In this paper, we considered a class of $m$-point boundary-value
problem of fractional differential equations at resonance with
$p$-Laplacian operator in the following:
\begin{eqnarray*}
\left\{
\begin{array}{ll}
D_{0^+}^\beta \varphi_p
(D_{0^+}^\alpha u(t)) =
f(t,u(t),D_{0^+}^{\alpha - 2}
u(t),D_{0^+}^{\alpha - 1} u(t),
D_{0^+}^\alpha u(t)),\quad t
\in (0,1), \\ u(0) =
u’(0)=D_{0^+}^\alpha u(0) =
0,\quad D_{0^+}^{\alpha - 2}
u(1) = \sum_{i = 1}^{m-2} {a_i
D_{0^+}^{\alpha - 2} u(\eta_i
)} , \end{array} \right.
\end{eqnarray*} where $2 <
\alpha \le 3$, $\eta_1
<\eta_2
<\cdots<
\eta_{m-2}$, $0 < \beta
\le 1$, $3 < \alpha +
\beta \le 4$, $\sum_{i
= 1}^{m-2}a_i\eta_i = 1$,
$D_{0^+}^\alpha$ denote the Riemann-Liouville
fractional derivative,
$\varphi_{p}(s)=|s|^{p-2}s$
is $p$-Laplacian operator. The existence of solutions to above problem
is obtained by using the extension of Mawhin’s continuation theorem. It
is note that our method dropped a usual condition in the process of
investigating above problem. So, in some sense, we got a new result
under weaker condition than previous ones