Oscillations of advanced differential inequalities (Q1121431)

The author considers the inequalities: \[ (N_ z)\quad (-1)^ zx^{(n)}(t)sgn x(t)\geq p(t)\prod^{m}_{i=1}| x(g_ i(t))|^{\alpha_ i}, \] \[ (L_ z)\quad (-1)^ zx^{(n)}(t)sgn x(t)\geq \sum^{m}_{i=1}p_ i(t)| x(g_ i(t))|, \] where \(n\geq 2\), \(z\in \{1,2,n,n+1\}\), \(\alpha_ i\) \((i=1,...,m)\) are real positive numbers with \(\alpha_ 1+...+\alpha_ m=1\); p, \(p_ i: R_+\to R_+\), \(g_ i: R_+\to R_+=[0,\infty)\) \((g_ i(t)\geq t\) on \(R_+)\), \(i=1,...,m\) are continuous functions. On the other assumptions on p, \(p_ i\), \(g_ i\) \((i=1,...,m)\) the author proves sufficient conditions under which a) all regular solutions of \((N_ z)\), \((L_ z)\) are oscillatory, b) every regular solution x of \((N_ z)\), \((L_ z)\) is either oscillatory or \(x^{(k)}(t)\to 0\) \((k=1,...,n)\) monotonically as \(t\to \infty\).