Existence of nontrivial solutions of a rational difference equation

摘要

We prove that the Putnam difference equation x(n+1) = 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]. (c) 2006 Elsevier Ltd. All rights reserved.