Multi-point boundary value problems on time scales at resonance

We study the nonlinear second order differential equation on a time scaleu▵∇(t)=f(t,u(t),u▵(t))+e(t),t∈(0,1)∩T, subject to the multi-point boundary conditionsu▵(0)=0,u(1)=∑i=1mαiu(ηi)andu(0)=0,u(1)=∑i=1mαiu(ηi), where TT is a time scale such that 0,1∈T0,1∈T, ηi∈(0,1)∩Tηi∈(0,1)∩T, i=1,…,mi=1,…,m, and f is a continuous function satisfying a Carathéodory-type growth condition and e is a Lebesgue integrable function. Our existence results are obtained by applying a coincidence degree theorem due to Mawhin.