Periodicity and Solutions for Some Systems of Nonlinear Rational Difference Equations

Abstract

In this paper, we investigated the periodic nature and the form of the solutions of nonlinear dierence equations systems of order threewith initial conditions x−2, x−1, x0, y−2, y−1 and y0 are nonzero real numbers.

On the solutions of a second-order difference equation in terms of generalized Padovan sequences

“…Recently, the problem of finding closed-form solutions of rational difference equations and systems of rational of difference equations have gained considerable interest from many mathematicians. In fact, countless papers have been published previously focusing on the aforementioned topic, see for example [5,6,7,16,20] and [21]. Interestingly, some of the solution forms of these equations are even expressible in terms of well-known integer sequences such as the Fibonacci numbers, Horadam numbers and Padovan numbers (see, e.g., [9,11,12,14,22,24,25,26,27,29,34]).…”

Investigating dynamical behaviors of the difference equation \(x_{n+1}= \frac{Cx_{n-5}}{A+Bx_{n-2}x_{n-5}}\)

“…Difference equations have potential applications in various fields of science such as game theory [2,3,5,6,17,18], mathematical biology [15,34], physics, and engineering [11]. Therefore, the study of the qualitative behavior of a nonlinear rational difference equations of higher orders [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] is of paramount importance because of its promising applications.…”

On the solutions of a higher‐order difference equation in terms of generalized Fibonacci sequences

“…Recently, the problem of finding closed-form solutions of rational difference equations and systems of rational of difference equations have gained considerable interest from many mathematicians. In fact, countless papers have been published previously focusing on the aforementioned topic, see for example [5,6,7,16,20] and [21]. Interestingly, some of the solution forms of these equations are even expressible in terms of well-known integer sequences such as the Fibonacci numbers, Horadam numbers and Padovan numbers (see, e.g., [9,11,12,14,22,24,25,26,27,29,34]).…”

“…In fact, as far as we know, there has no known general method to deal with different classes of difference equations solvable in closed-forms. Nevertheless, numerous studies have recently dealt with finding appropriate techniques in solving closed-form solutions of some systems of difference equations (see, e.g., [2,5,6,7,15,23]). …”