Existence of solutions for $m$-point fractional boundary value problems.

In this paper, by employing the Leggett-Williams fixed point theorem, we study the existence of three solutions or the following $m$-point fractional boundary value problem \begin{equation*} \begin{cases} {}^cD_{0^+}^\alpha u (t) = f (t, u (t), u' (t)), &    t \in (0, 1),\\ u''(0) = 0, \;\;\;u' (0) = \sum_{i = 1}^{m - 2} a_i u' (\xi_i), \;\;\; u (1) = \sum_{i = 1}^{m - 2} b_i u (\xi_i), \end{cases} \end{equation*} where $2 0$ for $1 \leq i \leq m - 2$ and $\sum_{i = 1}^{m - 2} a_i < 1$, $0 < \sum_{i = 1}^{m - 2} b_i  < 1$, $f \in C ([0, 1] \times [0, \infty); [0, \infty))$.