Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 215, pp. 1-27.

Title: Positive solutions of fractional differential 
       equations with derivative terms
        
Authors: Cuiping Cheng (Hangzhou Dianzi Univ., China)
         Zhaosheng Feng (Univ. of Texas-Pan American, Edinburg, TX, USA)
	 Youhui Su (Xuzhou Institute of Tech., Xuzhou, China)

Abstract:
 In this article, we are concerned with the existence of positive
 solutions for nonlinear fractional differential equation
 whose nonlinearity contains the first-order derivative,
 $$\displaylines{
 D_{0^+}^{\alpha}u(t)+f(t,u(t),u'(t))=0,\quad t\in (0,1),\;
 n-1<\alpha\leq n,\cr
 u^{(i)}(0)=0, \quad  i=0,1,2,\dots,n-2,\cr
 [D_{0^+}^{\beta}u(t)]_{t=1}=0, \quad 2\leq\beta\leq n-2,
 }$$
 where $n>4 $ $(n\in\mathbb{N})$, $D_{0^+}^{\alpha}$ is the
 standard Riemann-Liouville fractional derivative of order $\alpha$
 and $f(t,u,u'):[0,1] \times [0,\infty)\times(-\infty,+\infty)
 \to [0,\infty)$ satisfies the Caratheodory  type condition.
 Sufficient conditions are obtained for the existence of
 at least one or two positive solutions by using  the nonlinear
 alternative of the Leray-Schauder type and Krasnosel'skii's fixed
 point theorem. In addition, several other sufficient
 conditions are established for the existence of at least triple,
 n  or  2n-1 positive solutions. Two examples
 are given to illustrate our theoretical results.

Submitted August 15, 2012. Published November 29, 2012.
Math Subject Classifications: 34A08, 34B18, 34K37.
Key Words: Positive solution; equicontinuity; 
           fractional differential equation; fixed point theorem;
           Caratheodory type condition.