Classifications and existence of positive solutions of higher-order nonlinear neutral differential equations. (Q1415267)

Here, necessary and sufficient conditions are obtained for the existence and classification of positive solutions to higher-order nonlinear neutral differential equations of the form \[ \left[r(t)\left(x(t)-\sum_{i=1}^m P_i(t)x(t-\tau_i)\right)^{(n-1)}\right]'+f(t, x(t-\sigma_1), \ldots, x(t-\sigma_l))=0, \] where \(n\) is a positive integer, \(t\geq t_0\); \(r, P_i\in C([t_0, \infty), \mathbb R),\) \(r(t)>0,\) \(P_i\geq 0,\) \(\tau_i>0,\) \(\sigma_j\geq 0,\) \(i=1, \ldots, m,\) \(j=1, \ldots, l\); \(f(t, \mu_1, \ldots, \mu_l)\in C([t_0, \infty)\times \mathbb R^l, \mathbb R)\) and \(f>0\) as \(\mu_j>0,\) \(j=1, \ldots, l\).