Electron. J. Diff. Equ., Vol. 2012 (2012), No. 213, pp. 1-14.

Positive solutions to boundary-value problems of p-Laplacian fractional differential equations with a parameter in the boundary

Zhenlai Han, Hongling Lu, Shurong Sun, Dianwu Yang

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.

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Zhenlai Han
School of Mathematical Sciences
University of Jinan
Jinan, Shandong 250022, China
email: hanzhenlai@163.com
Hongling Lu
School of Mathematical Sciences
University of Jinan
Jinan, Shandong 250022, China
email: lhl4578@126.com
Shurong Sun
School of Mathematical Sciences
University of Jinan
Jinan, Shandong 250022, China
email: sshrong@163.com
Dianwu Yang
School of Mathematical Sciences
University of Jinan
Jinan, Shandong 250022, China
email: ss_yangdw@ujn.edu.cn

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