The Properties Of Solutions For Functional Partial Difference Equations And Dynamic Equations On Time Scales

With the development of modern science and technology, people have proposed a great many new problems about functional difference equations in natural science and social science, which starve for us to solve them with related mathematical theories. The partial difference equations occur frequently in the approximation of solutions of partial differential equations by finite difference methods, random walk problems, the study of molecular orbits, mathematical physics problems and etc. But the oscillatory theories get rapid development only in recent years. So it is a new investigative realm with blooming life-force. Because of the progress in modern science and technology, the studies about this new academic branch are not only a demand of the development of mathematics theories themselves, but also a demand of actual applications.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.This paper focuses on the research of frequent oscillation of partial difference equations, oscillation,asymptotic behavior and the existence of positive solutions of dynamic equations on time scales.Firstly, frequent oscillations of a class of partial difference equations with constant coefficients and nonlinear partial difference equations with coefficient which may change sign are studied respectively in this paper. Some sufficient conditions to satisfy frequent oscillations of these equations are gained. At the same time, some examples are given.Secondly, this paper discusses oscillations of solutions of second and third nonlinear neutral dynamic equations. By employing Riccati technique, some sufficient conditions for oscillations of all solutions of these equations are established. Meanwhile, some examples are given.Finally, consider the existence of positive solutions of second order self-adjoint dynamic equations, second order mixed and second order three point boundary value problems. Some sufficient conditions to satisfy existence of positive solutions of these equations are derived.