Generalized periodicity and applications to logistic growth

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:qN0→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.