Periodic Solutions Of Dynamic Equations On Time Scales

The theory of calculus on time scales, essentially introduced in seminal work of Stefan Hilger, is a new studying field and has tremendous potential for applications. The two main features of the calculus on time scales are unification and extension. A dynamic equation on a time scales is not only related to continuous process or discrete process but those pertaining to the mixture of continuousness and discreteness. Many results concerning differential equations (continuousness) carry over quite easily to cor-responding results for difference equations (discreteness), while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on general time scales can reveal such discrepancies and help avoid proving results twice-once for differential equations (continuousness) and once again for difference equations (discreteness). Recently, in the studying of dynamic equations on time scales, most of problems have been focused on boundary value and oscillation. However, there are few studies to consider the effects of periodicity. Therefore, the principle aim of this paper is to explore periodic solutions of dynamic equations on time scales.In this paper. we systematically explore the periodicity of some non-autonomous dynamic equations on time scales, which incorporate as special cases many population models (e.g., predator-prey systems, competition systems, single species systems and feedback control systems) in mathematical biology governed by differential equations and difference equations. The main approach is based on a continuation theorem in coin-cidence degree theory, which has been extensively applied in studying existence problems in differential equations and difference equations but rarely applied in dynamic equations on time scales. Explicit verifiable sufficient criteria are established for the existence of periodic solutions of such dynamic equations, which generalize many known results for continuous and discrete population models. This study shows that it is unnecessary to explore the existence of periodic solutions of continuous and discrete population models in separate way. and one can unify such studies in the sense of dynamic equations on general time scales. And second, with the help of the contraction mapping principle, and based on a kind of semi-linear dynamic equations on time scales, an asymptotically stable periodic solution is obtained. In addition, we also study the boundedness and asymptotic stability of solutions. Finally, as applications of exponential dichotomies of linear dynamic equations on time scales, we investigate the existence of periodic solutions of semi-linear and nonlinear higher-dimensional dynamic equations on time scales, and obtain new suf-ficient criteria for the existence of periodic solutions for such systems. Meanwhile, we also explore some basic properties of exponential dichotomies on time scales, establish some necessary and sufficient criteria for the existence of an exponential dichotomy, and present perturbation theorems on the roughness of exponential dichotomies. An explicit sufficient criteria for linear dynamic equations to be exponentially dichotomous will be developed.