Electron. J. Diff. Eqns., Vol. 2007(2007), No. 23, pp. 1-10.

Multiple positive solutions for fourth-order three-point p-Laplacian boundary-value problems

Hanying Feng, Meiqiang Feng, Ming Jiang, Weigao Ge

Abstract:
In this paper, we study the three-point boundary-value problem for a fourth-order one-dimensional $p$-Laplacian differential equation
$$ \big(\phi_p(u''(t))\big)''+ a(t)f\big(u(t)\big)=0, \quad t\in (0,1), $$
subject to the nonlinear boundary conditions:
$$\displaylines{ u(0)=\xi u(1),\quad u'(1)=\eta u'(0),\cr (\phi _{p}(u''(0))' =\alpha _{1}(\phi _{p}(u''(\delta))', \quad u''(1)=\sqrt[p-1]{\beta _{1}}u''(\delta), }$$
where $\phi_{p}(s)=|s|^{p-2}s$, $p>1$. Using the five functional fixed point theorem due to Avery, we obtain sufficient conditions for the existence of at least three positive solutions.

Submitted November 30, 2006. Published February 4, 2007.
Math Subject Classifications: 34B10, 34B15, 34B18.
Key Words: Fourth-order boundary-value problem; one-dimensional p-Laplacian; five functional fixed point theorem; positive solution

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Hanying Feng
Department of Mathematics
Beijing Institute of Technology
Beijing 100081, China.
email: fhanying@yahoo.com.cn
Meiqiang Feng
Department of Mathematics
Beijing Institute of Technology
Beijing 100081, China
email: meiqiangfeng@sina.com}
Ming Jiang
Department of Mathematics
Shijiazhuang Mechanical Engineering College
Shijiazhuang 050003, China
email: jiangming27@163.com
Weigao Ge
Department of Mathematics
Beijing Institute of Technology
Beijing 100081, China
email: gew@bit.edu.cn

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