On the Solutions of Four Rational Difference Equations Associated to Tribonacci Numbers

Abstract

In this study, we investigate the form of solutions, stability character and asymptotic behavior of the following rational difference equationwhere the inital values x −1 and x 0 and α, β and γ with γ = 0 are nonnegative real numbers. Its solutions are associated with generalized Tribonacci numbers.

On the Solutions of Four Second-Order Nonlinear Difference Equations

“…Difference equation or discrete dynamical system is a diverse field which impact almost every branch of pure and applied mathematics. Lately, there has been great interest in the study of solving difference equations and systems of difference equations, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. In these studies, the authors deal with the closed-form, stability, periodicity, boundedness and asymptotic behavior of solutions of nonlinear difference equations and systems of difference equations.…”

A Review on the Solutions of Difference Equations via Integer Sequences such as Fibonacci Numbers and Tribonacci Numbers

“…if n = 1, 3, 5, ..., Next, they in [22] studied the following difference equation…”

“…Then, Okumuş and Soykan in [19] considered the following four difference equations…”

“…Also, in [21], they examined the following systems of difference equations…”

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.…”