A unified concept of periodicity on any time scale and applications

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

doi:10.3934/math.2025956

AIMS Mathematics

2025, Volume 10, Issue 9: 21512-21532. doi: 10.3934/math.2025956

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A unified concept of periodicity on any time scale and applications

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

Received: 18 April 2025 Revised: 07 September 2025 Accepted: 12 September 2025 Published: 17 September 2025
MSC : 34N05, 39A05

We introduce a novel definition of periodicity on arbitrary time scales, dependent on a strictly increasing and differentiable function. This removes the commonly used and restrictive assumption of a periodic time scale to define periodic functions. Our new definition furthermore allows for a wider class of functions to be studied using the theory of periodic systems. After providing crucial properties of these periodic functions, such as the translation invariance of integrals of periodic functions, we apply the concept of this new periodicity to linear dynamic equations. We provide necessary and sufficient conditions for a linear dynamic equation to have such a periodic solution and discuss its uniqueness.
- periodicity,
- time scales,
- existence,
- uniqueness,
- global stability,
- linear dynamic equations
Citation: Martin Bohner, Jaqueline G. Mesquita, Sabrina H. Streipert. A unified concept of periodicity on any time scale and applications[J]. AIMS Mathematics, 2025, 10(9): 21512-21532. doi: 10.3934/math.2025956

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Abstract

We introduce a novel definition of periodicity on arbitrary time scales, dependent on a strictly increasing and differentiable function. This removes the commonly used and restrictive assumption of a periodic time scale to define periodic functions. Our new definition furthermore allows for a wider class of functions to be studied using the theory of periodic systems. After providing crucial properties of these periodic functions, such as the translation invariance of integrals of periodic functions, we apply the concept of this new periodicity to linear dynamic equations. We provide necessary and sufficient conditions for a linear dynamic equation to have such a periodic solution and discuss its uniqueness.

References

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