Positive solutions for a nonlocal multi-point boundary-value problem of fractional and second order

In this article we study the existence of positive solutions for the nonlocal multi-point boundary-value problem $$displaylines{ u''(t)+f(t, ^{c}D^{alpha}u(t))=0, quad alpha in(0, 1), ext{ a.e. } tin(0, 1), cr u(0)=0, quad u(1)=sum_{k=1}^m a_k u(au_k), quad au_kin(a, b)subset (0, 1). }$$ We also consider the corresponding integral condition, and the two special cases $alpha = 0 $ and $ alpha = 1$