Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 234, pp. 1-11.

Title: Existence and uniqueness of positive solutions to higher-order
       nonlinear fractional differential equation with integral 
       boundary conditions
        
Author: Chenxing Zhou (Changchun Normal Univ., Jilin, China)

Abstract:
 In this article, we consider the nonlinear fractional order
 three-point boundary-value problem
 $$\displaylines{
 D_{0+}^{\alpha} u(t) + f(t,u(t))=0, \quad 0 < t < 1,\cr
 u(0) = u'(0) = \dots = u^{(n-2)}(0)=0, \quad u^{(n-2)}(1) =
 \int_0^\eta u(s)ds,
 }$$
 where $D_{0+}^{\alpha}$ is the standard Riemann-Liouville fractional
 derivative, $n-1 < \alpha \leq n$, $n \geq 3$. By using a fixed-point
 theorem  in partially ordered sets,  we obtain sufficient conditions
 for the existence and uniqueness of positive and nondecreasing
 solutions to  the above boundary value problem. 

Submitted September 19, 2012. Published December 21, 2012.
Math Subject Classifications: 26A33, 34B18, 34B27.
Key Words: Partially ordered sets; fixed-point theorem; positive solution.