An oscillation test for a class of linear neutral differential equations (Q807830)

The author concerns with the problem of oscillation of the first-order neutral differential equations of the form \[ (A_ j)\quad \frac{d}{dt}[x(t)-h(t)x(\tau (t))]+(-1)^ jp(t)x(g(t))=0\quad (j=1,2), \] where h and \(p:[t_ 0<\infty)\to (0,\infty)\) are continuous, \(\tau\) and g: [t\({}_ 0,\infty)\to R\) are continuous, \(\tau\) is an increasing function, and \(\lim_{t\to \infty}\tau (t)=\infty\), \(\lim_{t\to \infty}g(t)=\infty.\) First, the author shows that the possible non-oscillatory solutions of \((A_ 2)\) (or \((A_ 1))\) can be partitioned into disjoint subsets \(N^+\) and \(N^-\) of the set N of all non-oscillatory solutions according to whether \[ (1)\quad x(t)y(t)>0\quad or \] \[ (2)\quad x(t)y(t)<0\quad (y(t)=x(t)-h(t)x(\tau (t))). \] The main results of this paper establish the nonexistence of non-oscillatory solutions of \((A_ j)\) \((j=1,2)\) satisfying (1) or (2). Combining these results, the sufficient conditions for all proper solutions of \((A_ 2)\) (or \((A_ 1))\) to be oscillatory are presented.