On oscillation of solutions of nth-order delay differential equations (Q789595)

Oscillatory behaviour of solutions of the nth-order delay differential equation \(L_ nx(t)+q(t)f(x[g(t)])=0,\) where \(L_ 0x(t)=x(t), L_ kx(t)=a_ k(L_{k-1}x(t))', a_ 0(t)=a_ n(t)=1\) is discussed. The following four cases are studied separately: 1) n even, \(q\geq 0\), 2) n odd, \(q\geq 0\), 3) n even, \(q\leq 0\), 4) n odd, \(q\leq 0\). It is assumed that \(g:[0,\infty)\to [0,\infty)\) is continuous and nondecreasing, g(t)\(\leq t\) and \(\lim_{t\to \infty}g(t)=\infty\). Besides this some additional conditions on \(a_ i\), q and f are supposed. The results obtained are an extension of some results by \textit{W. J. Kim} [Proc. Am. Math. Soc. 62, 77-82 (1977; Zbl 0376.34020)] for equation \(x^{(n)}+px=0.\)