Consider the equation \[ {d^ n\over dt^ n}(x(t)\pm\lambda x(t- r))+f(t,x(g_ 1(t)),\ldots,x(g_ N(t)))=0, (A_ \pm) \] \(n\geq 1\), \(\lambda>0\), \(r>0\), \(g,f\), continuous, \(\lim_{t\to\infty}g_ i(t)=\infty\), \(| f(t,u_ 1,\ldots,u_ N)|\leq F(t,| u_ 1|,\ldots,| u_ N|)\), \(v_ i\mapsto F(t,v_ 1,\ldots,v_ N)\) nondecreasing. Assume \(0<\lambda \leq 1\) and existence of \(\mu\in(0,\lambda)\), \(a>0\) such that \[ \int^ \infty t^{n-1}\mu^{- t/r}F(t,a\lambda^{g_ 1(t)/r},\cdots,a\lambda^{g_ N(t)/r})dt<\infty. \] Then (i) for any continuous, periodic, oscillatory function \(\omega_ -\) with period \(r\), equation \((A_ -)\) has a bounded oscillatory solution \(x_ -\) such that \(x_ - (t)=\lambda^{t/r}\omega_ -(t)+o(\lambda^{t/r})\), \(t\to\infty\); (ii) for any continuous, oscillatory function \(\omega_ +\) with \(\omega_ +(t+r)=-\omega_ +(t)\), equation \((A_ +)\) has a bounded oscillatory solution \(x_ +\) such that \(x_ +(t)=\lambda^{t/r}\omega_ +(t)+o(\lambda^{t/r})\), \(t\to\infty\). Similar results are obtained for \(\lambda>1\), for unbounded oscillatory solutions \((x_ \pm(t)=\lambda^{t/r}\omega_ \pm(t)+o(1)\), \(t\to\infty)\).