Electron. J. Diff. Eqns., Vol. 2007(2007), No. 112, pp. 1-7.

Multiple positive solutions for nonlinear third-order three-point boundary-value problems

Li-Jun Guo, Jian-Ping Sun, Ya-Hong Zhao

Abstract:
This paper concerns the nonlinear third-order three-point boundary-value problem
$$\displaylines{ u'''(t)+h(t)f(u(t))=0, \quad t\in (0,1), \cr u(0)=u'(0)=0, \quad u'(1)=\alpha u'(\eta ), }$$
where $0<\eta <1$ and $1<\alpha <\frac 1\eta $. First, we establish the existence of at least three positive solutions by using the well-known Leggett-Williams fixed point theorem. And then, we prove the existence of at least $2m-1$ positive solutions for arbitrary positive integer $m$.

Submitted April 18, 2007. Published August 18, 2007.
Math Subject Classifications: 34B10, 34B18.
Key Words: Third-order boundary value problem; positive solution; three-point boundary value problem; existence; cone; fixed point.

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Li-Jun Guo
Department of Applied Mathematics
Lanzhou University of Technology
Lanzhou, Gansu, 730050, China
email: school520@lut.cn
Jian-Ping Sun
Department of Applied Mathematics
Lanzhou University of Technology
Lanzhou, Gansu, 730050, China
email: jpsun@lut.cn
Ya-Hong Zhao
Department of Applied Mathematics
Lanzhou University of Technology
Lanzhou, Gansu, 730050, China
email: zhaoyahong88@sina.com

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