Linearized oscillations in nonautonomous delay differential equations (Q1261643)

Linearized oscillation results are established for a general nonautonomous equation of the form \[ \dot x+f(t,x(t-\tau_ 1(t)),\ldots,x(t-\tau_ m(t))=0,\tag{1} \] where \(f\in C[[t_ 0,\infty)\times R^ m,R]\), \(\lim_{t\to\infty}[t-\tau_ i(t)]=\infty\), \(\tau_ i\in C[[t_ 0,\infty),(0,\infty)]\) and \[ \begin{cases} f(t,u_ 1,\ldots,u_ m)\geq 0, & \text{ for } u_ 1,\ldots,u_ m\geq 0,\\ f(t,u_ 1,\ldots,u_ m)\leq 0, & \text{ for } u_ 1,\ldots,u_ m\leq 0.\end{cases} \] It is shown that under appropriate hypotheses the oscillatory behavior of equation (1) is characterized by the oscillatory behavior of the associated linearized equation with variable coefficients of the form \[ \dot y+\sum_{i=1}^ m p_ i(t)y(t-\tau_ i(t))=0.\tag{2} \] The main result is stated in the following: Theorem 1. Assume that the following hypothesis holds: (H) For any function \(u\in C[[t_ 0,\infty),R]\), which either eventually decreases to a positive limit or eventully increases to a negative limit, \[ \int_{t_ 0}^ \infty| f(t,u(t-\tau(t)),\ldots,u(t-\tau_ m(t)))| dt=\infty. \] Assume also that there exists a positive constant \(\delta\) and functions \(p_ 1,\dots,p_ m\in C[[t_ 0,\infty),R^ +]\) such that either \[ \begin{cases} f(t,u_ 1,\ldots,u_ m)\geq\sum_{i=1}^ m p_ i(t)u_ i>0 & \text{ for } 0<u_ 1,\ldots,u_ m\leq\delta,\\ f(t,u_ 1,\ldots,u_ m)\leq\sum_{i=1}^ m p_ i(t)u_ i<0 & \text{ for } 0>u_ 1,\ldots,u_ m\geq-\delta\end{cases} \] or \[ \lim_{{(u_ 1,\ldots,u_ m)\to 0\atop u_ i u_ j>0, i,j=1,\ldots,m}}{f(t,u_ 1,\ldots,u_ m)\over p_ 1(t)u_ 1+\cdots+p_ m(t)u_ m}\equiv 1. \] Suppose that every solution of equation (2) oscillates. Then every solution of equation (1) also oscillates.