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Oscillation of a class of higher order neutral differential equations. (English) Zbl 1116.34333

The authors consider a higher order nonlinear neutral differential equation of the form

x ( t ) - p ( t ) x ( t - τ ) (n) +q(t)fx (g 1 (t)) , , x (g m (t))=0,tt 0 ,(1)

where n2 is even, τ>0 is a constant; pC([t 0 ,+[;), 0p(t)1; qC([t 0 ,+[; + ) is not identically zero on any ray [t 1 ,+[, t 1 >t 0 ; g i C([t 0 ,+[;), and lim t+ g i (t)=+; fC( m ;) is nondecreasing in each of the variables.

In Theorem 1, there are established conditions under which every proper solution of (1) is oscillatory. This result is reformulated as Theorem 2 for the special case, where

fx (g 1 (t)) , , x (g m (t))= i=1 m f i x ( g i (t) ),

f i C(,), xf i (x)>0 for x0 (i=1,,m). The result of Theorem 1 is illustrated by an example.

34K11Oscillation theory of functional-differential equations
34K40Neutral functional-differential equations