The global attractivity of the rational difference equation yn = 1+ yn-k/yn-m

Abstract

This paper studies the behavior of positive solutions of the recursive equation yn = 1+ yn-k/yn-m, n = 0, 1, 2,., with y-s, y-s+1,.,y-1ε (0,ε) and k,m. {1, 2, 3, 4,.}, where s = max{k,m}. We prove that if gcd(k,m) = 1, with k odd, then yn tends to 2, exponentially. When combined with a recent result of E. A. Grove and G. Ladas (Periodicities in Nonlinear Difference Equations, Chapman & Hall/CRC Press, Boca Raton (2004)), this answers the question when y = 2 is a global attractor.