On the global attractivity of two nonlinear difference equations

Abstract

The main objective of this paper is to study some qualitative behavior of the solutions of the two difference equations $$ {x_{n + 1}} = {{{a{x_n} - b{x_{n - k}}}} \left/ {{\left( {c{x_n} - d{x_{n - k}}} \right)}} \right.}\quad n = 0,\,1,\,2, \ldots, $$and $$ {x_{n + 1}} = {{{a{x_{n - k}} + b{x_n}}} \left/ {{\left( {c{x_n} - d{x_{n - k}}} \right)}} \right.}\quad n = 0,\,1,\,2, \ldots, $$where the initial conditions x − k, ⋯, x − 1, x0 are arbitrary positive real numbers and the coefficients a, b, c, and d are positive constants, while k is a positive integer number.