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

Abstract

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 ) $x_{n+1}=f(x_{n-l}, x_{n-k})$ (Adv. Differ. Equ. 2018:447, 2018), is incorrect.

Keywords

• Difference equation
• Unbounded solutions
• Nondecreasing function
• Homogeneous function