Electron. J. Diff. Equ., Vol. 2014 (2014), No. 58, pp. 1-10.

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

Tengfei Shen, Wenbin Liu, Taiyong Chen, Xiaohui Shen

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.

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Tengfei Shen
College of Sciences
China University of Mining and Technology
Xuzhou 221008, China
email: shentengfei1987@126.com
Wenbin Liu
College of Sciences
China University of Mining and Technology
Xuzhou 221008, China
email: wblium@163.com
Taiyong Chen
College of Sciences
China University of Mining and Technology
Xuzhou 221008, China
email: taiyongchen@cumt.edu.cn
Xiaohui Shen
College of Sciences
China University of Mining and Technology
Xuzhou 221008, China
email: shenxiaohuicool@163.com

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