On the Periodicity of a Max-type Fuzzy Difference Equations

Our aim in this paper is to discuss the periodicity and boundedness of a max-type fuzzy difference equation. When studying the periodicity of the solution to the max-fuzzy difference equation, the equation is first converted into a difference system composed of two related difference equations through the cut set theory of the fuzzy number, then the periodicity of each solution sequence in the system is obtained by means of inequality technique, mathematical induction and other theoretical methods, thus the periodicity of the solution is proved. As researching the boundedness of the solution for the fuzzy difference equation, the difference system is also obtained through the cut set theory of the fuzzy number, then analyze the boundedness to each solution sequence according to the periodicity with the solution sequence, through examining the value of the finite subsequence in each solution sequence, the boundedness with these subsequences can be obtained, and then the boundedness for each solution sequence made up of complete subsequences can be known, thus the boundedness of the solution is proved. Finally, the results obtained in this paper are simulated by using the software package MATLAB 2016, the numerical results not only show the dynamic behavior of the solutions to the fuzzy difference systems, but also verify the effectiveness of the theoretical results.


Introduction
The difference equation is a kind of mathematical model describing discrete dynamic system. As an important branch of the dynamic system, the difference equation theory is widely used in population statistics and analysis, economic finance, control science, communication science and computer science (see, e.g., [1][2][3][4][5] and the reference therein). With the progress and development of science and technology, the difference equation will become a more and more popular research topic. As an important part of the difference equation, the max-type difference equation has important applications in many fields, such as models of automatic control theory and so on (see, e.g., [6][7][8][9][10] and thereference therein). Because of this, scholars have studied many theories and applications of the max-type difference equation in recent years. Since periodical research of differential equation theory has been very mature and perfect, some scholars try to investigate the periodicity of difference equation with the theory of differential equation (see, e.g., [11][12][13][14][15][16][17] and the reference therein).
Making a historical flash back for the equation we study in this paper, we need to mention that in 2006, Yang et al. [18] studied the following second order max-type difference equation with index parameters At the same time we could see that in 2012, S. Stevic [19] studied the periodicity of the solutions for the difference equations (2), (1) (1) (1) ,1 ,2 , 1 . When the parameters are permanent, then the final period value of the solutions to the equations (2) is s. In addition, S. Stevic also dissected the periodicity of the min-type difference equations corresponding to the equations (2).
In addition, in 2002, Voulov [20] analyzed the periodicity of the positive solution of the max-type difference equation (3), Where are arbitrary positive integers, the parameters and the initial conditions are arbitrary positive real numbers, in which . The periodicity of the solution for the equation is proved by the thought of classified discussions.
Recently, many scholars have further expanded the form of the equation ( Besides, in 1999, Szalkai [22] presented the periodicity of the solution for the max-type difference equation (5), 1 1 max{ , , , }, 0,1, where k is a positive integer, the parameters A and the initial conditions In this paper, the author proved that every positive solution of the equation is periodic with 2 k + . With the sustainable development of the max-type difference equation theory, scholars have combined it with the fuzzy sets theory (see, e.g., [23][24][25][26][27][28][29][30] and the reference therein), thus creating the max-type fuzzy difference equation, and at the same time doing numerous studies on the max-type fuzzy difference equation. In 2004, Stefanidou [31] generalized the difference equation (5) from the real number range to the fuzzy number range, that is, the parameter A and the initial values 0 { } i i k x =− are positive fuzzy numbers, then the solutions of the equation are also positive fuzzy numbers. In this paper, the author gave the corresponding conditions for the solution of the equation to be periodic, unbounded and unsustainable. Meanwhile, the dynamic properties of the max-type fuzzy difference equation (6), a special form of the equation (3), are also discussed in reference [31] where the parameter 0 1 , A A and the initial values 1 0 , x x − are positive fuzzy numbers given arbitrarily. In this paper, the author obtained that the solution of the equation (6) is periodic, unbounded and non-persistent. In 2006, Stefanidou [32] further extended the form of the equation (6), and considered the periodicity of the solution for the max-type type fuzzy difference equation (7), where the parameter are positive fuzzy numbers given arbitrarily.
Inspired by the discussions above, in this paper, we will study the periodic property of the solutions to the following fuzzy difference equation (8) where k is a positive integer, the parameters A and the initial conditions , are positive fuzzy numbers.

Preliminaries and Notations
For the convenience of readers, we need to give the following notations, definitions and preliminary results, see [33][34][35][36][37]. Definition 2.1 For a set B we denote by B the closure of B , we say that a function : (2) A is a fuzzy convex set, i.e., where the α − cuts of A are closed intervals, define as  we denote the following metric   , , , , Considering the following difference equation where x I is a real number interval and 1 : k x x f I I + → is a continuous function. 1 1 1

Main Results and Proofs
According to α − cut set theory, the max-type fuzzy difference equation (8) can be transformed into the following ordinary difference system consisting of two ordinary difference equations Proof: According to the difference system (11), we can gain the following several inequalities ( 1) ( 1)  ( 1) k + . And once again, according to the difference system (11), the following some inequalities can be obtained ( 1) ( 1) 1 , , From the synthesis, we can get the following inequalities ( 1) ( 1) , ,( 0,1, ,( 1)) n n k n i n i k n i k n i The inequalities above reflect the size relations of the algebraic expressions on the right-hand side of another equation in the difference system (11), then through the value of the size relationship and the definition of the equation in the system (11), we can gain

Numerical Simulation
In order to verify the conclusions obtained in this paper, some numerical simulations are carried out by using the software package MATLAB 2016.
Example 4.1 Considering the case of the difference system (11) at 4 k = , i.e. the following max-type fuzzy difference equation The following figure shows the simulation results of the numerical solutions for the difference system (12).

Conclusion
In this paper, we study the periodicity and boundedness of the solutions to a max-type difference equation. Firstly, by using iteration method, inequality skills and mathematical induction, we proved that the max-type difference equation has period of ( 1) k + when the parameters and the initial conditions are all positive fuzzy numbers. Secondly, it is proved that the solutions for the max-type difference equation is bounded and persistent when the parameters and the initial conditions are all positive fuzzy numbers. Finally, the effectiveness of the theoretical results is verified by the numerical experiments with the software package MATLAB 2016.