| Summary: | In this article, we consider the multi-point boundary-value problem
for nonlinear fractional differential equations with $p$-Laplacian operator:
$$\displaylines{
D_{0^+}^\beta \varphi_p (D_{0^+}^\alpha u(t))
= f(t,u(t),D_{0^+}^{\alpha - 2} u(t),D_{0^+}^{\alpha - 1} u(t),
D_{0^+}^\alpha u(t)),\quad t \in (0,1), \cr
u(0) = u'(0)=D_{0^+}^\alpha u(0) = 0,\quad
D_{0^+}^{\alpha - 1} u(1) = \sum_{i = 1}^m
{\sigma_i D_{0^+}^{\alpha - 1} u(\eta_i )} ,
}$$
where $2 < \alpha \le 3$, $0 < \beta \le 1$, $3 < \alpha + \beta \le 4$,
$\sum_{i = 1}^m {\sigma_i } = 1$, $D_{0^+}^\alpha$ is the standard
Riemann-Liouville fractional derivative. $\varphi_{p}(s)=|s|^{p-2}s$ is
p-Laplacians operator. The existence of solutions for above fractional
boundary value problem is obtained by using the extension of Mawhin's
continuation theorem due to Ge, which enrich konwn results.
An example is given to illustrate the main result.
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