Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 36, pp. 1-12.

Title: Positive solutions for boundary-value problems of nonlinear
fractional differential equations
       
Author: Shuqin Zhang (Univ. of Mining and Tech., Beijing, China)

Abstract: 
 In this paper, we  consider the existence and multiplicity
 of positive solutions for the nonlinear fractional differential
 equation boundary-value problem
 $$\displaylines{
 \mathbf{D}_{0+}^\alpha u(t)=f(t,u(t)),\quad 0<t<1\cr
 u(0)+u'(0)=0,\quad   u(1)+u'(1)=0
 }$$
 where $1<\alpha\leq 2$ is a real number, and
 $\mathbf{D}_{0+}^\alpha$ is the Caputo's fractional derivative,
 and $f:[0,1]\times[0,+\infty)\to [0,+\infty)$ is
 continuous. By means of a fixed-point theorem on cones, some
 existence and multiplicity results of positive solutions are
 obtained.
 
Submitted November 30, 2005. Published March 21, 2006.
Math Subject Classifications: 34B15.
Key Words: Caputo's fractional derivative;
 fractional differential equation; boundary-value problem; 
 positive solution; fractional Green's function; fixed-point theorem.

An addendum as attached on November 9, 2009. It corrects the definition of n in Lemmas 2.2 and 2.3. See the last page of this article.