On the Solutions of Four Second-Order Nonlinear Difference Equations

Abstract

This paper deals with the form, the stability character, the periodicity and the global behavior of solutions of the following four rational difference equations x n+1 = ±1 x n (x n−1 ± 1) − 1 x n+1 = ±1 x n (x n−1 ∓ 1) + 1. .

Solutions formulas for some general systems of difference equations

“…The papers, [1], [2] and especially [15] are our main motivation in the present work. The obtained results considerably generalize some existing results in the literature, see [1], [2], [3], [4], [5], [12], [11], [13], [16], [15], [14], [17], [18], [21].…”

“…which is a particular case of system (3.14) with j = k = 0. Noting that system (3.19), generalize the studies in [12], [11], [13]. Then, putting j = k = 0 in the formulas of well-defined solutions of system (3.14), we obtain the following result.…”

“…and these are the formulas given in [12] for the solutions of (3.41). ( 5) Consider a = b = c = 1, system (3.34) becomes (3.43).…”

“…It is clear that if we want to understand our models, we need to know the behavior of the solutions of the equations of the models, and this fact will be possible if we can solve in closed form these equations. One can find in the literature a lot of works on difference equations where explicit formulas of the solutions are given, see for instance [1], [2], [5], [8], [9], [10], [7], [12], [11], [13], [16], [15], [14], [17], [18], [21], [20], [22]. Such type of difference equations and systems is called solvable difference equations.…”