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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
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