A necessary and sufficient condition for the oscillation in a class of even order neutral differential equations

S. Tanaka, Hachinohe National College of Technology, Hachinohe, Japan

E. J. Qualitative Theory of Diff. Equ., No. 4. (2000), pp. 1-27.

Abstract: The even order neutral differential equation
$$\frac{d^n}{dt^n} [ x(t) + \lambda x(t-\tau) ] + f(t,x(g(t))) = 0\leqno{(1.1)}$$
is considered under the following conditions: $n\ge 2$ is even; $\lambda>0$; $\tau>0$; $g \in C[t_0,\infty)$, $\lim_{t\to\infty} g(t) = \infty$; $f \in C([t_0,\infty) \times {\bf R})$, $u f(t,u) \ge 0$ for $(t,u) \in [t_0,\infty) \times {\bf R}$, and $f(t,u)$ is nondecreasing in $u \in {\bf R}$ for each fixed $t\ge t_0$. It is shown that equation (1.1) is oscillatory if and only if the non-neutral differential equation
$$x^{(n)}(t) + \frac{1}{1+\lambda} f(t,x(g(t))) = 0$$
is oscillatory.

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