Positive solutions for a nonlocal multi-point boundary-value problem of fractional and second order. (English) Zbl 1291.34009
Summary: We study the existence of positive solutions for the nonlocal multi-point boundary-value problem
\[
\begin{gathered} u''(t)+ f(t,^cD^\alpha u(t))= 0,\quad \alpha\in (0,1),\text{ a.e. }t\in (0,1),\\ u(0)= 0,\quad u(1)= 0,\quad u(1)= \sum^m_{k=1} a_k u(\tau_k),\quad\tau_k\in (a,b)\subset (0,1).\end{gathered}
\]
We also consider the corresponding integral condition, and the two special cases \(\alpha=0\) and \(\alpha=1\).
MSC:
34A08 | Fractional ordinary differential equations |
34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |