Electron. J. Diff. Equ., Vol. 2012 (2012), No. 54, pp. 1-10.

Existence of solutions for multi-point nonlinear differential equations of fractional orders with integral boundary conditions

Gang Wang, Wenbin Liu, Can Ren

Abstract:
In this article, we study the multi-point boundary-value problem of nonlinear fractional differential equation
$$\displaylines{ D^\alpha_{0+}u(t)=f(t,u(t)),\quad 1<\alpha\leq 2,\; t\in[0,T],\; T>0,\cr I_{0+}^{2-\alpha}u(t)|_{t=0}=0,\quad D_{0+}^{\alpha-2}u(T)=\sum_{i=1}^ma_i I_{0+}^{\alpha-1}u(\xi_i), }$$
where $D_{0^+}^\alpha$ and $I_{0^+}^\alpha$ are the standard Riemann-Liouville fractional derivative and fractional integral respectively. Some existence and uniqueness results are obtained by applying some standard fixed point principles. Several examples are given to illustrate the results.

Submitted November 27, 2011. Published April 5, 2012.
Math Subject Classifications: 34B15.
Key Words: Fractional differential equation; boundary value problem; fixed point theorem; existence and uniqueness.

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Gang Wang
Department of mathematics
University of Mining and Technology
Xuzhou 221008, China
email: wangg0824@163.com
Wenbin Liu
Department of mathematics
University of Mining and Technology
Xuzhou 221008, China
email: wblium@163.com
Can Ren
Department of mathematics
University of Mining and Technology
Xuzhou 221008, China
email: rencan0502@163.com

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