Electronic Journal of Differential Equations,
Conference 03 (1999), pp. 63-73.

Title:  Uniqueness implies existence for discrete fourth order Lidstone 
boundary-value problems

Authors:  Johnny Henderson (Auburn Univ., Auburn, AL, USA) 
  Alvina M. Johnson (Auburn Univ., Auburn, AL, USA)
 
Abstract: 
We study the fourth order difference equation
$$u(m+4) = f(m, u(m), u(m+1),u(m+2), u(m+3))\,,$$
where $f: \mathbb {Z} \times {\mathbb R} ^4 \to {\mathbb R}$ 
is continuous and the equation $u_5 = f(m, u_1, u_2, u_3,$ $ u_4)$ 
can be solved for $u_1$ as a continuous function of $u_2, u_3, u_4, u_5$ 
for each $m \in {\mathbb Z}$. It is shown that the uniqueness of
solutions implies the existence of solutions for Lidstone boundary-value 
problems on ${\mathbb Z}$. To this end we use shooting and topological 
methods.

Published July 10, 2000.
Math Subject Classifications: 39A10, 34B10, 34B15.
Key Words: Difference equation; uniqueness; existence.