- PII
- 10.31857/S0374064123040064-1
- DOI
- 10.31857/S0374064123040064
- Publication type
- Article
- Status
- Published
- Authors
- Volume/ Edition
- Volume 59 / Issue number 4
- Pages
- 494-500
- Abstract
- We consider homogeneous and inhomogeneous systems of linear difference equations with coefficients that are -periodic functions of discrete time. For homogeneous systems, sufficient conditions for the existence of periodic and almost periodic solutions are obtained. For inhomogeneous systems, it is shown that a necessary and sufficient condition for the existence of an N-periodic solution is the existence of a bounded solution. Necessary and sufficient conditions for theN orthogonality of the fundamental matrix of the homogeneous system are established. Illustrative examples are given.
- Keywords
- Date of publication
- 19.09.2025
- Year of publication
- 2025
- Number of purchasers
- 0
- Views
- 11
References
- 1. Chen S., Liu X. Stability analysis of discrete-time coupled systems with delays // J. of the Franklin Institute. 2020. № 357. P. 9942-9959.
- 2. Игнатьев А.О. Метод функций Ляпунова в системах разностных уравнений: устойчивость относительно части переменных // Дифференц. уравнения. 2022. Т. 58. № 3. С. 407-415.
- 3. Elaydi S. An Introduction to Difference Equations. New York, 2005.
- 4. Lakshmikantham V., Trigiante D. Theory of Difference Equations: Numerical Methods and Applications. New York, 2002.
- 5. Agarwal R., Popenda J. Periodic solutions of first order linear difference equations // Math. Comput. Modelling. 1995. V. 22. № 1. P. 11-19.
- 6. Савченко А.Я., Игнатьев А.О. Некоторые задачи устойчивости неавтономных динамических систем. Киев, 1989.
- 7. Giang D.V. Linear difference equations and periodic sequences over finite fields // Acta Math. Vietnam. 2016. V. 41. № 1. P. 171-181.
- 8. Hasil P., Vesely M. Limit periodic homogeneous linear difference systems // Appl. Math. Comput. 2015. V. 265. P. 958-972.
- 9. Janglajew K., Schmeidel E. Periodicity of solutions of nonhomogeneous linear difference equations // Adv. Difference Equat. 2012. V. 195.
- 10. Agarwal R. Difference Equations and Inequalities. Theory, Methods, and Applications. V. 228. New York, 2000.
- 11. Agarwal R., Wong P. Advanced Topics in Difference Equations. Dordrecht, 1997.
- 12. Gasull A. Difference equations everywhere: some motivating examples // Difference Equations, Discrete Dynamical Systems and Applications / Eds. S. Elaydi et al. 2019. V. 287. P. 129-167.
- 13. Elaydi S., Sacker R. Periodic difference equations, population biology and the Cushing-Henson conjectures // Math. Biosci. 2006. V. 201. № 1-2. P. 195-207.
- 14. Elaydi S., Sacker R. Global stability of periodic orbits of non-autonomous difference equations and population biology // J. Differ. Equat. 2005. V. 208. № 1. P. 258-273.
- 15. Ignatyev A.O., Ignatyev O.A. On the stability of discrete systems // Integral Methods in Science and Engineering. Boston, 2006. P. 105-116.
- 16. Ignatyev A.O., Ignatyev O.A. On the stability in periodic and almost periodic difference systems // J. Math. Anal. Appl. 2006. V. 313. № 2. P. 678-688.
- 17. Zhang S., Liu P., Gopalsamy K. Almost periodic solutions of nonautonomous linear difference equations // Appl. Analysis: an Int. J. 2002. V. 81. № 2. P. 281-301.
- 18. Деменчук А.К. О сильно нерегулярных периодических решениях линейного дискретного уравнения первого порядка // Весцi НАН Беларусi. Сер. фiз.-мат. навук. 2020. Т. 56. № 1. С. 30-35.
- 19. Popenda J., Schmeidel E. Asymptotically periodic solution of some linear difference equations // Facta Univ. Ser. Math. Inform. 1999. V. 14. P. 31-40.
- 20. Clark M.E., Gross L.J. Periodic solutions to nonautonomous difference equations // Math. Biosci. 1990. V. 102. № 1. P. 105-119.
- 21. Vesely M. Construction of almost periodic sequences with given properties // Electron. J. Differ. Equat. 2008. V. 126.
- 22. Vesely M. Almost periodic homogeneous linear difference systems without almost periodic solutions // J. Difference Equat. Appl. 2012. V. 18. № 10. P. 1623-1647.
- 23. Massera J.L. The existence of periodic solutions of systems of differential equations // Duke Math. J. 1950. V. 17. № 4. P. 457-475.
- 24. Makay G. On some possible extensions of Massera's theorem // Electronic J. of Qualit. Theory of Differ. Equat. 2000. V. 16. P. 1-8.
- 25. Zubelevich O. A note on theorem of Massera // Regul. Chaotic Dyn. 2006. V. 11. № 4. P. 475-481.
- 26. Li Y., Lin Z., Li Z. A Massera type criterion for linear functional differential equations with advanced and delay // J. Math. Anal. Appl. 1996. V. 200. P. 717-725.
- 27. Corduneanu C. Almost Periodic Functions. New York, 1989.
- 28. Левитан Б.М. Почти периодические функции. М., 1953.
- 29. Мишина А.П., Проскуряков И.В. Справочная математическая библиотека. Высшая алгебра. М., 1965.
- 30. Vleck F.S.V. A note on the relation between periodic and orthogonal fundamental solutions of linear systems // Amer. Math. Monthly. V. 71. № 4. P. 406-408.
