Oscillatory Theorems Of Neutral Delay Differential Equations

Neutral delay differential equations are often utilized to describe many natural phe-nomena and applied to many fields such as physics, biology, and ecology. Neutral differen-tial equations are used to act as their mathematical model in many case. So it is significantto investigate the neutral delay differential equations. However, oscillation behaviour issome main researsh fields in neutral differential equations. Recently, many scholars ab-orad study the oscillation behaviour for first order, higher order, linear, nonlinear, one de-lay, several delays neutral differential equations. This paper includes two chapters. Wemainly consider the oscillation of first order and higher order neutral delay differentialequations.In the first chanpter, we consider the oscillation behaviour of solutions for first orderneutral differential equation with variable coefficiens d/dt[y(t)-p(t)y(t-Ï„)]+sum from i=1 to m q_i(t)y(t-σ_i)=0, t≥t₀, (1.1.1)and first order neutral differential equation witn positive and negative coefficiens respec-tively d/dt[y(t)-sum from i=1 to l r_i(t)y(t-r_i)]+p(t)y(t-Ï„)-q(t)y(t-σ)=0, t≥t₀, (1.2.1)some sufficient conditions for oscillation behaviour of solutions are obtained respectively,we extend results in the associated paper.In the second chapter, we consider the oscillation behaviour for even order neutralequations with distributed deviating argumengts respectively dⁿ/dtⁿ[a(t)y(t)+sum from i=1 to m c_i(t)y(t-Ï„_i)]+integral from n=a to b f(t,ξ, y[g(t,ξ)])dσ(ξ)=0, t≥t₀, (2.1.1)and higher order neutral type nonlinear forced differential equation with oscillating coef-ficients dⁿ/dtⁿ[a(t)y(t)+sum from i=1 to m p_i(t)y(Ï„_i(t))]+sum from j=1 to l q_j(t)f_j(t, y(t,σ_j(t)))=s(t), t≥t₀, (2.2.1)some sufficient conditions for oscillation behaviour of solutions are obtained respectively,we extend results in the associated paper.