A multi-point boundary value problem with two critical conditions

We apply a coincidence degree theorem due to Mawhin to show the existence of at least one solution of the nonlinear second order multi-point boundary value problem u″(t)=f(t,u(t),u′(t)),t∈(0,1),u′(0)=u′(η),∑i=1nαiu(ηi)=u(1), where 0<η≤1, 0<ηi<1, i=1,…,n, ∑i=1nαi=∑i=1nαiηi=1, n≥2, and f:[0,1]×R2→R satisfying the Carathéodory conditions. In our setting both boundary conditions are responsible for resonance.