Equivalence of oscillation of a class of neutral differential equations and ordinary differential equations
It is shown that a necessary and sufficient condition for equation \[ d/dt\left(x(t)-x(t-r )\right)^{(n)}+p(t)x(t-\sigma )=0,\tag{1} \] to oscillate is that the ordinary differential equation \[ x^{(n+1)}(t)+p(t)x(t)/r=0 \] is oscillatory. Here \(r>0\), \(\sigma \in \mathbb{R}\) are constants, \(n\geq 0\) is an odd integer and \(p:[0,\infty )\to [0,\infty )\) is a continuous function. Some comparison theorems are established. It is shown that the deviation \(\sigma\) has no affection on the oscillation of the equation (1).
scientific article; zbMATH DE number 1045999