Solvability of some fractional-order three point boundary value problems

Abstract

In this work, we prove the existence of at least one solution of the two fractional-order three point boundary value problems:
$$
\left\{\begin{array}{c}
D^\beta u(t)+\lambda a(t) f(u(t))=0, \beta \in(1,2], t \in(0,1), \\
u(0)=0, \alpha u(\eta)=u(1), 0<\eta<1,0 \leq \alpha \eta<1 .
\end{array}\right.
$$
and
$$
\left\{\begin{array}{c}
D^\beta u(t)+\lambda a(t) f(u(t))=0, \beta \in(1,2], t \in(0,1), \\
u^{\prime}(0)=0, \alpha u^{\prime}(\eta)=u(1), 0<\eta<1,0 \leq \alpha \eta<1 .
\end{array}\right.
$$