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Oscillations of higher-order neutral equations. (English) Zbl 0566.34055

Consider the neutral delay differential equation of order n (*)(d n /dt n )[y(t)+py(t-τ)]+qy(t-σ)=0, tt 0 where q is a positive constant, the delays τ and σ are nonnegative constants and the coefficient p is a real number. Theorem 1. (a) Assume that n is odd and that p<-1. Then every nonoscillatory solution of (*) tends to + or - as t. (b) Assume that n is odd or even and that p>-1. Then every nonoscillatory solution of (*) tends to zero as t. Theorem 2. Assume that n is odd. Then each of the following four conditions implies that every solution of (*) oscillates: (i) p<-1 and (-q/(1+p)) 1/n (τ-σ)/n>1/e; (ii) p=-1; (iii) p>-1 and (q/(1+p)) 1/n (σ-τ)/n>1/e; (iv) -1<p<0 and q 1/n σ/n>1/e·

Theorem 3. Assume that n is even. Then each of the following two conditions implies that all solutions of (*) oscillate: (i) p0; (ii) -1p<0 and (q/p) 1/n (σ-τ)/n>1/e. Theorem 4. Assume that n is even, p<-1, and (q/p) 1/n (σ-τ)/n>1/e. Then every bounded solution of (*) oscillates.


MSC:
34K99Functional-differential equations
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory