Electron. J. Diff. Eqns., Vol. 2008(2008), No. 45, pp. 1-12.

Multiple positive solutions for nonlinear second-order m-point boundary-value problems with sign changing nonlinearities

Fuyi Xu, Zhenbo Chen, Feng Xu

Abstract:
In this paper, we study the nonlinear second-order m-point boundary value problem
$$\displaylines{ u''(t)+f(t,u)=0,\quad 0\leq t \leq 1, \cr \beta u(0)-\gamma u'(0)=0,\quad u(1)=\sum _{i=1}^{m-2}\alpha_{i} u(\xi_{i}), }$$
where the nonlinear term $f$ is allowed to change sign. We impose growth conditions on $f$ which yield the existence of at least two positive solutions by using a fixed-point theorem in double cones. Moreover, the associated Green's function for the above problem is given.

Submitted December 27, 2007. Published March 29, 2008.
Math Subject Classifications: 34B15.
Key Words: m-point; boundary-value problem; Green's function; fixed point theorem in double cones.

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Fuyi Xu
School of Mathematics and Information Science
Shandong University of Technology
Zibo, Shandong, 255049, China
email: zbxufuyi@163.com
Zhenbo Chen
School of Mathematics and Information Science
Shandong University of Technology
Zibo, Shandong, 255049, China
email: czb56@sdut.edu.cn
Feng Xu
School of Mathematics and Information Science
Shandong University of Technology
Zibo, Shandong, 255049, China
email: zbxf878@126.com

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