Investigation On Dynamic Equations On Time Scales

Theory of dynamic equations on time scales, founded by Stenfan Hilger in 1998, is undergoing a rapid development as it provides a powerful tool not only to unify the existing theories of both differential equations and finite differ-ence equations but also to generalize the discussion to a wide class of equations on time scales. In addition, the study of dynamic equations on time scales has lead to several important applications related to mathematical modeling of real phenomena. For instance, investigation of population dynamics, financial assumption process, and epidemic models have been involved with the concept of dynamic equations on time scales. Clearly, theory of dynamic equations on time scales has been and are being proved to be of great importance and potential application.In this thesis, we investigate the dynamics of dynamic equations on time scales, obtaining some important analytical and numerical results. In partic-ular, the generalized invariance principle of autonomous dynamic equations are rigorously established. And chaos synchronization of a class of nonlin-ear system on time scale, as well as one-dimensional p-Laplacian boundary value problems, is investigated. Moreover, the existence of periodic solution and eigenvalue problems in a class of functional dynamic equations are further studied. As a matter of fact, all the results obtained in the thesis not only unify the existing results for the conventional continuous system and discrete system but also generalize those results to a more extensive class of systems on time scale.In the first chapter, we review the history of the development of the time scales theory. Particularly, we construct a concrete example to show that dynamic equation is not a trivial combination of the conventional ordinary differential equation and the difference equation, probably exhibiting totally different dynamical phenomenon. In addition, a brief introduction to the the-ories of calculus on time scales, as well as a summary of this thesis, is made in this chapter.In the second chapter, generalized principle of stability and their pos-sible applications are discussed. The Lasalle-like invariance principle is gener-alized to a class of autonomous dynamic equations on time scale. A concrete example is provided to show the possible application of our theoretical result. Furthermore, chaos synchronization of a class of nonlinear system on time scale is novelly discussed. Criteria of complete synchronization are rigorously established, and then are applied to the coupled Chua's systems on time scales. Differences between our results and the existing condition for the continuous Chua's systems are further illustrated.In chapter three, one-dimensional dynamic p-Lpalacian boundary prob-lem is studied. By virtue of the Avery-Henderson's fixed point theorem and the Legget-Williams's fixed point theorem, two sufficient conditions on the existence of at least two or three positive solutions for p-Lpalacian m-point boundary problem are rigorously obtained, which further generalize the ex-isting results on the existence theory of one-dimensional dynamic p-Lpalacian boundary value. Furthermore, a class of p-Laplacian nonlocal boundary prob-lems are discussed. And some criteria on the existence of at least one (positive) solutions are derived in light of the fixed point theorem for the combined two operators.In the last chapter, sufficient conditions on the existence of at least one or two positive solution of the functional dynamic equations by the fixed the-orem point in cone. And then, eigenvalue problem for the functional dynamic equations are investigated by using of the fixed point index theorem. Finally, asymptotic property of the solutions, as well as the existence of the periodic solutions, is discussed in a class of scalar functional dynamic equations. All the results obtained in this chapter nontrivially generalize the existing results on the conventional functional differential equations and functional difference equations.