Existence of nontrivial solutions of a rational difference equation

Abstract

We prove that the Putnam difference equation x n + 1 = frac(x n + x n - 1 + x n - 2 x n - 3 , x n x n - 1 + x n - 2 + x n - 3 ), n = 0, 1, ... has a positive solution which is not eventually equal to 1. This provides positive confirmation of a conjecture due to G. Ladas [Open problems and conjectures, J. Difference Equ. Appl. 4 (1998) 497-499].