Existence of solutions to functional boundary value problem of second-order nonlinear differential equation

In this paper, a functional boundary value problem is studied. Based on Mawhin's coincidence degree theory, some existence theorems are obtained in the case of non-resonance and the cases of and at resonance. The results not only generalize and improve some known results of multi-point and integral boundary value problems, but also give some existence theorems for boundary value problems that all their boundary value conditions are relied on both x and x′.