Classifications and existence of positive solutions of higher-order nonlinear delay differential equations (Q1567530)

The equation \[ (r(t) x^{(n- 1)}(t))'+ f(t, x(t-\tau))= 0,\quad t\geq t_0,\tag{1} \] is considered, with \(\tau\in [0,+\infty)\), \(r(t)> 0\) for \(t\geq t_0\) and \(f(t,x)x> 0\) for \(x>0\), \(t\geq t_0\). A classification of positive monotone solutions to equation (1) is given, and sufficient conditions for this equation to have different kind of monotone solutions are established.