Abstract and Applied Analysis

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

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

Full-text: Open access

Abstract

We investigate the periodic nature of the positive solutions of the fuzzy max-difference equation x n + 1 = max A n / x n - m , x n - k , n = 0,1 , , where k , m { 1,2 , } , A n is a periodic sequence of fuzzy numbers, and x - d , x - d + 1 , , x 0 are positive fuzzy numbers with d = m , k . We show that every positive solution of this equation is eventually periodic with period k + 1 .

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 760247, 4 pages.

Dates
First available in Project Euclid: 27 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1425049093

Digital Object Identifier
doi:10.1155/2014/760247

Mathematical Reviews number (MathSciNet)
MR3214451

Zentralblatt MATH identifier
07023031

Citation

He, Qiuli; Tao, Chunyan; Sun, Taixiang; Liu, Xinhe; Su, Dongwei. Periodicity of the Positive Solutions of a Fuzzy Max-Difference Equation. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 760247, 4 pages. doi:10.1155/2014/760247. https://projecteuclid.org/euclid.aaa/1425049093


Export citation

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.
  • 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.
  • 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.
  • 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.
  • T. Sun, B. Qin, H. Xi, and C. Han, “Global behavior of the max-type difference equation ${x}_{n+1}=\text{max}\{1/{x}_{n},{A}_{n}/{x}_{n-1}\}$,” Abstract and Applied Analysis, vol. 2009, Article ID 152964, 10 pages, 2009.
  • T. Sun, H. Xi, C. Han, and B. Qin, “Dynamics of the max-type difference equation ${x}_{n}=\text{max}\left\{1/{x}_{n-m},{A}_{n}/{x}_{n-r}\right\}$,” Journal of Applied Mathematics and Computing, vol. 38, no. 1-2, pp. 173–180, 2012.
  • T. Sun, H. Xi, and B. Qin, “Global behavior of the max-type difference equation ${x}_{n+1}=\text{max}\left\{A/{x}_{n-m},1/{x}_{n-k}^{\alpha }\right\}$,” Journal of Concrete and Applicable Mathematics, vol. 10, no. 1-2, pp. 32–39, 2012.
  • E. M. Elsayed and S. Stević, “On the max-type equation ${x}_{n+1}=\text{max}\{A/{x}_{n},{x}_{n-2}\}$,” Nonlinear Analysis: Theory, Methods & Applications A: Theory and Methods, vol. 71, no. 3-4, pp. 910–922, 2009.
  • B. D. Iričanin and E. M. Elsayed, “On the max-type difference equation ${x}_{n+1}=\text{max}\{A/{x}_{n},{x}_{n-3}\}$,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 675413, 13 pages, 2010.
  • 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.
  • B. Qin, T. Sun, and H. Xi, “Dynamics of the max-type difference equation ${x}_{n+1}=\text{max}\{A/{x}_{n},{x}_{n-k}\}$,” Journal of Computational Analysis and Applications, vol. 14, no. 5, pp. 856–861, 2012.
  • 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).
  • 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.
  • 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.
  • 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.
  • Q. Zhang, L. Yang, and D. Liao, “On first order fuzzy Ricatti difference equation,” Information Sciences, vol. 270, pp. 226–236, 2014.
  • 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.
  • H. T. Nguyen and E. A. Walker, A First Course in Fuzzy Logic, CRC Press, Boca Raton, Fla, USA, 1997.
  • 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.
  • G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic, Prentice Hall PTR, Upper Saddle River, NJ, USA, 1995.
  • 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.
  • 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. \endinput