Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 174, pp. 1-11.
Title: Uniqueness of positive solutions for fractional q-difference
boundary-value problems with p-Laplacian operator
Authors: Fenghua Miao (Changchun Normal Univ., Changchun, Jilin, China)
Sihua Liang (Changchun Normal Univ., Changchun, Jilin, China)
Abstract:
In this article, we study the fractional q-difference boundary-value
problems with p-Laplacian operator
$$\displaylines{
D_{q}^{\gamma}(\phi_p(D_{q}^{\alpha}u(t))) + f(t,u(t))=0, \quad
0 < t < 1, \; 2 < \alpha < 3,\cr
u(0) = (D_qu)(0) = 0, \quad (D_qu)(1) = \beta (D_qu)(\eta),
}$$
where $0 < \gamma < 1$, $2 < \alpha < 3$,
$0<\beta\eta^{\alpha-2}<1$, $D_{0+}^{\alpha}$ is the
Riemann-Liouville fractional derivative, $\phi_p(s)=|s|^{p-2}s$,
p>1. By using a fixed-point theorem in partially ordered sets, we
obtain sufficient conditions for the existence and uniqueness of
positive and nondecreasing solutions.
Submitted June 15, 2013. Published July 29, 2013.
Math Subject Classifications: 39A13, 34B18, 34A08.
Key Words: Fractional q-difference equations; partially ordered sets;
fixed-point theorem; positive solution.