Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 40, pp. 1-7.
Title: Existence of solutions to higher-order discrete three-point problems
Author: Douglas R. Anderson (Concordia College, Moorhead, MN, USA)
Abstract:
We are concerned with the higher-order discrete three-point boundary-value
problem
$$\displaylines{
(\Delta^n x)(t)=f(t,x(t+\theta)), \quad t_1\le t\le t_3-1,
\quad -\tau\le \theta\le 1\cr
(\Delta^i x)(t_1)=0, \quad 0\le i\le n-4, \quad n\ge 4 \cr
\alpha (\Delta^{n-3}x)(t)-\beta (\Delta^{n-2}x)(t)=\eta(t),
\quad t_1-\tau-1\le t\le t_1 \cr
(\Delta^{n-2}x)(t_2)=(\Delta^{n-1}x)(t_3)=0.
}$$
By placing certain restrictions on the nonlinearity and the distance
between boundary points, we prove the existence of at least one
solution of the boundary value problem by applying the Krasnoselskii
fixed point theorem.
Submitted August 19, 2002. Published April 15, 2003.
Math Subject Classifications: 39A10.
Key Words: Difference equations; boundary-value problem; Green's function;
fixed points; cone.