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), 2024. Published by Science Publishing Group |
Difference Equations, Qualitative Properties, Periodicity, Semicycle, Boundedness, Global Stability
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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
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
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
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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}
}
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 -
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