On the Solutions of Three-Dimensional Rational Difference Equation Systems

In this paper, we are interested in a technique for solving some nonlinear rational systems of difference equations of third order, in three-dimensional case as a special case of the following system:xn+1 � (ynzn− 1/yn ± xn− 2), yn+1 � (znxn− 1/zn ± yn− 2), and zn+1 � (xnyn− 1/xn ± zn− 2) with initial conditions x− 2, x− 1, x0, y− 2, y− 1, y0, z− 2, z− 1, and z0 are nonzero real numbers. Moreover, we study some behavior of the systems such as the boundedness of solutions for such systems. Finally, we present some numerical examples by giving some numerical values for the initial values of each case. Some figures have been given to explain the behavior of the obtained solutions in the case of numerical examples by using the mathematical program MATLAB to confirm the obtained results.


Introduction
We believe that difference equations, also referred to as recursive sequence, are a hot topic here as there has been increasing interest in the study of qualitative analysis of difference equations and systems of difference equations. Difference equations appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations having applications in biology, ecology, economics, physics, computer sciences, and so on. Although difference equations are very simple in form, it is extremely difficult to understand thoroughly the behaviors of their solution (see [1][2][3][4][5][6][7][8][9] and the references cited therein). Recently, a great effort has been made in studying the qualitative analysis of rational difference equations and rational difference system (see ).
e study of the nonlinear rational difference equations is quite challenging and rewarding [2,8]. e results about these equations offer prototypes towards the development of the basic theory of the global behavior of nonlinear difference equations [1]. Recently, many researchers have investigated the behavior of the solution of difference equations. Difference equations arise in the situations in which the discrete values of the independent variable involve. Many practical phenomena are modeled with the help of difference equations [1]. In engineering, difference equations arise in control engineering, digital signal processing, electrical networks, etc. In social sciences, difference equations arise to study the national income of a country and then its variation with time, Cobweb phenomenon in economics, etc.
ere are many papers related to the difference equation system, for example, the periodicity of the positive solutions of the rational difference equations system: Khan et al. [6] studied the equilibrium points, local asymptotic stability of an equilibrium point, instability of equilibrium points, periodicity behavior of positive solutions, and global character of an equilibrium point of a fourth-order system of rational difference equations of the form: x n+1 � αx n− 3 β + cy n y n− 1 y n− 2 y n− 3 , Elabbasy et al. [7] has obtained the solution of particular cases of the following general system of difference equations: x n+1 � a 1 + a 2 y n a 3 z n + a 4 x n− 1 z n , In [36], Elsayed et al. dealt with the solutions of the systems of the difference equations: x n+1 � 1 x n− p y n− p , y n+1 � x n− p y n− p x n− q y n− q , x n+1 � 1 x n− p y n− p z n− p , y n+1 � x n− p y n− p z n− p x n− q y n− q z n− q , z n+1 � x n− q y n− q z n− q x n− r y n− r z n− r . (4) Kurbanli [13][14][15] investigated the behavior of the solutions of the difference equation systems: In [21], Yalcinkaya et al. studied the periodic character of the following two systems of difference equations: where the initial values are nonzero real numbers for x (1) 0 , x (2) 0 , . . . , x (k) 0 ≠ 1. In [37], Zhang et al. studied the boundedness, the persistence, and global asymptotic stability of the positive solutions of the system of difference equations: Zkan and Kurbanli [38] have investigated the periodical solutions of the following system of third-order rational difference equations: Journal of Mathematics Similar to the references above, this paper is devoted to obtain the form of the solutions of the following third-order systems of rational difference equations: where the initial conditions x − 2 , x − 1 , x 0 , y − 2 , y − 1 , y 0 , z − 2 , z − 1 , and z 0 are nonzero real numbers.

The System:
x n+1 = � (y n z n−1 /y n + x n−2 ), y n+1 = � (z n x n−1 / z n + y n−2 ), and z n+1 = � (x n y n−1 /x n + z n−2 ) We obtain the form of solutions of the following system of difference equations: Theorem 1. If x n , y n , z n be the solution of (10), then

Journal of Mathematics
Proof. Obviously, results are true for n � 0. Suppose that it is also true for n − 1, i.e., Journal of Mathematics From equation (10), it follows that

Journal of Mathematics
Now, one can see that Moreover, from (10), one has en, 6

Journal of Mathematics
Finally, from equation (10), us, In a similar way, other relations can also be proved. □ Lemma 1. If x n , y n , z n is the +ve solution of (10), then it is bounded as well as converge to zero.

The System:
x n+1 = � (y n z n−1 /y n + x n−2 ), y n+1 = � (z n x n−1 / z n + y n−2 ), and z n+1 = � (x n y n−1 /x n − z n−2 ) Here, we will discuss solutions of the following system: where n ∈ N 0 and nonzero initial conditions such that Theorem 2. If x n , y n , z n are solutions of (21), then solutions of (21) are represented by the following formulas for n � 0, 1, . . . :