On a special case of a difference equation with powers

Abstract

Investigation of the long-term behaviour of solutions to the nonlinear difference equation (Formula Presented), where A, p, q ∈ R, k, m ∈ N<inf>0</inf>, k ̸= m, was proposed by S. Stević about twenty years ago. A very special case of the equation (p = 1, r = 2, m = 0) has been recently considered in [J. Appl. Math. Comput. 67(2021), 423–437]. We show that the main results therein are known or have some inaccuracies. Among other things, we show that the boundedness result therein is a consequence of some known results and using one of our previous methods we give a better upper bound for positive solutions to the equation, show that the proof of the global convergence result therein is not correct and provide a complete proof of a generalization, and also show that the results on semi-cycles of positive solutions are not correct and present some correct ones. Several comments are also given and some analyses are conducted.