# Global Analysis and the Periodic Character of a Class of Difference Equations

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Periodic Solutions with Period $\mathit{p}$

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Remark**

**1.**

## 3. Stability and Boundedness

**Case (1)**: If $\alpha +\beta =\gamma $, then the only positive equilibrium point is

**Case (2)**: If $\alpha +\beta \ne \gamma $, then the only positive equilibrium point is

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Proof.**

**Theorem**

**6.**

**Proof.**

## 4. Application and Discussion

**Corollary**

**2.**

**Remark**

**2.**

**Example**

**1.**

n | … | 89 | 90 | 91 | 92 | … |

J_{n} | … | 3.63746 | 3.63745 | 3.63745 | 3.63745 | … |

α | 0.5 | 1 | 3 | 7 |

N_{e} | 150 | 90 | 43 | 28 |

J_{e} | 3.08114 | 3.63745 | 5.76040 | 9.85514 |

**Remark**

**3.**

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Abdelrahman, M.A.E.; Chatzarakis, G.E.; Li, T.; Moaaz, O. On the difference equation J
_{n+1}= aJ_{n−l}+ bJ_{n−k}+ f(J_{n−l}, J_{n−k}). Adv. Differ. Equ.**2018**, 431, 2018. [Google Scholar] - Agarwal, R.P.; Elsayed, E.M. Periodicity and stability of solutions of higher order rational difference equation. Adv. Stud. Contemp. Math.
**2008**, 17, 181–201. [Google Scholar] - Ahlbrandt, C.D.; Peterson, A.C. Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1996. [Google Scholar]
- Ahmad, S. On the nonautonomous Volterra-Lotka competition equations. Proc. Am. Math. Soc.
**1993**, 117, 199–204. [Google Scholar] [CrossRef] - Allman, E.S.; Rhodes, J.A. Mathematical Models in Biology: An Introduction; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Andres, J.; Pennequin, D. Note on Limit-Periodic Solutions of the Difference Equation x
_{t+1}− [h(xt) + λ]x = rt,λ > 1. Axioms**2019**, 8, 19. [Google Scholar] [CrossRef] - Din, Q.; Elsayed, E.M. Stability analysis of a discrete ecological model. Comput. Ecol. Soft.
**2014**, 4, 89–103. [Google Scholar] - Grove, E.A.; Ladas, G. Periodicities in Nonlinear Difference Equations; Chapman & Hall/CRC: Boca Raton, FL, USA, 2005; Volume 4. [Google Scholar]
- Elabbasy, E.M.; El-Metwally, H.; Elsayed, E.M. On the difference equation J
_{n+1}= (aJ_{n−l}+ bJ_{n−k})/(cJ_{n−l}+ dJ_{n−k}). Acta Math. Vietnam.**2008**, 33, 85–94. [Google Scholar] - Elabbasy, E.M.; Elsayed, E.M. Dynamics of a rational difference equation. Chin. Ann. Math. Ser. B
**2009**, 30, 187–198. [Google Scholar] [CrossRef] - El-Dessoky, M.M. On the difference equation J
_{n+1}= aJ_{n−l}+ bJ_{n−k}+ cJ_{n−s}/(dJ_{n−s}− e). Math. Methods Appl. Sci.**2016**, 1, 082579. [Google Scholar] - Elettreby, M.F.; El-Metwally, H. On a system of difference equations of an economic model. Discret. Dyn. Nat. Soc.
**2013**, 2013, 405628. [Google Scholar] [CrossRef] - Elsayed, E.M. Dynamics and behavior of a higher order rational difference equation. J. Nonlinear Sci. Appl.
**2015**, 9, 1463–1474. [Google Scholar] [CrossRef] - Elsayed, E.M. New method to obtain periodic solutions of period two and three of a rational difference equation. Nonlinear Dyn.
**2015**, 79, 241–250. [Google Scholar] [CrossRef] - Elsayed, E.M.; Iricanin, B.D. On a max-type and a min-type difference equation. Appl. Math. Comput.
**2009**, 215, 608–614. [Google Scholar] [CrossRef] - Elsayed, E.M.; El-Dessoky, M.M. Dynamics and behavior of a higher order rational recursive sequence. Adv. Differ. Equ.
**2012**, 69, 2012. [Google Scholar] [CrossRef] - Foupouagnigni, M.; Mboutngam, S. On the Polynomial Solution of Divided-Difference Equations of the Hypergeometric Type on Nonuniform Lattices. Axioms
**2019**, 8, 47. [Google Scholar] [CrossRef] - Foupouagnigni, M.; Koepf, W.; Kenfack-Nangho, M.; Mboutngam, S. On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices. Axioms
**2013**, 2, 404–434. [Google Scholar] [CrossRef] - Gil, M. Solution Estimates for the Discrete Lyapunov Equation in a Hilbert Space and Applications to Difference Equations. Axioms
**2019**, 8, 20. [Google Scholar] [CrossRef] - Haghighi, A.M.; Mishev, D.P. Difference and Differential Equations with Applications in Queueing Theory; John Wiley & Sons Inc.: Hoboken, NJ, USA, 2013. [Google Scholar]
- Kalabusic, S.; Kulenovic, M.R.S. On the recursive sequnence J
_{n+1}= (αJ_{n−1}+ βJ_{n−2})/(γJ_{n−1}+ δJ_{n−2}). J. Differ. Equ. Appl.**2003**, 9, 701–720. [Google Scholar] - Kelley, W.G.; Peterson, A.C. Difference Equations: An Introduction with Applications, 2nd ed.; Harcour Academic: New York, NY, USA, 2001. [Google Scholar]
- Kocic, V.L.; Ladas, G. Global Behavior of Nonlinear Difference Equations of Higher Order with Applications; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1993. [Google Scholar]
- Kulenovic, M.R.S.; Ladas, G. Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures; Chapman & Hall/CRC Press: Boca Raton, FL, USA, 2001. [Google Scholar]
- Liu, X. A note on the existence of periodic solutions in discrete predator-prey models. Appl. Math. Model.
**2010**, 34, 2477–2483. [Google Scholar] [CrossRef] - Ma, W.-X. Global behavior of a new rational nonlinear higher-order difference equation. Complexity
**2019**, 2019, 2048941. [Google Scholar] [CrossRef] - Migda, M.; Migda, J. Nonoscillatory Solutions to Second-Order Neutral Difference Equations. Symmetry
**2018**, 10, 207. [Google Scholar] [CrossRef] - Moaaz, O. Comment on new method to obtain periodic solutions of period two and three of a rational difference equation [Nonlinear Dyn 79: 241–250]. Nonlinear Dyn.
**2017**, 88, 1043–1049. [Google Scholar] [CrossRef] - Moaaz, O. Dynamics of difference equation J
_{n+1}= f(J_{n−l}, J_{n−k}). Adv. Differ. Equ.**2018**, 447, 2018. [Google Scholar] - Moaaz, O.; Chalishajar, D.; Bazighifan, O. Some Qualitative Behavior of Solutions of General Class of Difference Equations. Mathematics
**2019**, 7, 585. [Google Scholar] [CrossRef] - Pogrebkov, A. Hirota Difference Equation and Darboux System: Mutual Symmetry. Symmetry
**2019**, 11, 436. [Google Scholar] [CrossRef] - Stevic, S. On the recursive sequance x
_{n+1}= α + ${x}_{n-1}^{p}$/${x}_{n}^{p}$. J. Appl. Math. Comput.**2005**, 18, 229–234. [Google Scholar] - Stevic, S.; Kent, C.; Berenaut, S. A note on positive nonoscillatory solutions of the differential equation x
_{n+1}= α + ${x}_{n-1}^{p}$/${x}_{n}^{p}$. J. Diff. Eqs. Appl.**2006**, 12, 495–499. [Google Scholar] - Stevic, S. On the recursive sequence x
_{n+1}= α_{n}+ x_{n−1}/x_{n}. Dynam. Contin. Discret. Impuls. Syst. Ser. A Math. Anal.**2003**, 10, 911–917. [Google Scholar] - Stevic, S. A note on periodic character of a difference equation. J. Differ. Equ. Appl.
**2004**, 10, 929–932. [Google Scholar] [CrossRef] - Stevic, S. A short proof of the Cushing–Henson conjecture. Discret. Dyn. Nat. Soc.
**2006**, 4, 37264. [Google Scholar] [CrossRef] - Stevic, S. Global stability and asymptotics of some classes of rational difference equations. J. Math. Anal. Appl.
**2006**, 316, 60–68. [Google Scholar] [CrossRef] - Stevic, S. Asymptotics of some classes of higher order difference equations. Discret. Dyn. Nat. Soc.
**2007**, 2007, 56813. [Google Scholar] [CrossRef] - Stevic, S. Asymptotic periodicity of a higher order difference equation. Discret. Dyn. Nat. Soc.
**2007**, 2007, 13737. [Google Scholar] [CrossRef] - Stevic, S. Existence of nontrivial solutions of a rational difference equation. Appl. Math. Lett.
**2007**, 20, 28–31. [Google Scholar] [CrossRef] - Taousser, F.Z.; Defoort, M.; Djemai, M.; Djouadi, S.M.; Tomsovic, K. Stability analysis of a class of switched nonlinear systems using the time scale theory. Nonlinear Anal. Hybrid Syst.
**2019**, 33, 195–210. [Google Scholar] [CrossRef] - Wang, C.; Agarwal, R.P. Almost periodic solution for a new type of neutral impulsive stochastic Lasota–Wazewska timescale model. Appl. Math. Lett.
**2017**, 70, 58–65. [Google Scholar] [CrossRef] - Yang, C. Positive Solutions for a Three-Point Boundary Value Problem of Fractional Q-Difference Equations. Symmetry
**2018**, 10, 358. [Google Scholar] [CrossRef] [Green Version]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chatzarakis, G.E.; Elabbasy, E.M.; Moaaz, O.; Mahjoub, H.
Global Analysis and the Periodic Character of a Class of Difference Equations. *Axioms* **2019**, *8*, 131.
https://doi.org/10.3390/axioms8040131

**AMA Style**

Chatzarakis GE, Elabbasy EM, Moaaz O, Mahjoub H.
Global Analysis and the Periodic Character of a Class of Difference Equations. *Axioms*. 2019; 8(4):131.
https://doi.org/10.3390/axioms8040131

**Chicago/Turabian Style**

Chatzarakis, George E., Elmetwally M. Elabbasy, Osama Moaaz, and Hamida Mahjoub.
2019. "Global Analysis and the Periodic Character of a Class of Difference Equations" *Axioms* 8, no. 4: 131.
https://doi.org/10.3390/axioms8040131