Exact Solutions of Three-Dimensional Max-Type System of Difference Equations

Dierence equations and systems which do not stem from the dierential ones have attracted some attention in the last few decades (see, e.g., [1–26]). Some of the systems that are of interest are symmetric or those retrieved from symmetric by alteration of their parameters. Recently, there has been enormous interest in various types of nonlinear dierence equations and the references cited therein. One of the reasons for this is a necessity for some techniques which can be used in inspecting equations arising in mathematical models illustrating real life situations in biology, control theory, economics, physics, sociology, and so on (see, e.g., [21, 23, 24]). Among these equations, the so-called max-type dierence equations also attracted some attention. e study of the system of max-type dierence equations attracted recently a considerable attention; see, for example, [1–16], and the references listed therein. is type of difference equations stemming from, for example, certain models are useful in automatic control theory (see [27]). In the beginning, on the study of these equations, experts have been focused on the investigation of the behavior of some particular cases. In [3], Berenhaut et al. explained the boundedness nature of positive solutions of the following max-type dierence equation system:


Introduction
Di erence equations and systems which do not stem from the di erential ones have attracted some attention in the last few decades (see, e.g., ). Some of the systems that are of interest are symmetric or those retrieved from symmetric by alteration of their parameters. Recently, there has been enormous interest in various types of nonlinear di erence equations and the references cited therein. One of the reasons for this is a necessity for some techniques which can be used in inspecting equations arising in mathematical models illustrating real life situations in biology, control theory, economics, physics, sociology, and so on (see, e.g., [21,23,24]). Among these equations, the so-called max-type di erence equations also attracted some attention.
In [3], Berenhaut et al. explained the boundedness nature of positive solutions of the following max-type di erence equation system: where A n and B n are periodic parameters. In 2020, Balibrea et al. [7] obtained in an elegant way the general solution of the following max-type system of di erence equations: where A n , B n ∈ (0, +∞), t, s ∈ 1, 2, . . . x n max A, y n−1 x n−2 , y n max B, where A n , B n ∈ (0, +∞) are periodic sequences with period 2.
In 2015, Grove et al. [13] obtained the solution of the following max-type difference equation system: e parameter A is positive real number.
In [15], Su et al. obtained the solution of the following max-type difference equation system with a period 3 parameter: In [26], C.M. Kent and M.A. Radin explained the boundedness nature of positive solutions of the following difference equation: where A n and B n are periodic parameters. Motivated by the above study, our purpose in this paper is to evaluate the eventual periodicity of the following maxtype 3 D-system of difference equations: where n ∈ N 0 , N 0 � N ∪ 0 { }, (A n ) n ∈ N 0 , (B n ) n ∈ N 0 , and (C n ) n ∈ N 0 are positive periodic sequences and initial en, the following statements hold: Proof.
(1) From the following conditions: we have 2 Mathematical Problems in Engineering By induction, we obtained formula as follows: which are formulas of odd terms in (4)- (6). Hence, it remains only to prove the formulas for even terms in (4)- (6). Similarly, we can find for even terms.
By induction, (2) Because u 0 ≥ A 1 /v 0 , v 0 ≥ B 1 /w 0 , and w 0 ≤ C 1 /u 0 , we have Mathematical Problems in Engineering By induction, we obtain formulas for even terms as given in (11).
So, we have By induction, we obtain formulas as given in (26): e proof is completed.

Mathematical Problems in Engineering
Proof.
(1) From the conditions By induction, we obtained formula as follows: Similarly, we can find the proof for even terms.