Qualitative Study of Solutions of Some Difference Equations

Abstract

We obtain in this paper the solutions of the following recursive sequences $x_{n+1}=x_n{x}_{n-3}/x_{n-2}\left(\pm 1\pm x_n{x}_{n-3}\right)$, $n=0,1,\dots ,$ where the initial conditions are arbitrary real numbers and we study the behaviors of the solutions and we obtained the equilibrium points of the considered equations. Some qualitative behavior of the solutions such as the boundedness, the global stability, and the periodicity character of the solutions in each case have been studied. We presented some numerical examples by giving some numerical values for the initial values and the coefficients of each case. Some figures have been given to explain the behavior of the obtained solutions in the case of numerical examples by using the mathematical program Mathematica to confirm the obtained results.