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

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.

Keywords

• Fractional differential equation
• boundary value problem
• p-Laplacian operator
• Coincidence degree theory
• Resonance