Positive solutions of nonlinear fractional three-point boundary-value problem

Abstract

In this paper, we study the existence of positive solutions to the boundary-value problem with fractional order
\begin{eqnarray*}
\begin{split}
(^{C}_{a}D^{\alpha}y)(t)+q(t)f(y)&=0, \hskip 0.5cm 0\leq a<t<b, \hskip 0.5cm 1<\alpha <2,\\
\\&y(a)=0, \hskip 0.5 cm y(b)=\beta y(\eta),
\end{split}
\end{eqnarray*}
where $a<\eta<b$ and $\beta (\eta-a)-b+a\neq 0$. We prove the existence of at least one positive
solution when $f$ is either superlinear or sublinear using the well-known Guo's fixed point theorem in
cones. Moreover, the convexity and concavity of the solutions are investigated with respect to the behavior of
the function $ q $.