Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 54, pp. 1-10.
Title: Existence of solutions for multi-point nonlinear differential
equations of fractional orders with integral boundary conditions
Authors: Gang Wang (Univ. of Mining and Technology, Xuzhou, China)
Wenbin Liu (Univ. of Mining and Technology, Xuzhou, China)
Can Ren (Univ. of Mining and Technology, Xuzhou, China)
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 05, 2012.
Math Subject Classifications: 34B15.
Key Words: Fractional differential equation; boundary value problem;
fixed point theorem; existence and uniqueness.