Electron. J. Diff. Equ., Vol. 2010(2010), No. 84, pp. 1-6.

Uniqueness and parameter dependence of solutions of fourth-order four-point nonhomogeneous BVPs

Jian-Ping Sun, Xiao-Yun Wang

Abstract:
In this article, we investigate the fourth-order four-point nonhomogeneous Sturm-Liouville boundary-value problem
$$\displaylines{ u^{(4)}(t)=f(t,u(t)),\quad t\in [0,1], \cr \alpha u(0)-\beta u'(0)=\gamma u(1)+\delta u'(1)=0, \cr au''(\xi _1)-bu'''(\xi _1)=-\lambda ,\quad cu''(\xi _2)+du'''(\xi _2)=-\mu , }$$
where $0\leq \xi _1<\xi _2\leq 1$ and $\lambda$ and $\mu $ are nonnegative parameters. We obtain sufficient conditions for the existence and uniqueness of positive solutions. The dependence of the solution on the parameters $\lambda$ and $\mu $ is also studied.

Submitted September 21, 2009. Published June 18, 2010.
Math Subject Classifications: 34B08, 34B10.
Key Words: Nonhomogeneous; fourth-order; four-point; Sturm-Liouville; boundary-value problem; positive solution; uniqueness; dependence on parameter

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Jian-Ping Sun
Department of Applied Mathematics, Lanzhou University of Technology
Lanzhou, Gansu 730050, China
email: jpsun@lut.cn
Xiao-Yun Wang
Department of Applied Mathematics, Lanzhou University of Technology
Lanzhou, Gansu 730050, China
email: catherine699@163.com

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