Electronic Journal of Differential Equations,
Vol. 2014 (2014), No. 58, pp. 1-10.

Title: Solvability of fractional multi-point boundary-value problems with
       p-Laplacian operator at resonance

Authors: Tengfei Shen (China University of Mining and Tech., Xuzhou, China)
         Wenbin Liu   (China University of Mining and Tech., Xuzhou, China)
         Taiyong Chen (China University of Mining and Tech., Xuzhou, China)
         Xiaohui Shen (China University of Mining and Tech., Xuzhou, China)

Abstract:
 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.

Submitted January 13, 2013. Published February 28, 2014.
Math Subject Classifications: 34A08, 34B15.
Key Words: Fractional differential equation;  boundary value problem;
           p-Laplacian operator; Coincidence degree theory; Resonance.