We establish necessary and sufficient conditions for the exponential dichotomy of the solutions of linear difference equations with piecewise constant operator coefficients.
Similar content being viewed by others
References
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).
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1968).
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.
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.
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).
I. M. Gelfand, D. A. Raikov, and G. E. Shilov, Commutative Normed Rings [in Russian], Fizmatgiz, Moscow (1960).
N. Dunford and J. T. Schwartz, Linear Operators. Part 1: General Theory, Interscience, New York (1958).
S. V. Coffman and J. J. Schaffer, “Dichotomies for linear difference equations,” Math. Ann., 172, 139–166 (1967).
A. Halanay and D. Wexler, Teoria Calitativ˘a a Sistemelor cu Impulsuri, Editura Academiei Republicii Socialiste România, Bucure¸sti (1968).
A. N. Sharkovskii, Yu. L. Maistrenko, and E. Yu. Romanenko, Difference Equations and Their Applications [in Russian], Naukova Dumka, Kiev (1986).
A. Ya. Dorogovtsev, Periodic and Stationary Modes of Infinite-Dimensional Deterministic and Stochastic Dynamical Systems [in Russian], Vyshcha Shkola, Kiev (1992).
V. Yu. Slyusarchuk, Invertibility of Nonlinear Difference Operators [in Ukrainian], National University of Water Management and Utilization of Natural Resources, Rivne (2006).
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).
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).
V. E. Slyusarchuk, “Representations of bounded solutions of linear discrete equations,” Nelin. Kolyv., 22, No. 2, 262–279 (2019).
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).
A. G. Baskakov, “On the invertibility and Fredholm property of difference operators,” Mat. Zametki, 67, Issue 6, 816–827 (2000).
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).
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).
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).
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).
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).
Yu. L. Daletskii and M. G. Krein, Stability of Solutions of Linear Differential Equations in Banach Spaces [in Russian], Nauka, Moscow (1970).
M. A. Krasnosel’skii, V. Sh. Burd, and Yu. S. Kolesov Nonlinear Almost Periodic Oscillations [in Russian], Nauka, Moscow (1970).
J. L. Massera and J. J. Schaffer, Linear Differential Equations and Function Spaces, Academic Press, New York (1966).
P. Hartman, Ordinary Differential Equations, Wiley, New York (1964).
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin (1981).
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).
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).
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.
A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, Walter de Gruyter, Berlin (2016).
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).
Author information
Authors and Affiliations
Corresponding author
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
About this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-020-01835-5