| Peer-Reviewed

Qualitative Study of a Novel Nonlinear Difference Equation of a General Order

Received: 30 November 2022     Accepted: 26 December 2022     Published: 10 January 2023
Views:       Downloads:
Abstract

Difference equations play a key role in analyzing many natural phenomena. Difference equations have many applications in different areas such as economic, biological physics, engineering, ecology, physiology, population dynamics and social sciences. Difference equations could also be used to simplify the dynamical systems represented by differential equations. So there exist rapid interest in investing the dynamics of the solutions of the difference equations. There exist different forms of difference equations including rational, nonlinear, max type and system of difference equations. In this paper, a novel nonlinear difference equation of general order is introduced and some qualitative properties of its solutions are studied. The parameters and the initial conditions of the difference equation are assumed to be positive real numbers. New results concerning the periodicity, semicycles, boundedness and global asymptotically stability are established. We prove that the proposed difference equation has unique positive equilibrium point. The periodic solutions with period two are studied. The semicycle analysis of the proposed difference equation is provided. The boundedness of the solutions is investigated. We give upper and lower bounds on the solutions in terms of the parameters of the proposed difference equation. Moreover, the local and global stability are investigated. Some numerical examples are provided to illustrate our results. The proposed difference equation is of general order, so the obtained results could be used for many difference equations.

Published in American Journal of Applied Mathematics (Volume 11, Issue 1)
DOI 10.11648/j.ajam.20231101.11
Page(s) 1-6
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), 2023. Published by Science Publishing Group

Keywords

Difference Equations, Qualitative Properties, Periodicity, Semicycle, Boundedness, Global Stability

References
[1] Kocic, V. L., & Ladas, G. (1993). Global behavior of nonlinear difference equations of higher order with applications, Springer. https://doi.org/10.1007/978-94-017-1703-8.
[2] El-Moneam, M. A. (2014). On the dynamics of the higher order nonlinear rational difference equation, Math. Sci. Lett., 3, 121–129. http://dx.doi.org/10.12785/msl/030208
[3] Kocic, V. L., & Ladas, G. (1993). Global attractivity in a second-order nonlinear difference equation, Journal of Mathematical Analysis and Applications, 180, 144-150.
[4] Alotaibi, A. M., & El-Moneam, M. A. (2022). On the dynamics of the nonlinear rational difference equation , AIMS Math., 7, 7374–7384. https://doi.org/10.3934/math.2022411
[5] El-Moneam, M. A., & Zayed, E. M. E. (2013). On the qualitative study of the nonlinear difference equation , Fasciculi Mathematici, 50, 137–147.
[6] Zayed, E. M. E. (2015). On the dynamics of the nonlinear rational difference equation, Dynam. Cont. Dis. Discrete Impulsive Ser. A: Math. Anal., 22, 61–71.
[7] Elsayed, E. M., Alofi, B. S., & Khan, A. Q. (2022). Qualitative behavior of solutions of tenth-order recursive sequence equation, Math. Probl. Eng., 2022 https://doi.org/10.1155/2022/5242325
[8] Ibrahim, T. F., Khan, A. Q., & Ibrahim, A. (2021). Qualitative behavior of a nonlinear generalized recursive sequence with delay, Math. Probl. Eng., 2021. https://doi.org/10.1155/2021/6162320
[9] Cinar, C. (2004). On the positive solutions of the difference equation , Applied Mathematics and Computation, 150, 21–24.
[10] Jafar, A., & Saleh, M. (2018). Dynamics of nonlinear difference equation , J. Appl. Math. Comput., 57, 493–522.
[11] El-Moneam, M. A., & Zayed, E. M. E. (2014). Dynamics of the rational difference equation, Inf. Sci. Lett., 3, 45–53. http://dx.doi.org/10.12785/isl/030202
[12] El-Moneam, M. A., & Alamoudy, S. O. (2014). On study of the asymptotic behavior of some rational difference equations, Dynam. Cont. Dis. Ser. A: Math. Anal., 21, 89–109.
[13] Hamza, A. E., & Khalaf-Allah, R. (2007). Global behavior of a higher order difference equation, Journal of Mathematics and Statistics, 3, 17–20.
[14] Kulenović, M. R. S., Ladas, G., & Prokup, N. P. (2001). A rational difference equation, Computers & Mathematics with Applications, 41: 671-678.
[15] El-Moneam, M. A. (2015). On the dynamics of the solutions of the rational recursive sequences, Brit. J. Math. Comput. Sci., 5, 654–665.
[16] Zayed, E. M. E., & El-Moneam, M. A. (2014). Dynamics of the rational difference equation , Commun. Appl. Nonlinear Anal., 21, 43–53.
[17] Elsayed, E. M. (2011). Solution and attractivity for a rational recursive sequence, Discrete Dyn. Nat. Soc., 2011. https://doi.org/10.1155/2011/982309
[18] E. M. Elsayed. (2013). Behavior and expression of the solutions of some rational difference equations, J. Comput. Anal. Appl, 15, 73-81.
[19] Yingchao Hao, & Cuiping Li. (2022). Some Characters of a Generalized Rational Difference Equation. American Journal of Applied Mathematics. Vol. 10, No. 1, pp. 9-14. doi: 10.11648/j.ajam.20221001.12.
[20] Amleh, A. M., Camouzis, E., & Ladas, G. (2008). On the Dynamics of a Rational Difference Equation Part 1, Int. J. Difference Equ., 3, 1-35.
[21] Grove, E. A., & Ladas, G. (2004). Periodicities in Nonlinear Difference Equations, Chapman and Hall/CRC, Boca Raton, London.
[22] Khalaf-Allah, R. (2009). Asymptotic behavior and periodic nature of two difference equations, Ukrains’kyi Matematychnyi Zhurnal, 61, 834–838.
[23] Elsayed, E. M., Din, Q., & Bukhary, N. A. (2022). Theoretical and numerical analysis of solutions of some systems of nonlinear difference equations, AIMS Mathematics, 7, 15532-15549.
[24] Halim, Y., & Bayram, M. (2016). On the solutions of a higher-order difference equation in terms of generalized Fibonacci sequences, Mathematical Methods in the Applied Sciences, 39, 2974- 2982.
[25] Elsayed, E. M. (2016). Expression and behavior of the solutions of some rational recursive sequences, Mathematical Methods in the Applied Sciences, 39, 5682-5694.
[26] Stević, S. (2011). On a nonlinear generalized max-type difference equation, Journal of Mathematical Analysis and Applications, 376, 317-328.
[27] Ibrahim, T. F., & Touafek, N. (2014). Max-type system of difference equations with positive two-periodic sequences, Math. methods Appl. sci, 37, 2562-2569.
[28] Phong, M. N. (2019). Global dynamics of the system of two exponential difference equations, Electron J Math Anal Appl, 7, 256-266.
[29] Elsayed, E. M. (2012). Solutions of rational difference systems of order two, Mathematical and Computer Modelling, 55, 378-384.
[30] Khan, A. Q., & Qureshi, M. N. (2015). Global dynamics of a competitive system of rational difference equations, Mathematical Methods in the Applied Sciences, 38, 4786-4796.
[31] Khan, A. Q., & Qureshi, M. N. (2016). Qualitative behavior of two systems of higher order difference equations, Mathematical Methods in the Applied Sciences, 39, 3058-3074.
[32] Stević, S. (2019). Two-dimensional solvable system of difference equations with periodic coefficients, Mathematical Methods in the Applied Sciences, 42, 6757-6774.
[33] Stević, S. (2013). On the system , Applied Mathematics and Computation, 219, 4526-4534.
[34] Khan, A. Q.,& Qureshi, M. N. (2021). Dynamical properties of some rational systems of difference equations, Mathematical Methods in the Applied Sciences, 44, 3485-3508.
[35] Kulenović, M. R. S., & Ladas, G. (2001). Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, London.
[36] Camouzis, E., & Ladas, G. (2007). Dynamics of Third-Order Rational Difference Equations With Open Problems and Conjectures, Chapman & Hall/CRC Press.
[37] Amleh, A. M., Grove, E. A., Ladas, G., & Georgiou, D. A. (1999). On the Recursive Sequence , Journal of Mathematical Analysis and Applications, 233, 790-798.
[38] Saleh, M., & Aloqeili, M. (2005). On the rational difference equation , Applied mathematics and computation., 171, 862-869.
[39] Hamzaa, A. E., & Morsyb, A. (2009). On the recursive sequence , Applied Mathematics Letters, 22, 91-95.
[40] Yalçinkaya, I. (2008). On the Difference Equation , Discrete Dynamics in Nature and Society, 2008.
[41] Wang, X. (2004). A simple proof of Descartes's rule of signs, Am Math Monthly; 111, 525–526.
Cite This Article
  • APA Style

    Ahmed Sayed Etman, Ahmed Essam Hammad. (2023). Qualitative Study of a Novel Nonlinear Difference Equation of a General Order. American Journal of Applied Mathematics, 11(1), 1-6. https://doi.org/10.11648/j.ajam.20231101.11

    Copy | Download

    ACS Style

    Ahmed Sayed Etman; Ahmed Essam Hammad. Qualitative Study of a Novel Nonlinear Difference Equation of a General Order. Am. J. Appl. Math. 2023, 11(1), 1-6. doi: 10.11648/j.ajam.20231101.11

    Copy | Download

    AMA Style

    Ahmed Sayed Etman, Ahmed Essam Hammad. Qualitative Study of a Novel Nonlinear Difference Equation of a General Order. Am J Appl Math. 2023;11(1):1-6. doi: 10.11648/j.ajam.20231101.11

    Copy | Download

  • @article{10.11648/j.ajam.20231101.11,
      author = {Ahmed Sayed Etman and Ahmed Essam Hammad},
      title = {Qualitative Study of a Novel Nonlinear Difference Equation of a General Order},
      journal = {American Journal of Applied Mathematics},
      volume = {11},
      number = {1},
      pages = {1-6},
      doi = {10.11648/j.ajam.20231101.11},
      url = {https://doi.org/10.11648/j.ajam.20231101.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20231101.11},
      abstract = {Difference equations play a key role in analyzing many natural phenomena. Difference equations have many applications in different areas such as economic, biological physics, engineering, ecology, physiology, population dynamics and social sciences. Difference equations could also be used to simplify the dynamical systems represented by differential equations. So there exist rapid interest in investing the dynamics of the solutions of the difference equations. There exist different forms of difference equations including rational, nonlinear, max type and system of difference equations. In this paper, a novel nonlinear difference equation of general order is introduced and some qualitative properties of its solutions are studied. The parameters and the initial conditions of the difference equation are assumed to be positive real numbers. New results concerning the periodicity, semicycles, boundedness and global asymptotically stability are established. We prove that the proposed difference equation has unique positive equilibrium point. The periodic solutions with period two are studied. The semicycle analysis of the proposed difference equation is provided. The boundedness of the solutions is investigated. We give upper and lower bounds on the solutions in terms of the parameters of the proposed difference equation. Moreover, the local and global stability are investigated. Some numerical examples are provided to illustrate our results. The proposed difference equation is of general order, so the obtained results could be used for many difference equations.},
     year = {2023}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Qualitative Study of a Novel Nonlinear Difference Equation of a General Order
    AU  - Ahmed Sayed Etman
    AU  - Ahmed Essam Hammad
    Y1  - 2023/01/10
    PY  - 2023
    N1  - https://doi.org/10.11648/j.ajam.20231101.11
    DO  - 10.11648/j.ajam.20231101.11
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
    SP  - 1
    EP  - 6
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20231101.11
    AB  - Difference equations play a key role in analyzing many natural phenomena. Difference equations have many applications in different areas such as economic, biological physics, engineering, ecology, physiology, population dynamics and social sciences. Difference equations could also be used to simplify the dynamical systems represented by differential equations. So there exist rapid interest in investing the dynamics of the solutions of the difference equations. There exist different forms of difference equations including rational, nonlinear, max type and system of difference equations. In this paper, a novel nonlinear difference equation of general order is introduced and some qualitative properties of its solutions are studied. The parameters and the initial conditions of the difference equation are assumed to be positive real numbers. New results concerning the periodicity, semicycles, boundedness and global asymptotically stability are established. We prove that the proposed difference equation has unique positive equilibrium point. The periodic solutions with period two are studied. The semicycle analysis of the proposed difference equation is provided. The boundedness of the solutions is investigated. We give upper and lower bounds on the solutions in terms of the parameters of the proposed difference equation. Moreover, the local and global stability are investigated. Some numerical examples are provided to illustrate our results. The proposed difference equation is of general order, so the obtained results could be used for many difference equations.
    VL  - 11
    IS  - 1
    ER  - 

    Copy | Download

Author Information
  • Engineering Mathematics and Physics Department, Engineering Faculty, Cairo University, Giza, Egypt

  • Engineering Mathematics and Physics Department, Engineering Faculty, Cairo University, Giza, Egypt

  • Sections