Massera's theorem on arbitrary discrete time domains

Martin Bohner; Jaqueline G. Mesquita; Sabrina Streipert; Martin Bohner; Jaqueline G. Mesquita; Sabrina Streipert

doi:10.3934/mbe.2025067

Mathematical Biosciences and Engineering

2025, Volume 22, Issue 7: 1861-1874. doi: 10.3934/mbe.2025067

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Research article

Massera's theorem on arbitrary discrete time domains

1.
Department of Mathematics and Statistics, Missouri S&T, 400 W. 12th St., Rolla, MO 65409, USA
2.
Departamento de Matemática, Universidade de Brasília, 70910-900 Brasília, DF, Brazil
3.
Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA

Received: 19 March 2025 Revised: 07 May 2025 Accepted: 19 May 2025 Published: 06 June 2025

We present a general version of Massera's theorems for arbitrary discrete domains, based on a newly introduced definition for both linear and nonlinear equations. For scalar nonlinear equations, we identify sufficient conditions that ensure each $ \mu $-bounded solution approaches a periodic solution asymptotically. In the case of linear systems, we prove that the presence of a $ \mu $-bounded solution necessarily leads to a periodic solution. We also provide some examples to show the practical implications of our findings.
- isolated time scales,
- linear dynamic equations,
- nonlinear dynamic equations,
- boundedness,
- periodicity
Citation: Martin Bohner, Jaqueline G. Mesquita, Sabrina Streipert. Massera's theorem on arbitrary discrete time domains[J]. Mathematical Biosciences and Engineering, 2025, 22(7): 1861-1874. doi: 10.3934/mbe.2025067

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Abstract

We present a general version of Massera's theorems for arbitrary discrete domains, based on a newly introduced definition for both linear and nonlinear equations. For scalar nonlinear equations, we identify sufficient conditions that ensure each $ \mu $-bounded solution approaches a periodic solution asymptotically. In the case of linear systems, we prove that the presence of a $ \mu $-bounded solution necessarily leads to a periodic solution. We also provide some examples to show the practical implications of our findings.

References

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