., 22, No. 2, 262–279 (2019).

    Google Scholar 

  16. M. F. Gorodnii, “Bounded and periodic solutions of a difference equation and its stochastic analog in Banach space,” Ukr. Mat. Zh., 43, No. 1, 42–46 (1991); English translation: Ukr. Math. J., 43, No. 1, 32–37 (1991).

  17. A. G. Baskakov, “On the invertibility and Fredholm property of difference operators,” Mat. Zametki, 67, Issue 6, 816–827 (2000).

    Article  MathSciNet  Google Scholar 

  18. V. Yu. Slyusarchuk, “Method of locally linear approximation of nonlinear difference operators by weakly regular operators,” Nelin. Kolyv., 15, No. 1, 112–126 (2012); English translation: J. Math. Sci., 187, No. 4, 494–510 (2012).

  19. V. Yu. Slyusarchuk, “Periodic and almost periodic solutions of difference equations in metric spaces,” Nelin. Kolyv., 18, No. 1, 112–119 (2015); English translation: J. Math. Sci., 215, No. 3, 387–394 (2016).

  20. I. V. Honchar, Bounded and Summable Solutions of Difference Equations with Jumps of the Operator Coefficient [in Ukrainian], Candidate-Degree Thesis (Physics and Mathematics), Kyiv (2018).

    Google Scholar 

  21. M. F. Horodnii and I. V. Honchar, “On bounded solutions of a difference equation with jumps of the operator coefficient,” Nelin. Kolyv., 20, No. 1, 66–73 (2017); English translation: J. Math. Sci., 229, No. 4, 403–411 (2018).

  22. V. Yu. Slyusarchuk, “Exponentially dichotomous difference equations with non-Lipschitz perturbations,” Nelin. Kolyv, 14, No. 4, 536–555 (2011); English translation: Nonlin. Oscillat., 14, No. 4, 568–588 (2012).

  23. Yu. L. Daletskii and M. G. Krein, Stability of Solutions of Linear Differential Equations in Banach Spaces [in Russian], Nauka, Moscow (1970).

    Google Scholar 

  24. M. A. Krasnosel’skii, V. Sh. Burd, and Yu. S. Kolesov Nonlinear Almost Periodic Oscillations [in Russian], Nauka, Moscow (1970).

    Google Scholar 

  25. J. L. Massera and J. J. Schaffer, Linear Differential Equations and Function Spaces, Academic Press, New York (1966).

    MATH  Google Scholar 

  26. P. Hartman, Ordinary Differential Equations, Wiley, New York (1964).

    MATH  Google Scholar 

  27. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin (1981).

    Book  Google Scholar 

  28. Yu. A. Mitropol’skii, A. M. Samoilenko, and V. L. Kulik, Investigations of the Dichotomy of Linear Systems of Differential Equations with the Use of the Lyapunov Function [in Russian], Naukova Dumka, Kiev (1990).

    Google Scholar 

  29. Yu. S. Kolesov, “Necessary and sufficient conditions for the exponential dichotomy of solutions of linear almost periodic equations with aftereffect,” Vestn. Yaroslav. Univ., Issue 5, 28–62 (1973).

    Google Scholar 

  30. V. G. Kurbatov, “On the dichotomy of solutions of equations of neutral type,” in: Investigations of Stability and the Theory of Oscillations [in Russian], Yaroslavl University (1977), pp. 156–166.

  31. A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, Walter de Gruyter, Berlin (2016).

    Book  Google Scholar 

  32. O. O. Pokutnyi, “Solutions of a linear difference equation in a Banach space bounded on the entire real axis,” Visn. Kyiv. Nats. Univ., No. 1, 182–188 (2006).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to V. Yu. Slyusarchuk.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 6, pp. 822–841, June, 2020. Ukrainian DOI: 10.37863/umzh.v72i6.1052.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Slyusarchuk, V.Y. Exponentially Dichotomous Difference Equations with Piecewise Constant Operator Coefficients. Ukr Math J 72, 953–977 (2020). https://doi.org/10.1007/s11253-020-01835-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-020-01835-5

Navigation