New oscillatory criteria for higher-order nonlinear neutral delay differential equation (Q938737)

The authors consider an \(n\)-th order nonlinear neutral delay differential equation of the form \[ (x(t)-p(t)x(t-\tau))^{(n)}+f(t,x(\sigma(t)))=0,\tag{1} \] where \(n\geq 3\) is an odd number, \(\tau >0\), \(p,\sigma\in C([t_0,\infty), \mathbb{R}^{+})\), \(\sigma(t)\leq t\), \(\lim_{t\to\infty}\sigma(t)=\infty\), \(f\in C([t_0,\infty)\times \mathbb{R}),\mathbb{R})\), \(f(t,u)\) is nondecreasing in \(u\). Additionally, the functions \(p\) and \(f\) satisfy also another conditions listed in the paper. Main results of the paper are sufficient conditions for every solution of (1) to be oscillatory.