On a system of difference equations of second order solved in a closed from

Abstract

In this work we solve in closed form the system of difference equationswhere the initial values x−1, x0, y−1 and y0 are arbitrary nonzero real numbers and the parameters a, b and c are arbitrary real numbers with c = 0. In particular we represent the solutions of some particular cases of this system in terms of Tribonacci and Padovan numbers and we prove the global stability of the corresponding positive equilibrium points. The result obtained here extend those obtained in some recent papers.

On a system of difference equations of third order solved in closed form

“…We will see that when a = b = c = d = 1 the solutions are expressed using the famous Teteranacci numbers. In particular, the results obtained here extend those in our work [1].…”

“…Solving difference equations and their systems is a subject that attract the attention of several researchers and a big number of papers is devoted to this line of research where various models are proposed. We can consult for example the papers [1]- [26], for some concrete models of such equations and systems where also we can understands the techniques used in solving these models.…”

“…Recently in [1] and as generalization of the equations and systems studied in [2], [8], [17], [26], we have solved in a closed form the system of difference equations…”

“…Clearly if d = 0, then system (1.2) is nothing other than system (1.1). For the readers interested in the solutions of this system, we refer to [1], where the system (1.1) was been completely solved. Noting also that the system (1.2) can be seen as a generalization of the equation…”

“…Here, we give a closed form for the well defined solutions of the system (1.2) with d = 0. To this end we will use the same change of variables as in [1] to transform the system (1.2) to a linear one and than following the same procedure as in [1] to obtain the closed form of the solutions. To get the solutions of the corresponding linear system we need to solve some fourth order linear difference equations.…”