Research On Oscillation For Several Classes Of Nonlinear Functional Differential Equations

With the rapid development of science and technology, many fields of science have some problems about differential equation, these issues have increasingly drawn attention. As is known to all, the oscillation of differential equations is a very important branch in differ-rential equation theory. It began in1836and the second order linear ordinary differential equations by the Sturm, this development laid the theoretical foundation oscillation differential equations. In recent years, many scholars do the research and exploration in the oscillation of differential equations, they get some conclusions and continuous improvement and the existing conclusion. What they have done there is an important meaning in theory also have practical value..This thesis can be divided into three parts:The first part describes the introduction. We back to review the history, research dynamic, start and development trend of the oscillation of differential equations. This part introduces the basic concept and summarized the main content and the structure of this article.The second chapter introduces the oscillation of second order nonlinear functional differential equations and mainly study the equations (a(t)(y’(t))(?))’+q(t)f(y((?)(t)))g(y’(t))=0,t(?)t0and equation (a(t)(x’(t))()?)’+p(x(t))x’(t)+q(t)f(x((?)(t)))g(x’(t))=0,t(?)t0oscillation of the solution,t(?)t0,(?)is constant, where t(?)t0, a(t)>0, q(t)(?)0, and q(t) is not constant be zero in final,(?)’(t)>0, and lim (?)(t)=+∞, the paper improves the result in existing literature.The third chapter introduces the oscillation of the third-order nonlinear neutral functional differential equations and mainly study the equations 〔a(t)[x(t)+p(t)x((?)(t))]"〕+q(t)f(x(g(t)))(?)(x’(t))=0,t(?)t0〔a(t)[x(t)+p(t)x((?)(t))]"〕+q(t)f(x(g(t)))=0, t(?)t0and equation 〔r(t)((a(t)+p(t)x((?)(t)))’)’)a〕+q(t)f(x((?)(t)))=0,t(?)t0there solution is vibration, the paper improves the result in existing literature.