# zbMATH — the first resource for mathematics

A necessary and sufficient condition for the oscillation in a class of even order neutral differential equations. (English) Zbl 0948.34042
Summary: The even-order neutral differential equation $\frac{d^n}{dt^n} [ x(t) + \lambda x(t-\tau) ] + f(t,x(g(t))) = 0\tag{1}$ is considered under the following conditions: $$n\geq 2$$ is even; $$\lambda>0$$; $$\tau>0$$; $$g \in C[t_0,\infty)$$, $$\lim_{t\to\infty} g(t) = \infty$$; $$f \in C([t_0,\infty) \times {\mathbb{R}})$$, $$u f(t,u) \geq 0$$ for $$(t,u) \in [t_0,\infty) \times {\mathbb{R}}$$, and $$f(t,u)$$ is nondecreasing in $$u \in {\mathbb{R}}$$ for each fixed $$t\geq t_0$$. It is shown that equation (1) is oscillatory if and only if the nonneutral differential equation $x^{(n)}(t) + \frac{1}{1+\lambda} f(t,x(g(t))) = 0$ is oscillatory.

##### MSC:
 34K11 Oscillation theory of functional-differential equations 34K40 Neutral functional-differential equations
Full Text: