Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 213, pp. 1-14.

Title: Positive solutions to boundary-value problems of p-Laplacian 
       fractional differential equations with a parameter in the boundary
        
Authors: Zhenlai Han (Univ. of Jinan, Shandong, China)
         Hongling Lu (Univ. of Jinan, Shandong, China)
	 Shurong Sun (Univ. of Jinan, Shandong, China)
	 Dianwu Yang (Univ. of Jinan, Shandong, China)

Abstract:
 In this article, we consider the following boundary-value problem
 of nonlinear fractional differential equation with
 $p$-Laplacian operator
 $$\displaylines{
 D_{0+}^\beta(\phi_p(D_{0+}^\alpha u(t)))+a(t)f(u)=0, \quad 0<t<1, \cr
 u(0)=\gamma u(\xi)+\lambda, \quad
 \phi_p(D_{0+}^\alpha u(0))=(\phi_p(D_{0+}^\alpha u(1)))'
 =(\phi_p(D_{0+}^\alpha u(0)))''=0,
 }$$
 where $0<\alpha\leqslant1$, $2<\beta\leqslant 3$ are real numbers,
 $D_{0+}^\alpha, D_{0+}^\beta$ are the standard Caputo fractional derivatives,
 $\phi_p(s)=|s|^{p-2}s$, $p>1$,
 $\phi_p^{-1}=\phi_q$, $1/p+1/q=1$, $0\leqslant\gamma<1$,
 $0\leqslant\xi\leqslant1$, $\lambda>0$ is a parameter,
 $a:(0,1)\to [0,+\infty)$ and $f:[0,+\infty)\to[0,+\infty)$ are continuous.
 By the properties of Green function and Schauder fixed point theorem,
 several  existence and nonexistence results for positive solutions,
 in terms of the parameter $\lambda$ are obtained.
 The uniqueness of positive solution on the parameter $\lambda$ is 
 also studied.  Some examples are presented to illustrate the main results.

Submitted September 5, 2012. Published November 27, 2012.
Math Subject Classifications: 34A08, 34B18, 35J05.
Key Words: Fractional boundary-value problem; positive solution; cone; 
           Schauder fixed point theorem; uniqueness; p-Laplacian operator.