Oscillation For Third Order Nonlinear Neutral Differential Equations On Time Scales

This thesis mainly studies the oscillation of third-order nonlinear neutral delay differential equations on time scales.By using Riccati transform technique to do further research on its oscillation,we obtain some new results.This paper is mainly composed of three chapters.The first chapter is the introduction.It mainly introduces the background of this topic,main tasks of this article and some calculus theory on time scales.By means of Riccati transformation technique,the following chapters two and three study respectively oscillatory behavior of third order nonlinear neutral delay dif-ferential equations.This paper studys oscillation of solutions of the third order nonlinear neutral delay differential equation on time scales (c(t){[a(t)(x(t)±p(t)x(r(t)))△]△})△+∫(t,x(τ(t)))=0,t∈T,t≥t0 In this paper,we consider time seales T have no upper bound: namely sup T +∞,and assume t0 ∈T, t0 >0 is a real number.We define time scales section [t0,+∞)T=[t0,+∞)∩ T,where γ≥1 is a quotient of odd positive intege rs.We assume the following conditions are satisfied throughout the whole article:(A1)c(t)、a(t)、p(t) are positive rd—continuous functions on T,and ∫∞t0 a(t)/Δt ∞,∫t0/∞ 1/cr(t)/Δt=∞;(A2)Delay functions r(t):T�'T and τ(t):T�'Tsatisfy r(t)≤t,r(t)≤l,r△(t)≥ 0 and lim /t�'∞ r(t)�'∞,lim/t�'∞r(t)�'∞ with(Tοσ)(t)=(σοτ)(t)；(A3)0≤p(t)≤p0<1;(A4)The function f:T×R�'R satisfies,uf(t,u)>0, and there is a positive rd-continuous function q(t) on time scales T which makes f(t,u)/ur≥q(t),u≠0we establish soine sufficient conditions which the solutions oscillate.