Oscillation And Asymptotic Behavior Of Solutions For Functional Differential Equations And Difference Equations

With the development of modern science and technology, people have proposed a great many new problems about functional differential equations and functional difference equations in natural science and social science, which starve for us to solve them with related mathematical theories. The oscillation theory of functional differential equations and functional difference equations is one of the central contents of the qualitative theory of functional differential equations and functional difference equations. It is valuable both in theory and practice to investigate such theories widely. The problem of oscillation theory can open out the things'essence more accurately and make the character of the oscillatory solutions more quantitative. It enriches the oscillation theory of differential equations greatly. So it is not only a demand of the development of mathematics theories themselves, but also a demand of practical applications to investigate such a theory thoroughly, systemically and widely.This paper focuses on the research of the oscillation and asymptotic behavior of solutions of second order neutral delay differential equations, higher order nonlinear neutral delay differential equations and neutral delay partial difference equations. Meantime, some oscillatory solutions of the equations are established and some sufficient conditions for oscillations of the all solutions of the equations are obtained. These results are illustrated by some examples.Firstly, oscillation of second order neutral delay differential equations with positive and negative variable coefficients and second order nonlinear neutral delay differential equations are studied respectively in this paper. Meanwhile, comparison theories for oscillation of the equations are established, and some new criteria for oscillation of the equations under different conditions are obtained.Secondly, this paper discusses the oscillation and asymptotic behavior for higher order nonlinear neutral delay differential equations with distributed deviating arguments and higher order nonlinear neutral delay differential equations with forced term. The relationship between oscillation property of equations and functional differential equations is established successfully. And some valuable necessary and sufficient conditions for oscillations of the equations are obtained.Finally, consider the oscillation and asymptotic behavior of the neutral delay partial difference equations. The oscillation problems for a class of special partial difference equations are solved triumphantly by function inequality transform. Some sufficient conditions to satisfy oscillations of the solutions of the equations are derived.