Existence of Nonoscillatory Solutions for a Second-Order Nonlinear Neutral Delay Differential Equation

A new second-order nonlinear neutral delay differential equation r(t) x(t) + P(t)x(t-τ) + cr(t) x(t)-x(t-τ) + F t,x(t-σ1),x(t-σ2),...,x(t-σn) = G(t),t ≥ t0,where τ > 0,σ1,σ2,...,σn≥ 0,P,r ∈ C([t0,+∞),R),F ∈ C([t0,+∞)×Rn,R),G ∈ C([t0,+∞),R) and c is a constant,is studied in this paper,and some sufficient conditions for existence of nonoscillatory solutions for this equation are established and expatiated through five theorems according to the range of value of function P(t).Two examples are presented to illustrate that our works are proper generalizations of the other corresponding results.Furthermore,our results omit the restriction of Q1(t) dominating Q2(t)(See condition C in the text).