Uniform boundedness of solutions to linear difference equations with periodic forcing functions

Rinko Miyazaki; Dohan Kim; Jong Son Shin; Rinko Miyazaki; Dohan Kim; Jong Son Shin

doi:10.3934/math.20231229

AIMS Mathematics

2023, Volume 8, Issue 10: 24116-24131. doi: 10.3934/math.20231229

Previous Article Next Article

Research article Special Issues

Uniform boundedness of solutions to linear difference equations with periodic forcing functions

1.
Graduate School of Engineering, Shizuoka University, Hamamatsu, Shizuoka 432-8561, Japan
2.
Graduate School of Engineering Sciences, Osaka University, Toyonaka, Osaka 560-8531, Japan
3.
Department of Mathematics, Seoul National University, Gwanak-gu, Seoul 08826, Korea

Received: 14 May 2023 Revised: 17 July 2023 Accepted: 20 July 2023 Published: 09 August 2023
MSC : Primary: 39A05, 39A06; Secondary: 15A18

In this paper we give criteria on the uniform boundedness of the solutions to linear difference equations (LEs) with periodic forcing functions. First, we give a necessary and sufficient condition that the sequence $ \{L^n\} $ of a square matrix $ L $ is bounded, from which a criterion on the uniform boundedness of the solutions to LEs is obtained. Second, a criterion on the uniform boundedness of the solutions for LEs with periodic forcing functions is given by applying a certain representation of solutions. In connection with LEs with delay, we give the characteristic equation of a matrix under the commuting condition.
- uniform boundedness,
- spectral decomposition,
- periodic forcing functions,
- linear difference equation,
- characteristic equation
Citation: Rinko Miyazaki, Dohan Kim, Jong Son Shin. Uniform boundedness of solutions to linear difference equations with periodic forcing functions[J]. AIMS Mathematics, 2023, 8(10): 24116-24131. doi: 10.3934/math.20231229

Related Papers:

Abstract

In this paper we give criteria on the uniform boundedness of the solutions to linear difference equations (LEs) with periodic forcing functions. First, we give a necessary and sufficient condition that the sequence $ \{L^n\} $ of a square matrix $ L $ is bounded, from which a criterion on the uniform boundedness of the solutions to LEs is obtained. Second, a criterion on the uniform boundedness of the solutions for LEs with periodic forcing functions is given by applying a certain representation of solutions. In connection with LEs with delay, we give the characteristic equation of a matrix under the commuting condition.

References

[1]	J. Kato, T. Naito, J. S. Shin, A characterization of solutions in linear differential equations with periodic forcing functions, J. Differ. Equ. Appl., 11 (2005), 1–19. http://dx.doi.org/10.1080/10236190412331317120 doi: 10.1080/10236190412331317120
[2]	T. Naito, J. S. Shin, On periodicizing functions, Bull. Korean Math. Soc., 43 (2006), 253–263. http://dx.doi.org/10.4134/BKMS.2006.43.2.253 doi: 10.4134/BKMS.2006.43.2.253
[3]	T. Naito, P. H. T. Ngoc, J. S. Shin, Representations and asymptotic behavior to periodic linear difference equations, Funkc. Ekvacioj, 51 (2008), 55–80. http://dx.doi.org/10.1619/fesi.51.55 doi: 10.1619/fesi.51.55
[4]	T. Naito, P. H. T. Ngoc, J. S. Shin, Floquet representations and asymptotic behavior of solutions to periodic linear difference equations, Hiroshima Math. J., 38 (2008), 135–154.
[5]	J. S. Shin, T. Naito, Representations of solutions, translation formulae and asymptotic behaviors in discrete and periodic continuous linear systems, Hiroshima Math. J., 44 (2014), 75–126. http://dx.doi.org/10.32917/hmj/1395061558 doi: 10.32917/hmj/1395061558
[6]	S. Stevic, Bounded and periodic solutions to the linear first-order difference equation on the integer domain, Adv. Differ. Equ., 2017 (2017), 283. http://dx.doi.org/10.1186/s13662-017-1350-8 doi: 10.1186/s13662-017-1350-8
[7]	E. Kreyszig, Introductory functional analysis with applications, New York: John Wiley & Sons, 1991.
[8]	T. Yoshizawa, Stability theory by Liapunov's second method, Tokyo, 1966.
[9]	S. Elaydi, An introduction to difference equations, 3 Eds., New York: Springer-Verlag, 2005.
[10]	F. Faá di Bruno, Note sur une nouvelle formule de calcul differentiel, Quart. J. Pure Appl. Math., 1 (1857), 359–360.

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)