(w, c)-periodic functions on time scales and application for families of delayed dynamic equations representing population models

Abstract

This paper delves into a novel class of functions defined on an invariant under translation time scale T, known as (w, c)-periodic functions. This class encompasses various types of functions, such as periodic, anti-periodic, Bloch, and unbounded functions. To equip this set of functions with a comprehensive mathematical framework, we introduce a suitable norm on T, transforming the space consisting of (ω, c)-periodic functions into a Banach space. Several fundamental properties of this newly established class are examined. Furthermore, we present a significant application of our theoretical findings. We investigate the unique existence of asymptotically exponentially stable (w, c)-periodic solutions for some families of generalized dynamic equations on time scales. These dynamic equations represent population models incorporating feedback and time-varying delays. Our study outlines sufficient conditions for the emergence of such solutions, highlighting their importance in understanding the dynamics of complex systems over time scales. In particular, we explore (w, c)-periodic solutions for prominent models, such as the Nicholsan model, Lasota- Wazewska model, and their mixed versions on time scales. Our investigations are groundbreaking, as they encompass both discrete and continuous cases, presenting a unified perspective on periodic behavior across various systems.