Solvability of a system of nonlinear difference equations of higher order

Abstract

In this paper, we show that the following higher-order system of nonlinear difference equations, xn = x n−k y n−k−l y n−l (an + bnx n−k y n−k−l) , yn = y n−k x n−k−l x n−l (αn + βny n−k x n−k−l) , n ∈ N0, where k, l ∈ N , (an) n∈N 0 , (bn) n∈N 0 , (αn) n∈N 0 , (βn) n∈N 0 and the initial values x−i, y−i , i = 1, k + l , are real numbers, can be solved and some results in the literature can be extended further. Also, by using these obtained formulas, we investigate the asymptotic behavior of well-defined solutions of the above difference equations system for the case k = 2, l = k .

Qualitative behavior and solution of a system of three‐dimensional rational difference equations

“…Our study concerns on analyzing and examining the system properties. For instance, Okumuş and Soykan 5 examined the boundedness, persistence, and periodicity of the nonnegative solutions and the global asymptotic stability of the positive equilibrium points of the system of difference equations Kara et al 6 solved and determined the asymptotic behavior of solutions for of the system of nonlinear difference equations …”

On a Solvable System of Non-Linear Difference Equations With Variable Coefficients

“…, j k l  are real numbers can be solved in [29]. Also, by using the solutions of system (1.6), we investigate the asymptotic behavior of well-defined solutions of the above difference equations system for the case 2, k  .…”

Solutions formulas for some general systems of difference equations

“…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.…”