Oscillation For Third Order Nonlinear Neutral Differential Equations

This paper is mainly composed of three chapters. Chapter1is the introduction. Chap-ter2applies suitable comparison and studies the oscillation criteria for third order nonlinear neutral delay differential equation. In chapter3, we further discuss the asymptotical behavior of non-oscillation solution for third order nonlinear neutral delay differential equation.Consider the third order nonlinear neutral delay differential equation (r2(t)((ri(t)(x(t)±(t)x(τ(t)))’)’)γ)’+q(t)f(x(t),x(σ(t)))=0t≥t0. where to is a positive real number, the function of equation satisfies the following conditions(H1)r1,r2∈C1[t0,+∞),r1(t)>0, r2(t)>0and r’1(t)>0, r’2(t)>0for all t∈[t0,+∞) and(H2) p,q∈C[t0,+∞), for alll [t0+∞,0≤p(t)≤b<1, q(t)>0, where b is a constant;(H3) τ,σ∈C[t0,+∞o),σ(t)≤t,τ(t)≤t,(H4) f∈C(R2,R), there exsits a positive constant k, such that y≠0,(f(x,y))/yr≥k;(H5) γ=m/n, where m, n is odd positive integers.We will establish some sufficient conditions for oscillation and non-oscillation solution of equation, where, Chapter2improve and extend the corresponding results in [Applied Mathematics letters,2011,24:466-470]. Chapter3improve and extend the corresponding results in [Mathematical and computer Modelling,2010,52:215-226].