The differential equation \((*)\;L_ nu(t)+p(t)f(u(g(t)))=0\), where \(L_ n=(d/dt)(r_{n-1}(t)(d/dt)\) \((r_{n-2}(t)(d/dt)\dots(d/dt)r_ 1(t)(d/dt)\dots)\dots )\) \(n\geq 2\), has property \((C)\) if it is oscillatory for \(n\) even and for \(n\) odd every nonoscillatory solution of \((*)\) has the following property: \(I_ k(t)L_ ku(t)\to 0\) as \(t\to\infty\), \(0\leq k\leq n-1\), where \(I_ 0(t)=1\), \(I_ k(t)=\int^ t_{t_ 0}(I_{k-1}(s)/r_ k(s))ds\), as \(1\leq k\leq n-1\). Assuming that \(f,p,r_ k\), \(g\in C[[t_ 0,\infty))\), \(r_ k(t),p(t)>0\), \(g(t)\to\infty\) as \(t\to\infty\) and \(xf(x)>0\) for \(x\neq 0\), the author gives sufficient conditions that the equation \((*)\) has the property \((C)\).