Fourth Mississippi State Conference on Differential Equations and Computational Simulations,
Electron. J. Diff. Eqns., Conf. 03, 1999, pp. 63-73.

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

Johnny Henderson & Alvina M. Johnson

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.

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Johnny Henderson
Department of Mathematics, Auburn University
Auburn, AL 36849, USA
email: hendej2@mail.auburn.edu
Alvina M. Johnson
Department of Mathematics, Auburn University
Auburn, AL 36849, USA
e-mail: johnso25@mail.fvsu.edu

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