Periodic Solutions: The Natural Setup

Abstract

Among time scales, periodic ones deserve a special interest since they enable researchers to develop a theory for the existence of periodic solutions of dynamic equations on time scales (see for example Bi et al. (Nonlinear Anal 68(5):1226–1245, 2008), Bohner and Warth (Appl Anal 86(8):1007–1015, 2007), Kaufmann and Raffoul (J Math Anal Appl 319(1):315–325, 2006; Electron J Differ Equ 2007(27):12, 2007)). This chapter is devoted to the study of periodic solutions based on the definition of periodicity given by Kaufmann and Raffoul (J Math Anal Appl 319(1):315–325, 2006) and Atici et al. (Dynamic Systems and Applications, vol. 3, pp. 43–48, 2001). In this section, we cover natural setup of periodicity. For the conventional definition of periodic time scale and periodic functions on periodic time scales we refer to Definitions 1.5.1 and 1.5.2, respectively. Based on the Definitions 1.5.1 and 1.5.2, periodicity and existence of periodic solutions of dynamic equations on time scales were studied by various researchers and for first papers on the subject we refer to Adivar and Raffoul (Ann Mat Pura Appl 188(4):543–559, 2009; Comput Math Appl 58(2):264–272, 2009), Bi et al. (Nonlinear Anal 68(5):1226–1245, 2008), Kaufmann and Raffoul (J Math Anal Appl 319(1):315–325, 2006; Electron J Differ Equ 2007(27):12, 2007), and Liu and Li (Nonlinear Anal 67(5):1457–1463, 2007). We begin by introducing the concept of periodicity and then apply the concept to finite delay neutral nonlinear dynamical equations and to infinite delay Volterra integro-dynamic equations of variant forms by appealing to Schaefer fixed point theorem. The a priori bound will be obtained by the aid of Lyapunov-like functionals. Effort will be made to expose the improvement when we set the time scales to be the set of reals or integers. Then we proceed to study the existence of periodic solutions of almost linear Volterra integro-dynamic equations by utilizing Krasnosel’skii fixed point theorem (see Theorem 1.1.17). We will be examining the relationship between boundedness and the existence of periodic solutions. We will also be giving existence results for almost automorphic solutions of the delayed neutral dynamic system. We will end this chapter by proving existence results by using a fixed point theorem based on the large contraction.