Integral averaging technique and oscillation of certain even order delay differential equations (Q1826774)

This paper is devoted to the oscillatory behaviour of even-order delay differential equations of the form \[ (| x^{(n-1)}(t)|^{\alpha- 1}x^{(n-1)}(t))'+ F(t, x[g(t)])= 0,\;n\text{ even}, \] where \(\alpha> 0\) is a constant, \(g\in C([t_0,\infty),\mathbb{R})\), \(\lim_{t\to\infty}\,g(t)= \infty\), and \(F\in C([t_0, \infty)\times \mathbb{R},\mathbb{R})\), \(\text{sgn\,}F(t, x)= \text{sgn\,}x\), \(t\geq t_0\). To this end the authors use the generalized Riccati technique and the averaging technique.