Electron. J. Diff. Eqns., Vol. 2008(2008), No. 87, pp. 1-10.

Positive solutions to nonlinear second-order three-point boundary-value problems for difference equation with change of sign

Chunli Wang, Xiaoshuang Han, Chunhong Li

Abstract:
In this paper we investigate the existence of positive solution to the discrete second-order three-point boundary-value problem
$$\displaylines{ \Delta^2 x_{k-1}+ h(k) f(x_k)=0, \quad k \in [1, n], \cr x_0 =0, \quad a x_l = x_{n+1}, }$$
where $n \in [2, \infty)$, $l \in [1, n]$, $0 < a < 1$, $(1-a)l \geq 2$, $(1+a)l\leq n+1$, $f \in C(\mathbb{R}^+,\mathbb{R}^+)$ and $h(t)$ is a function that may change sign on $[1, n]$. Using the fixed-point index theory, we prove the existence of positive solution for the above boundary-value problem.

Submitted March 6, 2008. Published June 11, 2008.
Math Subject Classifications: 39A05, 39A10.
Key Words: Boundary value problem; positive solution; difference equation; fixed point; changing sign coefficients.

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Chunli Wang
Institute of Information Technology
University of Electronic Technology
Guilin, Guangxi 541004, China
email: wangchunliwcl821222@sina.com
Xiaoshuang Han
Yanbian University of Science and Technology
Yanji, Jilin 133000, China
email: petty_hxs@hotmail.com
Chunhong Li
Department of Mathematics
Yanbian University
Yanji, Jilin 133000, China
email: abbccc2007@163.com

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