On the solvability of a certain boundary value problem for a nonlinear singular functional-differential equation (Q1901904)

The author considers a multipoint boundary value problem for a singular nonlinear functional-differential equation of the type \(\varphi (t) x^{(n)} (t) = (Fx) (t)\), \(t \in [a,b]\), \(x^{(j)} (a_i) = \alpha_{ij}\), \(j = 0,1, \dots, \rho_i - 1\), \(i = 1, \dots, m\), under the assumptions \(\varphi (t) = \prod^m_{i = 1} (t - \alpha_i)^{\mu_i}\), \(m \geq 2\), \(a = \alpha_1 < \cdots < \alpha_m = b\); \(n \geq 2\), \(\mu_i\), \(\rho_i \in \{0,1, \dots, n - 1\}\), \(0 < \rho_i + \mu_i \leq n\), \(i = 1, \dots, m\). In seven theorems he presents results on existence and uniqueness of solutions of the problem and on properties of its Green operator.