Note on forced oscillation of \(n\)th-order sublinear differential equations (Q1888218)

The authors consider the \(n\)th-order nonlinear differential equation \[ x^{(n)}+q(t)| x| ^{\lambda}\operatorname {sgn}x=e(t),\;t\in [t_{0},\infty ), \tag{E} \] where \(q(t)\) and \(e(t)\) are continuous maps on \([t_{0},\infty ),\) and \( \lambda \in (0,1)\) (sublinear case). The main result provides a sufficient condition for the oscillatory character of (E), that is, to ensure that all its solutions have arbitrarily large zeros. The statement is very involved to be described here, and its proof is based on similar arguments to that of \textit{R. P. Agarwal} and \textit{S. R. Grace} [Appl. Math. Lett. 13, 53--57 (2000; Zbl 0978.39012)] and \textit{C. H. Ou} and \textit{J. S. W. Wong} [J. Math. Anal. Appl. 262, 722--732 (2001; Zbl 0997.34059)]. Moreover, as an application of the main theorem, the authors study the oscillatory nature of \[ x^{\prime \prime }+t^{\alpha }\sin t | x| ^{\lambda } \operatorname{sgn}x=mt^{\beta }\cos t,\;t\geq 0, \tag{E\('\)} \] where \(\alpha \geq 0,\) \(\beta >0,\) \(m\) and \(0<\lambda <1\) are real constants. It is proved that if \(\beta >\frac{\alpha +2}{1-\lambda }, \) then (E\('\)) is oscillatory. This gives an analogue example to that one of \textit{A.H. Nasr} [Proc. Am. Math. Soc. 126, 123-125 (1998; Zbl 0891.34038)].