Averaging technique and oscillation for even order damped delay differential equations (Q2492140)

Oscillatory properties are studied for the even-order nonlinear damped delay differential equation \[ \begin{split} (\Phi(x^{(n-1)}(t)))' + p(t)\Phi(x^{(n-1)}(t)) + f(t, x[\tau_{01}(t)],\dots , x[\tau_{0m}(t)],\dots , \\ x^{(n-1)}[\tau_{n-11}(t)],\dots , x^{(n-1)}[\tau_{n-1m}(t)])=0,\quad t\geq t_0>0, \end{split} \tag{1} \] where \(\Phi(s)=s| s|^{\alpha-1}\) with \(\alpha>0\), and \(n\) is an even number. By using the averaging technique and introducing parameter functions \(H(t,s)\), \(\rho(s)\) and \(k(s)\), the author presents some sufficient conditions for equation (1) to be oscillatory.