Research Article
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Year 2023, , 56 - 66, 28.03.2023
https://doi.org/10.36753/mathenot.1243583

Abstract

References

  • [1] Elsayed, E. M.: New method to obtain periodic solutions of period two and three of a rational difference equation. Nonlinear Dynamics. 79 (1), 241-250 (2014).
  • [2] 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. 88, 1043-1049 (2017).
  • [3] Elaydi, S.: An introduction to difference equations. 3rd ed. Springer-Verlag. New York (2005).
  • [4] Kelley, W. G., Peterson, A. C.: Difference equations: An introduction with applications. Academic Press. New York (1991).
  • [5] Koci´c, V., Ladas, G.: Global behavior of non-linear difference equations of higher-order with applications. Kluwer Academic Publishers. Dordrecht (1993).
  • [6] Levin, S. A., May, R. M.: A note on difference-delay equations. Theoretical Population Biology. 9 (2), 178-187 (1976).
  • [7] Mickens, R. E.: Difference equations, theory and applications. Van Nostrand Rheinhold. (1990).
  • [8] Allen, L. J. S.: An introduction to mathematical biology. Pearson/Prentice Hall. New Jersey (2007).
  • [9] Murray, J. D.: Mathematical biology I: An introduction. 3rd ed. Springer. (2002).
  • [10] Pielou, E. C.: An introduction to mathematical ecology. Wiley Interscience. New York (1969).
  • [11] Oztepe, G. S.: An investigation on the Lasota-Wazewska model with a piecewise constant argument. Hacettepe Journal of Mathematics and Statistics. 50 (5), 1500-1508 (2021).
  • [12] Abo-Zeid, R.: Global attractivity of a higher-order difference equation. Discrete Dynamics in Nature and Society. 2012, 930410 (2012).
  • [13] Abo-Zeid, R.: Global behavior of a higher-order difference equation. Mathematica Slovaca. 64 (4), 931-940 (2014).
  • [14] Belhannache, F., Touafek, N., Abo-Zeid, R.: Dynamics of a third-order rational difference equation. Bulletin Mathematique de La Societe Des Sciences Mathematiques de Roumanie. 59(107) (1), 13-22 (2016).
  • [15] Gumus, M.: Global dynamics of solutions of a new class of rational difference equations. Konuralp Journal of Mathematics. 7 (2), 380-387 (2019).
  • [16] Gumus, M.: Analysis of periodicity for a new class of non-linear difference equations by using a new method. Electron. J. Math. Anal. Appl. 8, 109-116 (2020).
  • [17] Halim, Y., Touafek, N.,Yazlik, Y.: Dynamic behavior of a second-order non-linear rational difference equation. Turkish Journal of Mathematics. 39 (6), 1004-1018 (2015).
  • [18] Touafek, N., Halim, Y.: Global attractivity of a rational difference equation. Mathematical Sciences Letters. 2 (3), 161-165 (2013).
  • [19] Yalçınkaya, I.: On the difference equation xn+1 = + xn􀀀m=xk n: Discrete Dynamics in Nature and Society. 2008, 805460 (2008).
  • [20] Moaaz, O., Chalishajar, D., Bazighifan, O.: Some qualitative behavior of solutions of general class of difference equation. Mathematics. 7, 585 (2019).
  • [21] Moaaz, O.: Dynamics of difference equation xn+1 = f(xn􀀀l; xn􀀀k). Advances in Difference Equations. 2018, 447 (2018).
  • [22] Moaaz, O., Mahjoub, H., Muhib, A.: On the periodicity of general class of difference equations. Axioms. 9, 75 (2020).
  • [23] Abdelrahman, M. A. E.: On the difference equation zm+1 = f(zm; zm􀀀1; : : : ; zm􀀀k): Journal of Taibah University for Science. 13 (1), 1014-1021 (2019).
  • [24] Kulenovi´c, M. R. S., Ladas, G.: Dynamics of second-order rational difference equations. Chapman & Hall/CRC. (2001).
  • [25] Border, K. C.: Euler’s theorem for homogeneous function. Caltech Division of The Humanities and Social Sciences. 27, 16-34 (2017).
  • [26] Boulouh, M., Touafek, N., Tollu, D. T.: On the behavior of the solutions of an abstract system of difference equations. Journal of Applied Mathematics and Computing. 68, 2937-2969 (2022).

On the Qualitative Behavior of the Difference Equation $\delta _{m+1}=\omega +\zeta \frac{f(\delta _{m},\delta _{m-1})}{\delta _{m-1}^{\beta}}$

Year 2023, , 56 - 66, 28.03.2023
https://doi.org/10.36753/mathenot.1243583

Abstract

In this paper, we aim to investigate the qualitative behavior of a general class of non-linear difference equations. That is, the prime period two solutions, the prime period three solutions and the stability character are examined. We also use a new technique introduced in [1] by E. M. Elsayed and later developed by O. Moaaz in [2] to examine the existence of periodic solutions of these general equations. Moreover, we use homogeneous functions for the investigation of the dynamics of the aforementioned equations.

References

  • [1] Elsayed, E. M.: New method to obtain periodic solutions of period two and three of a rational difference equation. Nonlinear Dynamics. 79 (1), 241-250 (2014).
  • [2] 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. 88, 1043-1049 (2017).
  • [3] Elaydi, S.: An introduction to difference equations. 3rd ed. Springer-Verlag. New York (2005).
  • [4] Kelley, W. G., Peterson, A. C.: Difference equations: An introduction with applications. Academic Press. New York (1991).
  • [5] Koci´c, V., Ladas, G.: Global behavior of non-linear difference equations of higher-order with applications. Kluwer Academic Publishers. Dordrecht (1993).
  • [6] Levin, S. A., May, R. M.: A note on difference-delay equations. Theoretical Population Biology. 9 (2), 178-187 (1976).
  • [7] Mickens, R. E.: Difference equations, theory and applications. Van Nostrand Rheinhold. (1990).
  • [8] Allen, L. J. S.: An introduction to mathematical biology. Pearson/Prentice Hall. New Jersey (2007).
  • [9] Murray, J. D.: Mathematical biology I: An introduction. 3rd ed. Springer. (2002).
  • [10] Pielou, E. C.: An introduction to mathematical ecology. Wiley Interscience. New York (1969).
  • [11] Oztepe, G. S.: An investigation on the Lasota-Wazewska model with a piecewise constant argument. Hacettepe Journal of Mathematics and Statistics. 50 (5), 1500-1508 (2021).
  • [12] Abo-Zeid, R.: Global attractivity of a higher-order difference equation. Discrete Dynamics in Nature and Society. 2012, 930410 (2012).
  • [13] Abo-Zeid, R.: Global behavior of a higher-order difference equation. Mathematica Slovaca. 64 (4), 931-940 (2014).
  • [14] Belhannache, F., Touafek, N., Abo-Zeid, R.: Dynamics of a third-order rational difference equation. Bulletin Mathematique de La Societe Des Sciences Mathematiques de Roumanie. 59(107) (1), 13-22 (2016).
  • [15] Gumus, M.: Global dynamics of solutions of a new class of rational difference equations. Konuralp Journal of Mathematics. 7 (2), 380-387 (2019).
  • [16] Gumus, M.: Analysis of periodicity for a new class of non-linear difference equations by using a new method. Electron. J. Math. Anal. Appl. 8, 109-116 (2020).
  • [17] Halim, Y., Touafek, N.,Yazlik, Y.: Dynamic behavior of a second-order non-linear rational difference equation. Turkish Journal of Mathematics. 39 (6), 1004-1018 (2015).
  • [18] Touafek, N., Halim, Y.: Global attractivity of a rational difference equation. Mathematical Sciences Letters. 2 (3), 161-165 (2013).
  • [19] Yalçınkaya, I.: On the difference equation xn+1 = + xn􀀀m=xk n: Discrete Dynamics in Nature and Society. 2008, 805460 (2008).
  • [20] Moaaz, O., Chalishajar, D., Bazighifan, O.: Some qualitative behavior of solutions of general class of difference equation. Mathematics. 7, 585 (2019).
  • [21] Moaaz, O.: Dynamics of difference equation xn+1 = f(xn􀀀l; xn􀀀k). Advances in Difference Equations. 2018, 447 (2018).
  • [22] Moaaz, O., Mahjoub, H., Muhib, A.: On the periodicity of general class of difference equations. Axioms. 9, 75 (2020).
  • [23] Abdelrahman, M. A. E.: On the difference equation zm+1 = f(zm; zm􀀀1; : : : ; zm􀀀k): Journal of Taibah University for Science. 13 (1), 1014-1021 (2019).
  • [24] Kulenovi´c, M. R. S., Ladas, G.: Dynamics of second-order rational difference equations. Chapman & Hall/CRC. (2001).
  • [25] Border, K. C.: Euler’s theorem for homogeneous function. Caltech Division of The Humanities and Social Sciences. 27, 16-34 (2017).
  • [26] Boulouh, M., Touafek, N., Tollu, D. T.: On the behavior of the solutions of an abstract system of difference equations. Journal of Applied Mathematics and Computing. 68, 2937-2969 (2022).
There are 26 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Mehmet Gümüş 0000-0002-7447-479X

Şeyma Irmak Eğilmez 0000-0003-1781-5399

Publication Date March 28, 2023
Submission Date January 27, 2023
Acceptance Date March 11, 2023
Published in Issue Year 2023

Cite

APA Gümüş, M., & Eğilmez, Ş. I. (2023). On the Qualitative Behavior of the Difference Equation $\delta _{m+1}=\omega +\zeta \frac{f(\delta _{m},\delta _{m-1})}{\delta _{m-1}^{\beta}}$. Mathematical Sciences and Applications E-Notes, 11(1), 56-66. https://doi.org/10.36753/mathenot.1243583
AMA Gümüş M, Eğilmez ŞI. On the Qualitative Behavior of the Difference Equation $\delta _{m+1}=\omega +\zeta \frac{f(\delta _{m},\delta _{m-1})}{\delta _{m-1}^{\beta}}$. Math. Sci. Appl. E-Notes. March 2023;11(1):56-66. doi:10.36753/mathenot.1243583
Chicago Gümüş, Mehmet, and Şeyma Irmak Eğilmez. “On the Qualitative Behavior of the Difference Equation $\delta _{m+1}=\omega +\zeta \frac{f(\delta _{m},\delta _{m-1})}{\delta _{m-1}^{\beta}}$”. Mathematical Sciences and Applications E-Notes 11, no. 1 (March 2023): 56-66. https://doi.org/10.36753/mathenot.1243583.
EndNote Gümüş M, Eğilmez ŞI (March 1, 2023) On the Qualitative Behavior of the Difference Equation $\delta _{m+1}=\omega +\zeta \frac{f(\delta _{m},\delta _{m-1})}{\delta _{m-1}^{\beta}}$. Mathematical Sciences and Applications E-Notes 11 1 56–66.
IEEE M. Gümüş and Ş. I. Eğilmez, “On the Qualitative Behavior of the Difference Equation $\delta _{m+1}=\omega +\zeta \frac{f(\delta _{m},\delta _{m-1})}{\delta _{m-1}^{\beta}}$”, Math. Sci. Appl. E-Notes, vol. 11, no. 1, pp. 56–66, 2023, doi: 10.36753/mathenot.1243583.
ISNAD Gümüş, Mehmet - Eğilmez, Şeyma Irmak. “On the Qualitative Behavior of the Difference Equation $\delta _{m+1}=\omega +\zeta \frac{f(\delta _{m},\delta _{m-1})}{\delta _{m-1}^{\beta}}$”. Mathematical Sciences and Applications E-Notes 11/1 (March 2023), 56-66. https://doi.org/10.36753/mathenot.1243583.
JAMA Gümüş M, Eğilmez ŞI. On the Qualitative Behavior of the Difference Equation $\delta _{m+1}=\omega +\zeta \frac{f(\delta _{m},\delta _{m-1})}{\delta _{m-1}^{\beta}}$. Math. Sci. Appl. E-Notes. 2023;11:56–66.
MLA Gümüş, Mehmet and Şeyma Irmak Eğilmez. “On the Qualitative Behavior of the Difference Equation $\delta _{m+1}=\omega +\zeta \frac{f(\delta _{m},\delta _{m-1})}{\delta _{m-1}^{\beta}}$”. Mathematical Sciences and Applications E-Notes, vol. 11, no. 1, 2023, pp. 56-66, doi:10.36753/mathenot.1243583.
Vancouver Gümüş M, Eğilmez ŞI. On the Qualitative Behavior of the Difference Equation $\delta _{m+1}=\omega +\zeta \frac{f(\delta _{m},\delta _{m-1})}{\delta _{m-1}^{\beta}}$. Math. Sci. Appl. E-Notes. 2023;11(1):56-6.

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