ON SOLUTIONS OF THREE-DIMENSIONAL SYSTEM OF DIFFERENCE EQUATIONS WITH CONSTANT COEFFICIENTS

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Introduction
In recent years, many authors have been interested in non-linear difference equations and non-linear systems of difference equations [1-3, 5, 6, 8-10, 12-14, 20-23, 25-41].One of the important topics in this field is the solvability of non-linear difference equations or non-linear difference equations systems.There are different methods for obtaining solutions of non-linear difference equations and non-linear systems of difference equations (two-dimensional or three-dimensional).One of the methods for solving non-linear difference equations and non-linear difference equations systems is to use the change of variables.Then, aforementioned difference equations or their systems can be reduced to a linear difference equation with constant or variable coefficients.The other method is to use induction method.For instance, El-Metwally et al. solved the following non-linear difference equations by using induction method in [7].In addition, they investigated the behavior of the solutions of difference equations in (1).
In addition, Ibrahim et al. in [15] obtained the solutions of the following difference equation where initial conditions x −2 , x −1 , x 0 are non-zero real numbers and (a n ) n∈N0 , (b n ) n∈N0 are real two-periodic sequences.They used induction method to acquire the solutions of equation ( 2).
Ahmed et al. in [4], investigated the periodic character and the form of the solutions of the following two-dimensional difference equations systems by induction with x −j , y −j , j = 0, 2 are nonzero real numbers.
A few years ago, in [16], Kara and Yazlik showed that the following two-dimensional difference equations system where the initial conditions x −j , y −j , j ∈ {1, 2, 3} and the sequences (a n ) n∈N0 , (b n ) n∈N0 , (α n ) n∈N0 , (β n ) n∈N0 are non-zero real numbers can be solved in closedform.In addition, they acquired the forbidden set of the initial values x −j , y −j , j = 1, 3 for system (4) and gave a study of the long-term behavior of its solutions when for every n ∈ N 0 , all the sequences (a n ) , (b n ) , (α n ) , (β n ) are constant.They used the change of variables to acquire the solutions of system (4).
Recently, the authors of [11], obtained exact formulas for the solutions of the two-dimensional system of difference equations , n ∈ N 0 , (5) where and (d n ) n∈N0 are non-zero real sequences.Note that, system (4) can be obtained by taking k = 2 in system (5).
In addition, Kara and Yazlik showed that the following two-dimensional system of non-linear difference equations where and the initial values x −i , y −i , i = 1, k + l, are real numbers can be solved in [17].Also, by using these obtained formulas, they investigated the asymptotic behavior of well-defined solutions of system (6) for the case k = 2, l = k.They used the change of variables to obtain the solutions of system (6).
Quite recently, authors of [18] showed that three-dimensional system of difference equations , where the initial values x −j , y −j , z −j , j ∈ {1, 2, 3} and the sequences ( are non-zero real numbers, can be solved in closed form.They used the change of variables to acquire the solutions of system (7).
Finally, in [19], Kara et al. obtained explicit formulas for the well defined solutions of the following system of difference equations , where k ∈ N 0 , the initial conditions x −i , y −i , z −i , i = 0, 3k and the sequences They used change of variables to obtain the solutions of system (8).
In this paper, we study the following three-dimensional system of difference equations , where the initial values x −i , y −i , z −i , i = 1, 3 and the parameters a, b, c, d, e, f are non-zero real numbers.We solve system (9) in closed form by using convenient transformation.We obtain the solutions of system (9) in explicit form according to the parameters a, c and e are equal 1 or not equal 1.In addition, we get periodic solutions of system (9).Finally, we define the forbidden set of the initial conditions by using the obtained formulas.Note that system (9) is three-dimensional form of equation (2) and system (4).
Definition 1. (Periodicity) Let (x n , y n , z n ) n≥−3 be solution to difference equations system (9).The solution (x n , y n , z n ) n≥−3 is said to be eventually periodic p if is said that the solution is periodic with period p.

Particular Cases of System (9)
Now, we will examine the solutions in 8 different cases depending on whether the parameters a, c and e are equal 1 or not equal 1.
In this case, the solutions of system (9) can be written in the following form In this case, solutions of system (9) are as follows for m ≥ −1 and t ∈ {1, 2, 3}.
In this case, the solutions of system ( 9) can be written in the following form for m ≥ −1 and t ∈ {1, 2, 3}.
Proof.We have obtained that the set Therefore, we can determine the forbidden set of the initial values for system (9) by using system (13).We know that the statements where m ∈ N 0 , i ∈ {0, 1}, f (x) = ax + b, g (x) = cx + d and h (x) = ex + f , characterize the solutions of system (9).By using the conditions (20) and the statements ( 21)- (23), we have where m ∈ N 0 , i ∈ {0, 1} and abcdef ̸ = 0.This means that if one of the conditions in ( 24)-( 26) holds, then m−th iteration or (m + 1) −th iteration in system (9) can not be calculated.Consequently, we obtain the result in (19).□

Conclusion
In this paper, we have solved the following three-dimensional system of difference equations , where the initial values x −i , y −i , z −i , i = 1, 3 and the parameters a, b, c, d, e, f are non-zero real numbers.In addition, we have obtained the solutions of above system in explicit form according to the parameters a, c and e are equal 1 or not equal 1.Moreover, we have got periodic solutions of aforementioned system.Finally, we have identified the forbidden set of the initial conditions by using the acquired formulas.
Author Contribution Statements All authors contributed equally and significantly to this manuscript and they read and approved the final manuscript.

Declaration of Competing Interests
The authors declare that they have no competing interest.