Existence of solutions of multi-point boundary value problems on time scales at resonance

By applying the coincidence degree theorem due to Mawhin, we show the existence of at least one solution to the nonlinear second-order differential equation u(Delta del)(t) = f(t, u(t), u(Delta)(t)), t is an element of [0, 1](T), subject to one of the following multi-point boundary conditions: u(0) = Sigma(m)(i=1)alpha(j)u(xi(j)), u(1) =0, and u(0) = Sigma(m)(i=1)alpha(i)u(xi(j))(i) u(1)=0, where T is a time scale such that 0 is an element of T, 1 is an element of T-k, xi(i) is an element of (0, 1) boolean AND T, i = 1,2,..., m, f : [0, 1](T) x R-2 -> R is continuous and satisfies the Caratheodory-type growth conditions.