On property (B) of higher order delay differential equations

Abstract

In this paper we offer criteria for property (B) and additional asymptotic behavior of solutions of the $n$-th order delay differential equations \begin{equation*} \big (r(t)\big [x^{(n-1)}(t)\big ]^{\gamma }\big )^{\prime }=q(t)f\big (x(\tau (t))\big )\,. \end{equation*} Obtained results essentially use new comparison theorems, that permit to reduce the problem of the oscillation of the n-th order equation to the the oscillation of a set of certain the first order equations. So that established comparison principles essentially simplify the examination of studied equations. Both cases $\int ^{\infty } r^{-1/\gamma }(t)\,{t}=\infty $ and $\int ^{\infty } r^{-1/\gamma }(t)\,{t}<\infty $ are discussed.