On the Periodicity of a Max-type Fuzzy Difference Equations

Changyou Wang; Wei Wei; Qiang Yang; Yonghong Li

doi:doi:10.11648/j.ajea.20190702.11

| Peer-Reviewed

On the Periodicity of a Max-type Fuzzy Difference Equations

Changyou Wang, Wei Wei, Qiang Yang, Yonghong Li

Published in American Journal of Electromagnetics and Applications (Volume 7, Issue 2)

Received: 17 November 2019 Accepted: 2 December 2019 Published: 11 December 2019

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

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.

Published in	American Journal of Electromagnetics and Applications (Volume 7, Issue 2)
DOI	10.11648/j.ajea.20190702.11
Page(s)	13-18
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Fuzzy Difference Equation, Max-Type, Cut Theory, Periodicity, Boundedness

References

[1]	S. Ellner (1984). Asymptotic behavior of some stochastic difference equation population models. Journal of Mathematical Biology 19, 169-200.
[2]	Z. Jing, B. G. Zhang (2003). The positive solutions of difference equations relevant to an economic problem. Computers & Mathematics with Applications 45, 835-842.
[3]	A. J. Avello, B. M. Al-Hadithi, M. I. G. Garcia (2012). Difference equation matrix model (DEMM) for the control of wind turbines. Wind Energy 17, 57-74.
[4]	S. Papadimitriou, A. Bezerianos, T. Bountis (1997). Secure communication with chaotic systems of difference equations. IEEE Transactions on Computers 46, 27-38.
[5]	D. T. Tollu, Y. Yazlik, N. Taskara (2013). On the solutions of two special types of Riccati difference equation via Fibonacci numbers. Advances in Difference Equations, Article ID: 174.
[6]	I. Yalçinkaya, B. D. Iricanin, C. Çinar (2007). On a Max-Type Difference Equation. Discrete Dynamics in Nature and Society, Article ID: 47264.
[7]	Y. Yazlik, D. T. Tollu, N. Taskara (2015). On the solutions of a max-type difference equation system. Mathematical Methods in the Applied Sciences 38, 4388-4410.
[8]	J. L. Williams (2016). On a class of nonlinear max-type difference equations. Cogent Mathematics 3, Article ID: 1269596.
[9]	X. F. Yang, W. P. Liu, J. M. Liu (2011). Global Attractivity of a Family of Max-Type Difference Equations. Discrete Dynamics in Nature and Society, Article ID: 506373.
[10]	S. Stević (2013). On a symmetric system of max-type difference equations. Applied Mathematics and Computation 219, 8407-8412.
[11]	E. M. Elsayed (2014). On the solutions and periodic nature of some systems of difference equations. International Journal of Biomathematics 7, Article ID: 1450067.
[12]	C. Y. Wang, X. T. Jing, X. H. Hu, R. Li (2017). On the periodicity of a max-type rational difference equation. Journal of Nonlinear Sciences and Applications 10, 4648-4661.
[13]	M. M. El-Dessoky (2014). On the periodicity of solutions of max-type difference equation. Mathematical Methods in the Applied Sciences 38, 3295-3307.
[14]	T. X. Sun, H. J. Xi (2016). Dynamics of a max-type system of difference equations. Analysis and Mathematical Physics 6, 393-402.
[15]	T. X. Sun, J. Liu, Q. L. He (2014). Eventually Periodic Solutions of a Max-Type Difference Equation. The Scientific World Journal, Article ID: 319437.
[16]	S. Stević, M. A. Alghamdi, A. Alotaibi (2014). Long-term behavior of positive solutions of a system of max-type difference equations. Applied Mathematics and Computation 235, 567-574.
[17]	K. S. Berenhaut, J. D. Foley, S. Stević (2006). Boundedness character of positive solutions of a max difference equation. Journal of Difference Equations and Application 12, 1193-1199.
[18]	X. F. Yang, X. F. Liao, C. D. Li (2006). On a difference equation with maximum. Applied Mathematics and Computation 181, 1-5.
[19]	S. Stević (2012). On some periodic system of max-type difference equations. Applied Mathematics and Computation 218, 11484-11487.
[20]	H. D. Voulov (2002). On the periodic character of some difference equations. Journal of Difference Equations and Applications 8, 799-810.
[21]	C. Cengiz, A. Gelisken (2009). On the Global Attractivity of a Max-Type Difference Equations. Discrete Dynamics in Nature and Society, Article ID: 812674.
[22]	I. Szalkai (1999). On the periodicity of the sequence x_n+1=max{A/x_n, A/x_n-1, …A/x_n-k}, n=0,1,…. Journal of Difference Equations and Applications 5, 25-29.
[23]	P. Diamond (2000). Stability and periodicity in fuzzy differential equations. IEEE Transactions on Fuzzy Systems 8, 583-590.
[24]	C. Y. Wang, X. L. Su, P. Liu, X. H. Hu, R. Li (2017). On the dynamics of a five-order fuzzy difference equation. Journal of Nonlinear Sciences and Applications 10, 3303-3319.
[25]	L. A. Zadeh (1965). Fuzzy sets. Information Control 8, 338-353.
[26]	E. Y. Deeba, A. de Korvin (1995). On a fuzzy difference equation. IEEE Transactions on Fuzzy Systems 3, 469-473.
[27]	S. P. Mondal, M. Mandal, D. Bhattacharya (2017). Non-linear interval-valued fuzzy numbers and their application in difference equations. Granular Computing 3, 177-189.
[28]	A. Khastan (2017). New solutions for first order linear fuzzy difference equations. Journal of Computational and Applied Mathematics 312, 156-166.
[29]	Q. H. Zhang, Z. G. Luo, J. Z. Liu, Y. F. Shao (2015). Dynamical behaviour of second-order rational fuzzy difference equation. International Journal of Dynamical Systems and Differential Equations 5, 336-353.
[30]	Q. H. Zhang, L. H. Yang, D. X. Liao (2014). On first order fuzzy Ricatti difference equation. Information Sciences 270, 226-236.
[31]	G. Stefanidou, G. Papaschinopoulos (2005). Behavior of the positive solution of fuzzy max-difference equations. Advances in the Difference Equations, Article ID: 947038.
[32]	G. Stefanidou, G. Papaschinopoulos (2006). The periodic nature of the positive solutions of a non-linear fuzzy max-difference equations. Information Sciences 176, 3696-3710.
[33]	B. Bebe (2013). Mathematics of Fuzzy Sets and Fuzzy Logic. Study in Fuzziness and Soft Computing, Springer, Heidelberg.
[34]	P. Diamond, P. Kloeden (1990). Metric spaces of fuzzy sets. Fuzzy Sets and Systems 35, 241-249.
[35]	V. L. Kocic, G. Ladas (1993). Mathematics and Its Applications: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications. Kluwer Academic, Dordrecht.
[36]	E. Y. Deeba, A. D. Korvin, E. L. Koh (1996). A fuzzy difference equation with an application. Journal of Difference Equations and Applications 2, 365-374.
[37]	V. Lakshmikantham, A. S. Vatsala (2002). Basic theory of fuzzy difference equations. Journal of Difference Equations and Applications 8, 957-968.

Cite This Article

Plain Text BibTeX RIS

APA Style

Changyou Wang, Wei Wei, Qiang Yang, Yonghong Li. (2019). On the Periodicity of a Max-type Fuzzy Difference Equations. American Journal of Electromagnetics and Applications, 7(2), 13-18. https://doi.org/10.11648/j.ajea.20190702.11

Copy | Download

ACS Style

Changyou Wang; Wei Wei; Qiang Yang; Yonghong Li. On the Periodicity of a Max-type Fuzzy Difference Equations. Am. J. Electromagn. Appl. 2019, 7(2), 13-18. doi: 10.11648/j.ajea.20190702.11

Copy | Download

AMA Style

Changyou Wang, Wei Wei, Qiang Yang, Yonghong Li. On the Periodicity of a Max-type Fuzzy Difference Equations. Am J Electromagn Appl. 2019;7(2):13-18. doi: 10.11648/j.ajea.20190702.11

Copy | Download

@article{10.11648/j.ajea.20190702.11,
  author = {Changyou Wang and Wei Wei and Qiang Yang and Yonghong Li},
  title = {On the Periodicity of a Max-type Fuzzy Difference Equations},
  journal = {American Journal of Electromagnetics and Applications},
  volume = {7},
  number = {2},
  pages = {13-18},
  doi = {10.11648/j.ajea.20190702.11},
  url = {https://doi.org/10.11648/j.ajea.20190702.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajea.20190702.11},
  abstract = {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.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - On the Periodicity of a Max-type Fuzzy Difference Equations
AU  - Changyou Wang
AU  - Wei Wei
AU  - Qiang Yang
AU  - Yonghong Li
Y1  - 2019/12/11
PY  - 2019
N1  - https://doi.org/10.11648/j.ajea.20190702.11
DO  - 10.11648/j.ajea.20190702.11
T2  - American Journal of Electromagnetics and Applications
JF  - American Journal of Electromagnetics and Applications
JO  - American Journal of Electromagnetics and Applications
SP  - 13
EP  - 18
PB  - Science Publishing Group
SN  - 2376-5984
UR  - https://doi.org/10.11648/j.ajea.20190702.11
AB  - 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.
VL  - 7
IS  - 2
ER  -

Copy | Download

Author Information

Changyou Wang

College of Science, Chongqing University of Posts and Telecommunications, Chongqing, P. R. China
Wei Wei

College of Science, Chongqing University of Posts and Telecommunications, Chongqing, P. R. China
Qiang Yang

College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, P. R. China
Yonghong Li

College of Science, Chongqing University of Posts and Telecommunications, Chongqing, P. R. China

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Changyou Wang, Wei Wei, Qiang Yang, Yonghong Li. (2019). On the Periodicity of a Max-type Fuzzy Difference Equations. American Journal of Electromagnetics and Applications, 7(2), 13-18. https://doi.org/10.11648/j.ajea.20190702.11

Copy | Download

ACS Style

Changyou Wang; Wei Wei; Qiang Yang; Yonghong Li. On the Periodicity of a Max-type Fuzzy Difference Equations. Am. J. Electromagn. Appl. 2019, 7(2), 13-18. doi: 10.11648/j.ajea.20190702.11

Copy | Download

AMA Style

Changyou Wang, Wei Wei, Qiang Yang, Yonghong Li. On the Periodicity of a Max-type Fuzzy Difference Equations. Am J Electromagn Appl. 2019;7(2):13-18. doi: 10.11648/j.ajea.20190702.11

Copy | Download

@article{10.11648/j.ajea.20190702.11,
  author = {Changyou Wang and Wei Wei and Qiang Yang and Yonghong Li},
  title = {On the Periodicity of a Max-type Fuzzy Difference Equations},
  journal = {American Journal of Electromagnetics and Applications},
  volume = {7},
  number = {2},
  pages = {13-18},
  doi = {10.11648/j.ajea.20190702.11},
  url = {https://doi.org/10.11648/j.ajea.20190702.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajea.20190702.11},
  abstract = {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.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - On the Periodicity of a Max-type Fuzzy Difference Equations
AU  - Changyou Wang
AU  - Wei Wei
AU  - Qiang Yang
AU  - Yonghong Li
Y1  - 2019/12/11
PY  - 2019
N1  - https://doi.org/10.11648/j.ajea.20190702.11
DO  - 10.11648/j.ajea.20190702.11
T2  - American Journal of Electromagnetics and Applications
JF  - American Journal of Electromagnetics and Applications
JO  - American Journal of Electromagnetics and Applications
SP  - 13
EP  - 18
PB  - Science Publishing Group
SN  - 2376-5984
UR  - https://doi.org/10.11648/j.ajea.20190702.11
AB  - 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.
VL  - 7
IS  - 2
ER  -

Copy | Download