Oscillation And Existence Of Solutions Of Dynamic Equations On Time Scales

With the development of modern science and technology, people have proposed a great many problems about differential and difference equations in natural science and social science, and some results are gained. The theory of time scales can unify continuous and discrete cases, which pioneers a new mathematical area. This theory not only unifies differential and difference equations, reveals the essence of continuous and discrete cases, avoids repeat study, but also contains many other cases. The prominent peculiarity of time scales is unification and generalization, so the study of time scales has important significance in theory and application. In addition, theory of dynamic equations on time scales has been and are being proved to be of great importance and potential application as to embranchment of applied mathematic. In application, dynamic equations has a wide base in nature, for instance, epidemic models, neuronal network model and insectile numbers models. Except these applications in biology, the mathematical tool has been used to prove calculational mode about stock market.In this thesis, oscillation properties,existence properties of solutions of dynamic equations and boundary value problems on time scales are researched.Firstly, oscillation properties of solutions for a class of second order nonlinear neutral dynamic equations on time scales are researched, a series of sufficient conditions for oscillation of the solutions are established. These results are illustrated by examples.Secondly, this paper discusses oscillations of sulutions of a class of third order nonlinear neutral damped dynamic equations on time scales are obtained. These results are explained by examples.By using monotone methods, some necessary and sufficient conditions for the existence of a positive solution of the second order nonlinear neutral damped dynamic equations on time scales are derived.Finally, the existence of positive solutions for a class of second order and third order nonlinear boundary value problems on time scales and nonlinear singular m - point boundary value problems on the half-line on time scales are considered. Some sufficient conditions to satisfy existence of positive solutions of these equations are derived.