Note on difference equations with the right-hand side function nonincreasing in each variable

Abstract

We present an example of a difference equation of arbitrary order, possessing the right-hand side function that is homogeneous to a certain degree and nonincreasing in each variable, which has a unique positive equilibrium, as well as solutions that do not converge to the equilibrium. The example shows that the main result in the paper: O. Moaaz, Dynamics of difference equation x(n+1) = f (x(n-l), x(n-k)) (Adv. Differ. Equ. 2018:447, 2018), is incorrect.We present an example of a difference equation of arbitrary order, possessing the right-hand side function that is homogeneous to a certain degree and nonincreasing in each variable, which has a unique positive equilibrium, as well as solutions that do not converge to the equilibrium. The example shows that the main result in the paper: O. Moaaz, Dynamics of difference equation x(n+1) = f (x(n-l), x(n-k)) (Adv. Differ. Equ. 2018:447, 2018), is incorrect.