Título:	Generalized periodicity and applications to logistic growth
Autor(es):	Bohner, Martin Mesquita, Jaqueline Godoy Streipert, Sabrina
ORCID:	https://orcid.org/0000-0001-8310-0266 https://orcid.org/0000-0002-1255-9384 https://orcid.org/0000-0002-5380-8818
Afiliação do autor:	Missouri University of Science and Technology, Department of Mathematics and Statistics University of Brasilia, Department of Mathematics University of Pittsburgh, Department of Mathematics
Assunto:	Equações diferenciais lineares Existência e unicidade Conjectura de Cushing-Henson
Data de publicação:	28-Jun-2024
Editora:	Elsevier
Referência:	BOHNER, Martin; MESQUITA, Jaqueline Godoy; STREIPERT, Sabrina. Generalized periodicity and applications to logistic growth. Chaos, Solitons & Fractals, [S.l.], v. 186, e115139, 2024. DOI: https://doi.org/10.1016/j.chaos.2024.115139. Disponível em: https://www.sciencedirect.com/science/article/pii/S096007792400691X?via%3Dihub. Acesso em: 07 ago. 2025.
Abstract:	Classically, a continuous function 𝑓 ∶ R → R is periodic if there exists an 𝜔 > 0 such that 𝑓(𝑡 + 𝜔) = 𝑓(𝑡) for all 𝑡 ∈ R. The extension of this precise definition to functions 𝑓 ∶ Z → R is straightforward. However, in the so-called quantum case, where 𝑓 ∶ 𝑞 N0 → R (𝑞 > 1), or more general isolated time scales, a different definition of periodicity is needed. A recently introduced definition of periodicity for such general isolated time scales, including the quantum calculus, not only addressed this gap but also inspired this work. We now return to the continuous case and present the concept of 𝜈-periodicity that connects these different formulations of periodicity for general discrete time domains with the continuous domain. Our definition of 𝜈-periodicity preserves crucial translation invariant properties of integrals over 𝜈-periodic functions and, for 𝜈(𝑡) = 𝑡 + 𝜔, 𝜈-periodicity is equivalent to the classical periodicity condition with period 𝜔. We use the classification of 𝜈-periodic functions to discuss the existence and uniqueness of 𝜈-periodic solutions to linear homogeneous and nonhomogeneous differential equations. If 𝜈(𝑡) = 𝑡 + 𝜔, our results coincide with the results known for periodic differential equations. By using our concept of 𝜈-periodicity, we gain new insights into the classes of solutions to linear nonautonomous differential equations. We also investigate the existence, uniqueness, and global stability of 𝜈-periodic solutions to the nonlinear logistic model and apply it to generalize the Cushing–Henson conjectures, originally formulated for the discrete Beverton–Holt model.
Unidade Acadêmica:	Instituto de Ciências Exatas (IE) Departamento de Matemática (IE MAT)
Licença:	This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/).
DOI:	https://doi.org/10.1016/j.chaos.2024.115139
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