Electron. J. Diff. Equ., Vol. 2013 (2013), No. 174, pp. 1-11.

Uniqueness of positive solutions for fractional q-difference boundary-value problems with p-Laplacian operator

Fenghua Miao, Sihua Liang

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.

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Fenghua Miao
College of Mathematics, Changchun Normal University
Changchun 130032, Jilin, China
email: mathfhmiao@163.com
Sihua Liang
College of Mathematics, Changchun Normal University
Changchun 130032, Jilin, China
email: liangsihua@163.com, Phone 08613578905216

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