Oscillation criteria for functional differential inequalities with strongly bounded forcing term (Q1089823)

The author considers the functional differential inequalities of the form \[ (1)\quad x(t)\{L_ nx(t)+f(t,x(g_ 1(t)),...,x(g_ m(t)))-h(t)\}\leq 0, \] n even, and \[ (2)\quad x(t)\{L_ nx(t)-f(t,x(g_ 1(t)),...,x(g_ m(t)))-h(t)\}\geq 0, \] n odd, where \(n\geq 2\), \(L_ n\) is the general disconjugate differential operator defined recursively by \(L_ 0x(t)=a_ 0(t)x(t)\) and \(L_ kx(t)=a_ k(t)(L_{k-1}x(t))',\) \(k=1,2,...,n\). The functions \(a_ i(t)\), \(i=0,1,...,n\) are positive and continuous on \([t_ 0,\infty)\) and such that \(\int^{\infty}_{t_ 0}a_ j^{-1}(t)dt=\infty\), \(j=1,2,...,n-1\). In the paper, several sufficient conditions for oscillation of all solutions, or of all solutions with some property, of (1) and (2) are presented. The well- known lemmas of Kiguradze play an important role in these considerations.