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

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 $f: mathbb {Z} imes {mathbb R} ^4 o {mathbb R}$ 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.

Keywords

• Difference equation
• uniqueness
• existence.