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.