The global attractivity of the rational difference equation yn = A + (yn-k/ yn-m)p

Abstract

This paper studies the behavior of positive solutions of the recursive equation yn = A + (yn-k /yn-m) p, n= 0, 1, 2, . . ., with y-s, y-s+1, . . .,y-1 ε (0,∞) and k,m ε {1, 2, 3, 4, . . .}, where s = max{k,m}. We prove that if gcd(k,m) = 1, and p ≤ min{1, (A + 1)/2}, then yn tends to A + 1. This complements several results in the recent literature, including the main result in K. S. Berenhaut, J. D. Foley and S. Stević, The global attractivity of the rational difference equation yn = 1+ yn-k/ y n-m, Proc. Amer. Math. Soc., 135 (2007) 1133-1140.