Frequent Oscillatory For Difference Equations And Existence Of Positive Solutions For Dynamic Equations On Time Scales

Difference equations are obtained from the discretization of differential equations,but they have many different characteristics. Difference equations are widely used inecology, economics, physics etc, so it is attracting more and more attention. Now, thedifference equation has become an important aspect of the research in mathematics andhas significant value in theory and practice.Partial difference equations are difference equations that involve functions with twoor more independent integer variables. The partial difference equations occur frequently inthe approximation of solutions of partial differential equations by finite differencemethods, random walk problems, the study of molecular orbits, mathematical physics-problems etc. The oscillatory theories for partial difference equations get rapidly development only in recent years, so it is a new investigative realm with blooming life-force. Thestudies about this new academic branch are not only has important value in theory, butalso a demand of actual applications.Recently, the theory of dynamic equations on time scales has attracted lots ofattention and progress rapidly. First of all, the theory of time scale unify the continuousand discrete cases. It not only unifies the differential and difference equations, but alsoreveals the discrepancies of continuous and discrete cases. Second, the dynamic equationson time scales are widely used in epidemic models, neuronal network model and insectilenumbers models etc. As the theory of dynamic equations on time scales has greattheoretical significance and practical value, there are more and more researchers areattracted in this field.This paper focuses on the research of frequent oscillatory criteria for two sets ofdifference equations and the existence of positive solution for dynamic equations on timescales.Firstly, frequent oscillation for two sets of nonlinear neutral difference equations areresearched. By use of frequency measures, some sufficient conditions are established andsome examples are given.Secondly, study frequent oscillation for partial difference equations with positive andnegative coefficients and a set of nonlinear neutral partial difference equations.Finally, based on the theory of time scales and fixed point theorem, we discuss theexistence of positive solution for a higher order dynamic equations on time scales. In the end, some examples are given to explain our results.