Periodicity Using Shift Periodic Operators

Abstract

In Chap. 7, we had to require the time scale to be additive in order to show the existence of periodic solutions. We devote this chapter to the study of periodic solutions of functional delay dynamical systems. The periodic delays are expressed in terms of shift operators (see Definition 1.4.4) that were developed by Adivar (Electron J Qual Theory Differ Equ 2010(7):1–22, 2010). We begin the chapter with the discussion of periodicity in shift operators over different time scales and provide examples. We proceed to unifying Floquet theory for homogeneous and nonhomogeneous dynamical systems. Then we consider neutral dynamical systems using Floquet theory and Krasnosel’skii fixed point theorem. We move to the existence of almost automorphic solutions of delayed neutral dynamic system using exponential dichotomy and Krasnosel’skii fixed point theorem. This chapter should serve as the foundation and guidance for future research on periodicity using the well-defined shift operators, by the authors. Part of this chapter is new and the rest can be found in Adivar (Math Slovaca 63(4):817–828, 2013), Adivar and Koyuncuoglu (Appl Math Comput 273:1208–1233, 2016; Appl Math Comput 242:328–339, 2014), and Koyuncuoglu (q-Floquet Theory and Its Extensions to Time Scales Periodic in Shifts. Thesis (Ph.D.), 2016).