Some results on boundary value problems for functional differential equations (Q1912331)

The boundary value problem \((*)\) \(x''(t)+ f(t, x_t, x'(t))= 0\), \(t\in [0, T]\), \(a_0 x_0- b_0 x' (0)= \varphi\), \(c_0 x(T)+ d_0 x' (T)= \eta\), \(x_0= x(\theta)\), \(\theta\in [-r, 0]\), \(\varphi\in \mathbb{C}\), \(\eta\in \mathbb{R}^n\), is considered. Using a priori estimates and the nonlinear alternative of Leray-Schauder the existence of at least one solution to \((*)\) is demonstrated. The result is a generalization of a recent one for ODE and completes an earlier one on the same problem \((*)\).