Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 87, pp. 1-10.
Title: Positive solutions to nonlinear second-order three-point boundary-value
problems for difference equation with change of sign
Authors: Chunli Wang (Univ. of Electronic Technology, Guilin, Guangxi, China)
Xiaoshuang Han (Yanbian Univ. Yanji, Jilin, China)
Chunhong Li (Yanbian Univ. Yanji, Jilin, China)
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.