On the correspondence between periodic solutions of differential and dynamic equations on periodic time scales
-
Viktoriia Tsan
,
Oleksandr Stanzhytskyi
and
Olha Martynyuk
Abstract
This paper studies the relationship between the existence of periodic solutions of systems of dynamic equations on time scales and their corresponding systems of differential equations. We have established that, for a sufficiently small graininess function, if a dynamic equation on a time scale has an asymptotically stable periodic solution, then the corresponding differential equation will also have a periodic solution. A converse result has also been obtained, where the existence of a periodic solution of a differential equation implies the existence of a corresponding solution on time scales, provided that the graininess function is sufficiently small.
Funding source: National Research Foundation of Ukraine
Award Identifier / Grant number: F81/41743
Funding source: Government of Ukraine
Award Identifier / Grant number: 210BF38-01
Funding statement: The work of Oleksandr Stanzhytskyi was partially supported by the National Research Foundation of Ukraine No. F81/41743 and by the Ukrainian Government Scientific Research Grant No. 210BF38-01.
References
[1] E. Akin, M. Bohner, L. Erbe and A. Peterson, Existence of bounded solutions for second order dynamic equations, J. Difference Equ. Appl. 8 (2002), 389–401. 10.1080/1026190290017414Search in Google Scholar
[2] M. Bohner, K. Kenzhebaev, O. Lavrova and O. Stanzhytskyi, Pontryagin’s maximum principle for dynamic systems on time scales, J. Difference Equ. Appl. 23 (2017), no. 7, 1161–1189. 10.1080/10236198.2017.1284829Search in Google Scholar
[3] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser, Boston, 2001. 10.1007/978-1-4612-0201-1Search in Google Scholar
[4] M. Bohner and A. Peterson, Advances in Dynamical Equations on Time Scales, Birkhäuser, Boston, 2003. 10.1007/978-0-8176-8230-9Search in Google Scholar
[5] M. Bohner, O. M. Stanzhytskyi and A. O. Bratochkina, Stochastic dynamic equations on general time scales, Electron. J. Differential Equations 2013 (2013), Paper No. 57. Search in Google Scholar
[6] L. Bourdin, O. Stanzhytskyi and E. Trélat, Addendum to Pontryagin’s maximum principle for dynamic systems on time scales, J. Difference Equ. Appl. 23 (2017), no. 10, 1760–1763. 10.1080/10236198.2017.1363194Search in Google Scholar
[7] L. Bourdin and E. Trélat, General Cauchy–Lipschitz theory for Δ-Cauchy problems with Carathéodory dynamics on time scales, J. Difference Equ. Appl. 20 (2014), no. 4, 526–547. 10.1080/10236198.2013.862358Search in Google Scholar
[8] S. G. Georgiev, S. Doğru Akgöl and M. Eymen Kuş, Existence of solutions for odd-order multi-point impulsive boundary value problems on time scales, Georgian Math. J. 29 (2022), no. 4, 505–513. 10.1515/gmj-2022-2153Search in Google Scholar
[9] S. Hilger, Ein Maßkettenkalkül mit Anwendungen auf Zentrumsmannigfaltigkeiten, Ph.D. Thesis, Universität Würzburg, Würzburg, Germany, 1988. Search in Google Scholar
[10] Y. Hino, Stability properties for functional-differential equations with infinite delay, Tohoku Math. J. (2) 35 (1983), no. 4, 597–605. 10.2748/tmj/1178228954Search in Google Scholar
[11] O. Karpenko, O. Stanzhytskyi and T. Dobrodzii, The relation between the existence of bounded global solutions of the differential equations and equations on time scales, Turkish J. Math. 44 (2020), no. 6, 2099–2112. 10.3906/mat-2006-79Search in Google Scholar
[12] E. R. Kaufmann and Y. N. Raffoul, Periodic solutions for a neutral nonlinear dynamical equation on a time scale, J. Math. Anal. Appl. 319 (2006), no. 1, 315–325. 10.1016/j.jmaa.2006.01.063Search in Google Scholar
[13]
M. V. Pratsiovytyi, Y. V. Goncharenko, I. M. Lysenko and S. P. Ratushniak,
Continued
[14]
M. V. Pratsiovytyi and S. P. Ratushnyak,
Properties and distributions of values of fractal functions related to
[15] A. M. Samoĭlenko, N. N. Bogolyubov and nonlinear mechanics (in Russian), Uspekhi Mat. Nauk 49 (1994), no. 5(299), 103–146; translation in Russian Math. Surveys 49 (1994), no. 5, 109–154. Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Numerical approaches for solution of hyperbolic difference equations on circle
- A note on b-generalized (α,β)-derivations in prime rings
- Analytic solution to functional differential equations via Bell’s polynomials
- Floquet theory and stability for a class of first order differential equations with delays
- Representations of a number in an arbitrary base with unbounded digits
- V_a -deformed free convolution and variance function
- Generalized essential spectra involving the class of g-g-Riesz operators
- On perturbation of continuous frames in Hilbert C *-modules
- Almost measurable functions on probability spaces
- On φ-u-S-flat modules and nonnil-u-S-injective modules
- Busemann--Petty-type problem for μ-intersection bodies
- On the representation of solution for the perturbed quasi-linear controlled neutral functional-differential equation with the discontinuous initial condition
- A generalization of Hardy’s inequality to infinite tensors
- A note on maximal estimate for an oscillatory operator
- Some summation theorems and transformations for hypergeometric functions of Kampé de Fériet and Srivastava
- On the correspondence between periodic solutions of differential and dynamic equations on periodic time scales
