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 01$,
$\phi_p^{-1}=\phi_q$, $1/p+1/q=1$, $0\leqslant\gamma<1$,
$0\leqslant\xi\leqslant1$, $\lambda>0$ 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.