Nonlinear Dynamic Equations on Translation Time Scales

Abstract

Since time scales can be divided into several types depending on their translations, it is very important to study the dynamical behavior of solutions for nonlinear dynamic equations with the properties of corresponding translation functions. In this chapter, different forms of generalized solutions for nonlinear dynamic equations are discussed. In Sects. 6.1, the existence and uniqueness of almost periodic solutions and pseudo almost periodic solutions for nonlinear dynamic equations on complete-closed time scales are investigated, especially for delay dynamic equations on CCTS. In Sects. 6.2, some existence conditions of the weighted pseudo almost periodic solutions for abstract dynamic equations are derived in the sense of Π-semigroups on time scales. In Sects. 6.3, local periodicity on changing-periodic time scales is proposed and the Clh space is introduced, then some preliminary results on changing-periodic time scales are obtained. Based on this, some sufficient conditions on the existence of positive local-periodic solutions for functional dynamic equations with infinite delay (FDEID) are established.