Specific properties of solutions of differential equations with a deflecting argument (Q807822)

The following advanced functional differential equation is considered: \[ u^{(n)}(t)=\sum^{m}_{i=1}\rho_ i(t)u(\tau_ i(t)), \] where \(n\geq 1\), \(\tau_ i:\) \(R_+\to R_+\), \(i=1,2,...,m\), are continuous non-decreasing functions, \(\tau_ i(t)\geq t\) for \(t\in R_+\), \(i=1,2,...,m\), and the functions \(\rho_ i: R_+\to R\), \(i=1,2,...,m\), are summable on each finite interval. Let \(t_ 0\in R_+\). The function \(u:[t_ 0,+\infty)\to R\) is said to be a regular solution if it is absolutely continuous together with its derivatives up to the order n-1, it satisfies almost everywhere on \([t_ 0,+\infty)\) the equation and sup \(\{| u(t)|: s\leq t<\infty \}>0\) for \(s\geq t_ 0\). The regular solution is said to be oscillatory if it possesses a sequence of zeros tending to \(+\infty\), and non-oscillatory in the contrary. The main assertion in this paper is the following one: Assume: \[ 1.\quad \liminf_{t\to \infty}\frac{t}{\tau_ i(t)}>0,\quad i=1,2,...,m; \] \[ 2.\quad \inf \{\liminf_{t\to \infty} t^{-\lambda}\int^{t}_{t_ 0}(t-s)^{n-1}\sum^{m}_{i=1}\rho_ i(s)\tau_ i^{\lambda}(s)ds\quad:\;\lambda \in (n-1,+\infty)\}\quad >\quad (n-1)!. \] \[ 3.\quad \liminf_{t\to \infty}\int^{\tau_ i(t)}_{t}s^{n-k- 1}\rho_ i(s)ds>0,\quad i=1,2,...,m; \] 4. let for some \(k\in \{0,1,...,n-1\}\) be true \[ \sup \{t^{n-k}\rho_ i(t): t\in [0,+\infty)\}<+\infty)\}<+\infty,\quad i=1,2,...,m; \] 5. let for odd n for each \(l\in \{1,3,...,n-2\}\) and \(\lambda\in [l-1,l)\) (for even n for each \(l\in \{2,4,...,n-2\}\) and \(\lambda\in [l-1,l))\) there exists \(\epsilon\in (0,1)\) such that \[ \liminf_{t\to \infty} t^{l- \lambda}\int^{+\infty}_{t}\xi^{n-l-1}\sum^{m}_{i=1}\rho_ i(\xi)\tau_ i^{\lambda}(\xi)d\xi \quad >\quad \prod^{n- 1}_{i=0,i\neq l}| \lambda -i| +\epsilon. \] Then all the regular solutions are oscillatory (either oscillatory or satisfying the condition \(| u^{(i)}(t)| \downarrow 0\) for \(t\uparrow +\infty)\).