Studies On Oscillation Of The Solution Of Second-order Differential Equations And Difference Equations

The oscillation of partial differential equation and difference equation are two important branches of differential equations. In the two fields of modern applied mathematics, they have made considerable headway in recent years, because all the structure of theirs emergence have deep physical background and realistic mathematicalmodel. Many scholars take on the research of these fields, they have achieved many good results. With the increasing development of science and technology, there are many problems relating to partial differential equation and difference equation derived from lots of real applications and practice, such as whether partial differential equation and difference equation have a oscillating solution or not, and whether all of their solutions are oscillatory or not. In very resent years, great changes of these fields have taken place. Especially, the second order partial differential equation and the second order difference equation have been paid more attentions and investigated in various classes by using different methods(see [l]-[38]).The present paper employs a generalized Riccati technique, integral average and the monotone of functions to investigate the oscillation criteria for some class of second order partial differential equations with delays and some class of second order nonlinear difference equations, the results of which generalized and improved some known oscillation criteria.The thesis is divided into two sections according to contents.In chapter 1, this chapter is divided into four sections to investigate some new oscillation criteria and interval oscillation criteria for partial differential equations with delays of the form:where Ω is a bounded domain in R^N with a piecewise smooth boundary (?)Ω,With the following boundary condition:where υ is the unit exterior normal vector to (?)Ω and g(x, t) is a nonnegative continuous function on (?)Ω × [0, ∞).We obtain some new oscillation criteria for the problem (1.1.1), (1.1.2) in Section 1 and Section 2. By using a function Φ = Φ(t, s, l) in Section 3 and by introducing a function H = H(t, s) in Section 4, we also get some new oscillations and interval oscillations of solutions of the problem (1.1.1), (1.1.2).In chapter 2, some new sufficient criteria for the second order nonlinear difference equation of the formare established, where the operator Δ is defined by Δy_n = y_n+1 — y_n.