Specific tests for oscillation of solutions of linear differential equations with delayed argument (Q1089493)

The author proves specific tests for the oscillation of solutions for the equation (1) \(u^{(n)}(t)=(-1)^ n\sum^{m}_{i=1}p_ i(t)u(\tau_ i(t)),\) where \(p_ i: R_+\to R_+,\tau_ i: R_+\to R\) are continuous functions, \(\tau_ i(t)\leq t\) for \(t\in R_+\), \(\lim \tau_ i(t)=\infty\) as \(t\to \infty\), \(i=1,2,...,m\). The main result: Let \(t_ 0,c_ i,\delta_ i\), \(i=1,2,...,m\) be nonnegative numbers such that \(p_ i(t)\geq c_ i\), \(t-\tau_ i(t)\geq \delta_ i\) for \(t\geq t_ 0\), \(\sum^{m}_{i=1}c_ ie^{\delta_ i\alpha}-\alpha^ n>0\) for \(\alpha >0\). Then the equation (1) has the property \(\tilde A\) (i.e. every regular solution of (1) is oscillatory for n even and for n odd it is either oscillatory or \(| u^{(k)}(t)| \uparrow \infty\) for \(t\uparrow \infty\), \(k=0,...,n-1)\).