Maintenance of oscillations under the effect of a strongly bounded forcing term (Q1098988)

The oscillatory behaviour of solutions of the forced differential equation \((1)\quad L_ nx+f(t,x)=h(t),\) where \(L_ 0x(t)=a_ 0(t)x(t),\quad L_ kx(t)=a_ k(t)(L_{k-1}x(t))',\) \(i=1,...,n\), is examined by comparing with that of the associated unforced equation \((2)\quad L_ nx+f(t,x)=0.\) More precisely, it is shown that the oscillation of solutions of (1) follows from the oscillation of solutions of (2) provided that h, \(a_ i\in C([t_ 0,\infty))\), \(f\in C([t_ 0,\infty)\times R)\), \(xf(t,x)>0\) for \(x\neq 0\), \(t\geq t_ 0\), \(a_ i(t)>0\) for \(t\geq t_ 0\), \(\int^{\infty}_{t_ 0}a_ i^{- 1}(t)dt=\infty\), \(i=1,...,n-1\), f(t,x) is nondecreasing in the second variable and h(t) is the n-th ``quasi-derivative'' of the function p(t) for which \(L_ 0p(t)\) is strongly bounded in the sense that it assumes its maximum and minimum on every interval of the form [T,\(\infty)\), \(T\geq t_ 0\).