## Qualitative Analysis of Differential, Difference Equations, and Dynamic Equations on Time Scales

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# Periodicity of the Positive Solutions of a Fuzzy Max-Difference Equation

**Academic Editor:**Tongxing Li

#### Abstract

We investigate the periodic nature of the positive solutions of the fuzzy max-difference equation , where , is a periodic sequence of fuzzy numbers, and are positive fuzzy numbers with . We show that every positive solution of this equation is eventually periodic with period .

#### 1. Introduction

The max operator arises naturally in certain models in automatic control theory (see [1, 2]). In recent years, the discrete case involving difference equations with maximum has been receiving increasing attention (see [3–8]). Elsayed and Stević [9] considered the max-difference equation where and the initial conditions and showed that every well-defined solution of this equation is eventually periodic with period .

In [10], Iričanin and Elsayed investigated the max-difference equation where and the initial conditions and showed that every well-defined solution of this equation is eventually periodic with period .

Recently Xiao and Shi [11] studied the max-difference equation where and the initial conditions and showed that every well-defined solution of the above equation is eventually periodic with period .

In [12], we dealt with the max-difference equation where , and the initial conditions and showed that every well-defined solution of the above equation is eventually periodic with period , which extended the results of [9–11] to the general case.

Recently there has been an increase in interest in the study of fuzzy difference equations (see [13–15]). In [16], Stefanidou and Papaschinopoulos studied the periodicity of the positive solutions of the following fuzzy max-difference equation where , , and the initial conditions with are positive fuzzy numbers.

In [17], Zhang et al. dealt with the existence, the boundedness, and the asymptotic behavior of the positive solutions to a first order fuzzy Ricatti difference equation where , and the initial condition are positive fuzzy numbers.

In this note, our goal is to investigate the periodicity of the positive solutions of the fuzzy max-difference equation where , is a periodic sequence of fuzzy numbers, and are positive fuzzy numbers with . Our main result is the following theorem.

Theorem 1. *Let and be a periodic sequence of fuzzy numbers. Then every positive solution of (7) is eventually periodic with period .*

#### 2. Preliminaries

We need the following definitions. A function from into the interval is called a fuzzy number if the following statements hold (see [18]).(1) is normal (i.e., for some ).(2) is a convex fuzzy set (i.e., for any and any ).(3) is upper semicontinuous.(4)The support is compact,where (for any ) (which are said to be the -cuts of the fuzzy number ) and is the closure of set . We see from [19, Theorem and Theorem ] the -cuts of the fuzzy number are closed intervals.

A fuzzy number is said to be positive if . If , then is a positive fuzzy number (it is called a trivial fuzzy number also) with for any .

For some positive integer , let be fuzzy numbers and with Write Then we know from [20, Theorem 2.1] that there exists a fuzzy number such that By [21] and [22, Lemma 2.3] one can define

A sequence of positive fuzzy numbers is said to be a solution of (7) if it satisfies (7). If there exists a positive integer and such that, for all , then is said to be eventually periodic with period .

Proposition 2. *Let be a sequence of positive fuzzy numbers. Then there exists a unique positive solution of (7) with initial values .*

*Proof. *Assume that (for any ) and . Let be positive fuzzy numbers such that
and let be the unique positive solution of the following system of difference equations:
with initial values . Arguing as in Proposition 3.1 of [23] we may show that determines a sequence of positive fuzzy numbers with
and that is the unique positive solution of (7) with initial values . This completes the proof of the proposition.

#### 3. Proof of Theorem 1

Lemma 3. *Consider the system of difference equations
**
where are two periodic sequences of positive real numbers and the initial values are positive real numbers. Then every positive solution of (16) is eventually periodic of period .*

*Proof. *Let be a positive solution of (16). We have from (16) that, for any and any ,
Then and are increasing for every .

Now we show that is a constant sequence eventually for every . Indeed, if is not constant sequence eventually for some , then there exist such that and is a constant sequence for all since is a periodic sequence. Thus we have
From this we obtain that, for all ,
This is a contradiction.

In a similar fashion, we can show that is also a constant sequence eventually for every .

From the above we see that is eventually periodic with period . This completes the proof of Lemma 3.

* Proof of Theorem 1. *Let be a positive solution of (7) with initial values satisfying (13) and let (15) hold. We see from Proposition 2 that satisfies system (14). Using Lemma 3 we know that is eventually periodic with period . Therefore, it follows from (14) and Lemma 3 that is eventually periodic of period . This completes the proof of Theorem 1.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This project is supported by NNSF of China (11261005) and NSF of Guangxi (2012GXNSFDA276040).

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#### Copyright

Copyright © 2014 Qiuli He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.