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

He, Qiuli; Tao, Chunyan; Sun, Taixiang; Liu, Xinhe; Su, Dongwei

doi:https://doi.org/10.1155/2014/760247

Abstract and Applied Analysis

On this page

Abstract Introduction Preliminaries Acknowledgments References Copyright Related Articles

Special Issue

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

View this Special Issue

Research Article | Open Access

Volume 2014 | Article ID 760247 | https://doi.org/10.1155/2014/760247

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

Qiuli He,^1,2Chunyan Tao,³Taixiang Sun,^2,3Xinhe Liu,³and Dongwei Su^1,2

Academic Editor: Tongxing Li

Received20 Mar 2014

Accepted13 Apr 2014

Published22 May 2014

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).

References

A. D. Mishkis, “On some problems of the theory of differential equations with deviating argument,” Uspekhi Matematicheskikh Nauk, vol. 32, no. 2, pp. 173–202, 1977.
View at: Google Scholar
E. P. Popov, Automatic Regulation and Control, Nauka, Moscow, Russia, 1966 (Russian).
S. Stević, “On a symmetric system of max-type difference equations,” Applied Mathematics and Computation, vol. 219, no. 15, pp. 8407–8412, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
S. Stević, “On positive solutions of some classes of max-type systems of difference equations,” Applied Mathematics and Computation, vol. 232, pp. 445–452, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
S. Stević, M. A. Alghamdi, A. Alotaibi, and N. Shahzad, “Eventual periodicity of some systems of max-type difference equations,” Applied Mathematics and Computation, vol. 236, pp. 635–641, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
T. Sun, B. Qin, H. Xi, and C. Han, “Global behavior of the max-type difference equation $x_{n + 1} = max {1 / x_{n}, A_{n} / x_{n - 1}}$ ,” Abstract and Applied Analysis, vol. 2009, Article ID 152964, 10 pages, 2009.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
T. Sun, H. Xi, C. Han, and B. Qin, “Dynamics of the max-type difference equation $x_{n} = max \{1 / x_{n - m}, A_{n} / x_{n - r}\}$ ,” Journal of Applied Mathematics and Computing, vol. 38, no. 1-2, pp. 173–180, 2012.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
T. Sun, H. Xi, and B. Qin, “Global behavior of the max-type difference equation $x_{n + 1} = max \{A / x_{n - m}, 1 / x_{n - k}^{α}\}$ ,” Journal of Concrete and Applicable Mathematics, vol. 10, no. 1-2, pp. 32–39, 2012.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
E. M. Elsayed and S. Stević, “On the max-type equation $x_{n + 1} = max {A / x_{n}, x_{n - 2}}$ ,” Nonlinear Analysis: Theory, Methods & Applications A: Theory and Methods, vol. 71, no. 3-4, pp. 910–922, 2009.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
B. D. Iričanin and E. M. Elsayed, “On the max-type difference equation $x_{n + 1} = max {A / x_{n}, x_{n - 3}}$ ,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 675413, 13 pages, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Q. Xiao and Q.-h. Shi, “Eventually periodic solutions of a max-type equation,” Mathematical and Computer Modelling, vol. 57, no. 3-4, pp. 992–996, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
B. Qin, T. Sun, and H. Xi, “Dynamics of the max-type difference equation $x_{n + 1} = max {A / x_{n}, x_{n - k}}$ ,” Journal of Computational Analysis and Applications, vol. 14, no. 5, pp. 856–861, 2012.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Q. H. Zhang and J. Z. Liu, “The first-order fuzzy difference equation $x_{n + 1} = A x_{n} + B$ ,” Fuzzy Systems and Mathematics, vol. 23, no. 4, pp. 74–79, 2009 (Chinese).
View at: Google Scholar | Zentralblatt MATH | MathSciNet
Q. Zhang, L. Yang, and D. Liao, “On the fuzzy difference equation $x_{n + 1} = A + \sum_{i = 0}^{k} B / x_{n - i}$ ,” World Academy of Science, Engineering and Technology, vol. 75, pp. 1032–1037, 2011.
View at: Google Scholar
Q. H. Zhang, L. H. Yang, and D. X. Liao, “Behavior of solutions to a fuzzy nonlinear difference equation,” Iranian Journal of Fuzzy Systems, vol. 9, no. 2, pp. 1–12, 2012.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
G. Stefanidou and G. Papaschinopoulos, “The periodic nature of the positive solutions of a nonlinear fuzzy max-difference equation,” Information Sciences, vol. 176, no. 24, pp. 3694–3710, 2006.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Q. Zhang, L. Yang, and D. Liao, “On first order fuzzy Ricatti difference equation,” Information Sciences, vol. 270, pp. 226–236, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
G. Papaschinopoulos and B. K. Papadopoulos, “On the fuzzy difference equation $x_{n + 1} = A + x_{n} / x_{n - m}$ ,” Fuzzy Sets and Systems, vol. 129, no. 1, pp. 73–81, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. T. Nguyen and E. A. Walker, A First Course in Fuzzy Logic, CRC Press, Boca Raton, Fla, USA, 1997.
View at: MathSciNet
C. Wu and B. Zhang, “Embedding problem of noncompact fuzzy number space $E^{-}$ (I),” Fuzzy Sets and Systems, vol. 105, no. 1, pp. 165–169, 1999.
View at: Publisher Site | Google Scholar | MathSciNet
G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic, Prentice Hall PTR, Upper Saddle River, NJ, USA, 1995.
View at: MathSciNet
G. Papaschinopoulos and B. K. Papadopoulos, “On the fuzzy difference equation $x_{n + 1} = A + B / x_{n}$ ,” Soft Computing, vol. 6, pp. 456–461, 2002.
View at: Google Scholar
G. Stefanidou and G. Papaschinopoulos, “Behavior of the positive solutions of fuzzy max-difference equations,” Advances in Difference Equations, no. 2, pp. 153–172, 2005.
View at: Google Scholar | Zentralblatt MATH | MathSciNet

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.

PDF Download Citation

Download other formats

Order printed copies

Views

744

Downloads

1166

Citations