Existence of Nonoscillatory Solution of Second Order Nonlinear Neutral Delay Equations
Lin Shi-zhong, Qu Ying, YU Yuan-hong · Kyungpook mathematical journal · 2006
Abstract. In this paper, we study nonoscillatory solutions of a class of second ordernonlinear neutral delay differential equations with positive and negative coefficients. Somesufficient conditions for existence of nonoscillatory solutions are obtained. 1. IntroductionConsider the second order nonlinear neutral delay differential equation withpositive and negative coefficients[r(t)(x(t)+p(t)x(t−τ)) 0 ] 0 +Q 1 (t)f(x(t−σ 1 ))−Q 2 (t)g(x(t−σ 2 )) = 0, (E)where t ≥ t 0 , τ ∈ (0,∞), σ 1 ,σ 2 ∈ [0,∞), p,Q 1 ,Q 2 ,r ∈ C([t 0 ,∞),R), f,g ∈C(R,R). Throughout this paper, we assume that(c 1 ) f and g satisfy local Lipschitz Condition, and xf(x) > 0, xg(x) > 0, forx 6= 0.(c 2 ) r(t) > 0, Q i ≥ 0,R ∞ R(t)Q i (t)dt 0.Second order neutral delay differential equations have applications in problemsdealing with vibrating masses attaches to an elastic bar and in some variationalproblems (see Hale [5]).