Research Article
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The Qualitative Analysis of Some Difference Equations Using Homogeneous Functions

Year 2023, Volume: 6 Issue: 4, 218 - 231, 31.12.2023

Mehmet Gümüş , Şeyma Irmak Eğilmez

https://doi.org/10.33401/fujma.1336964
https://izlik.org/JA24SZ88HZ

Abstract

This article deals with the qualitative analysis of a general class of difference equations. That is, we examine the periodicity nature and the stability character of some non-linear second-order difference equations. Homogeneous functions are used while examining the character of the solutions of introduced difference equations. Moreover, a new technique available in the literature is used to examine the periodic solutions of these equations.

Keywords

Two periodic solutions three periodic solutions qualitative analysis locally asymptotic stable.

References

  • [1] R. Abo-Zeid, Global attractivity of a higher-order difference equation, Discrete Dyn. Nat. Soc., 2012 (2012), Article ID 930410. $\href{https://doi.org/10.1155/2012/930410}{\color{blue}{[\mbox{CrossRef}]}}$
  • [2] M. Gümüş¸, The periodicity of positive solutions of the non-linear difference equation $x_{n+1}=\alpha+(x_{n-k}^{p}/x_{n}^{q})$, Discrete Dyn. Nat. Soc., 2013 (2013), Article ID 742912. $\href{https://doi.org/10.1155/2013/742912}{\color{blue}{[\mbox{CrossRef}]}}$
  • [3] M. Gümüş, Global dynamics of solutions of a new class of rational difference equations, Konuralp J. Math., 2(7) (2019), 380-387. $\href{https://dergipark.org.tr/tr/download/article-file/844502}{\color{blue}{[\mbox{CrossRef}]}}$
  • [4] M. Gümüş, Analysis of periodicity for a new class of non-linear difference equations by using a new method, Electron. J. Math. Anal. Appl., 8(1) (2020), 109-116. $\href{https://journals.ekb.eg/article_312810.html}{\color{blue}{[\mbox{CrossRef}]}}$
  • [5] Y. Halim ,N. Touafek and Y. Yazlık, Dynamic behavior of a second-order non-linear rational difference equation, Turkish J. Math., 6(39) (2015), 1004-1018. $\href{https://doi.org/10.3906/mat-1503-80}{\color{blue}{[\mbox{CrossRef}]}}$
  • [6] O. Moaaz, Comment on ”New method to obtain periodic solutions of period two and three of a rational difference equation”, Nonlinear Dyn., 88 (2017), 1043-1049. $\href{https://doi.org/10.1007/s11071-016-3293-0}{\color{blue}{[\mbox{CrossRef}]}}$
  • [7] O. Moaaz and A.A. Altuwaijri, The dynamics of a general model of the nonlinear difference equation and its applications, Axioms, 12(6) (2023), 598. $\href{https://doi.org/10.3390/axioms12060598}{\color{blue}{[\mbox{CrossRef}]}}$
  • [8] N. Touafek and Y. Halim, Global attractivity of a rational difference equation, Math. Sci. Lett., 3(2) (2013), 161-165. $\href{http://dx.doi.org/10.12785/msl/020302}{\color{blue}{[\mbox{CrossRef}]}}$
  • [9] İ. Yalçınkaya, On the recursive sequence $x_{n+1}=\alpha +x_{n-m}/x_{n}^{k}$, Discrete Dyn. Nat. Soc., 2008 (2008), Article ID 805460. $\href{http://dx.doi.org/10.1155/2008/805460}{\color{blue}{[\mbox{CrossRef}]}}$
  • [10] L.J.S. Mallen, An Introduction to Mathematical Biology, Pearson/Prentice Hall, (2007).
  • [11] C.W. Clark, A delayed recruitment model of population dynamics with an application to baleen whale populations, J. Math. Biol., 3 (1976), 381-391. $\href{https://doi.org/10.1007/BF00275067}{\color{blue}{[\mbox{CrossRef}]}}$
  • [12] L. Edelstein-Keshet, Mathematical Models in Biology, The Random House/Birkhauser Mathematical Series, New York, (1988).
  • [13] H.I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker Inc., New York, (1980).
  • [14] F.C. Hoppensteadt, Mathematical Models of Population Biology, Cambridge University Press, Cambridge, (1982). $\href{https://doi.org/10.1017/CBO9780511624087}{\color{blue}{[\mbox{CrossRef}]}}$
  • [15] R.M. May and G.F. Oster, Bifurcations and dynamic complexity in simple ecological models, Am. Nat., 110(974) (1976), 573-599. $\href{https://www.jstor.org/stable/2459579}{\color{blue}{[\mbox{CrossRef}]}}$
  • [16] R.M. May, Biological populations obeying difference equations: stable points, stable cycles, and chaos, J. Theor. Biol., 51(2) (1975), 511- 524. $\href{https://doi.org/10.1016/0022-5193(75)90078-8}{\color{blue}{[\mbox{CrossRef}]}}$
  • [17] E.C. Pielou, An Introduction to Mathematical Ecology, Wiley Interscience, New York, (1969).
  • [18] S. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, (2005). $\href{https://doi.org/10.1007/0-387-27602-5}{\color{blue}{[\mbox{CrossRef}]}}$
  • [19] V. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, (1993). $\href{https://doi.org/10.1007/978-94-017-1703-8}{\color{blue}{[\mbox{CrossRef}]}}$
  • [20] M.R.S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations, Chapman & Hall/CRC, (2001). $\href{https://doi.org/10.1201/9781420035384}{\color{blue}{[\mbox{CrossRef}]}}$
  • [21] E.M. Elsayed, New method to obtain periodic solutions of period two and three of a rational difference equation, Nonlinear Dyn., 1(79) (2014), 241-250. $\href{https://doi.org/10.1007/s11071-014-1660-2}{\color{blue}{[\mbox{CrossRef}]}}$
  • [22] O. Moaaz, D. Chalishajar and O. Bazighifan, Some qualitative behavior of solutions of general class of difference equation, Mathematics, 77 (2019), 585. $\href{https://doi.org/10.3390/math7070585}{\color{blue}{[\mbox{CrossRef}]}}$
  • [23] O. Moaaz, Dynamics of difference equation $x_{n+1}=f(x_{n-l},x_{n-k})$, Adv. Differ. Equ., 1 2018447 (2018). $\href{https://doi.org/10.1186/s13662-018-1896-0}{\color{blue}{[\mbox{CrossRef}]}}$
  • [24] S. Stevic, B. Iricanin, W. Kosmola and Z. Smarda, Note on difference equations with the right-hand side function nonincreasing in each variable, J. Inequal. Appl., 2022 (2022), 25. $\href{https://doi.org/10.1186/s13660-022-02761-9}{\color{blue}{[\mbox{CrossRef}]}}$
  • [25] M.A.E. Abdelrahman, G.E. Chatzarakis, T. Li and O. Moaaz, On the difference equations $x_{n+1}=ax_{n-l}+bx_{n-k}+f(x_{n-l},x_{n-k})$, Adv. Differ. Equ., 1 (2018), 1-14. $\href{https://doi.org/10.1186/s13662-018-1880-8}{\color{blue}{[\mbox{CrossRef}]}}$
  • [26] M.A.E. Abdelrahman, On the difference equation $z_{m+1}=f(z_{m},z_{m-1},\ldots ,z_{m-k}).$, J. Taibah Univ. Sci., 1(13) (2019), 1014-1021. $\href{https://doi.org/10.1080/16583655.2019.1678866}{\color{blue}{[\mbox{CrossRef}]}}$
  • [27] O. Moaaz, H. Mahjoub and A. Muhib, On the periodicity of general class of difference equations, Axioms, 9(3) (2019), 75. $\href{https://doi.org/10.3390/axioms9030075}{\color{blue}{[\mbox{CrossRef}]}}$
  • [28] 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, 1(11) (2023), 56-66. $\href{https://doi.org/10.36753/mathenot.1243583}{\color{blue}{[\mbox{CrossRef}]}}$

Year 2023, Volume: 6 Issue: 4, 218 - 231, 31.12.2023

Mehmet Gümüş , Şeyma Irmak Eğilmez

https://doi.org/10.33401/fujma.1336964
https://izlik.org/JA24SZ88HZ

Abstract

References

  • [1] R. Abo-Zeid, Global attractivity of a higher-order difference equation, Discrete Dyn. Nat. Soc., 2012 (2012), Article ID 930410. $\href{https://doi.org/10.1155/2012/930410}{\color{blue}{[\mbox{CrossRef}]}}$
  • [2] M. Gümüş¸, The periodicity of positive solutions of the non-linear difference equation $x_{n+1}=\alpha+(x_{n-k}^{p}/x_{n}^{q})$, Discrete Dyn. Nat. Soc., 2013 (2013), Article ID 742912. $\href{https://doi.org/10.1155/2013/742912}{\color{blue}{[\mbox{CrossRef}]}}$
  • [3] M. Gümüş, Global dynamics of solutions of a new class of rational difference equations, Konuralp J. Math., 2(7) (2019), 380-387. $\href{https://dergipark.org.tr/tr/download/article-file/844502}{\color{blue}{[\mbox{CrossRef}]}}$
  • [4] M. Gümüş, Analysis of periodicity for a new class of non-linear difference equations by using a new method, Electron. J. Math. Anal. Appl., 8(1) (2020), 109-116. $\href{https://journals.ekb.eg/article_312810.html}{\color{blue}{[\mbox{CrossRef}]}}$
  • [5] Y. Halim ,N. Touafek and Y. Yazlık, Dynamic behavior of a second-order non-linear rational difference equation, Turkish J. Math., 6(39) (2015), 1004-1018. $\href{https://doi.org/10.3906/mat-1503-80}{\color{blue}{[\mbox{CrossRef}]}}$
  • [6] O. Moaaz, Comment on ”New method to obtain periodic solutions of period two and three of a rational difference equation”, Nonlinear Dyn., 88 (2017), 1043-1049. $\href{https://doi.org/10.1007/s11071-016-3293-0}{\color{blue}{[\mbox{CrossRef}]}}$
  • [7] O. Moaaz and A.A. Altuwaijri, The dynamics of a general model of the nonlinear difference equation and its applications, Axioms, 12(6) (2023), 598. $\href{https://doi.org/10.3390/axioms12060598}{\color{blue}{[\mbox{CrossRef}]}}$
  • [8] N. Touafek and Y. Halim, Global attractivity of a rational difference equation, Math. Sci. Lett., 3(2) (2013), 161-165. $\href{http://dx.doi.org/10.12785/msl/020302}{\color{blue}{[\mbox{CrossRef}]}}$
  • [9] İ. Yalçınkaya, On the recursive sequence $x_{n+1}=\alpha +x_{n-m}/x_{n}^{k}$, Discrete Dyn. Nat. Soc., 2008 (2008), Article ID 805460. $\href{http://dx.doi.org/10.1155/2008/805460}{\color{blue}{[\mbox{CrossRef}]}}$
  • [10] L.J.S. Mallen, An Introduction to Mathematical Biology, Pearson/Prentice Hall, (2007).
  • [11] C.W. Clark, A delayed recruitment model of population dynamics with an application to baleen whale populations, J. Math. Biol., 3 (1976), 381-391. $\href{https://doi.org/10.1007/BF00275067}{\color{blue}{[\mbox{CrossRef}]}}$
  • [12] L. Edelstein-Keshet, Mathematical Models in Biology, The Random House/Birkhauser Mathematical Series, New York, (1988).
  • [13] H.I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker Inc., New York, (1980).
  • [14] F.C. Hoppensteadt, Mathematical Models of Population Biology, Cambridge University Press, Cambridge, (1982). $\href{https://doi.org/10.1017/CBO9780511624087}{\color{blue}{[\mbox{CrossRef}]}}$
  • [15] R.M. May and G.F. Oster, Bifurcations and dynamic complexity in simple ecological models, Am. Nat., 110(974) (1976), 573-599. $\href{https://www.jstor.org/stable/2459579}{\color{blue}{[\mbox{CrossRef}]}}$
  • [16] R.M. May, Biological populations obeying difference equations: stable points, stable cycles, and chaos, J. Theor. Biol., 51(2) (1975), 511- 524. $\href{https://doi.org/10.1016/0022-5193(75)90078-8}{\color{blue}{[\mbox{CrossRef}]}}$
  • [17] E.C. Pielou, An Introduction to Mathematical Ecology, Wiley Interscience, New York, (1969).
  • [18] S. Elaydi, An Introduction to Difference Equations, Springer-Verlag, New York, (2005). $\href{https://doi.org/10.1007/0-387-27602-5}{\color{blue}{[\mbox{CrossRef}]}}$
  • [19] V. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, (1993). $\href{https://doi.org/10.1007/978-94-017-1703-8}{\color{blue}{[\mbox{CrossRef}]}}$
  • [20] M.R.S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations, Chapman & Hall/CRC, (2001). $\href{https://doi.org/10.1201/9781420035384}{\color{blue}{[\mbox{CrossRef}]}}$
  • [21] E.M. Elsayed, New method to obtain periodic solutions of period two and three of a rational difference equation, Nonlinear Dyn., 1(79) (2014), 241-250. $\href{https://doi.org/10.1007/s11071-014-1660-2}{\color{blue}{[\mbox{CrossRef}]}}$
  • [22] O. Moaaz, D. Chalishajar and O. Bazighifan, Some qualitative behavior of solutions of general class of difference equation, Mathematics, 77 (2019), 585. $\href{https://doi.org/10.3390/math7070585}{\color{blue}{[\mbox{CrossRef}]}}$
  • [23] O. Moaaz, Dynamics of difference equation $x_{n+1}=f(x_{n-l},x_{n-k})$, Adv. Differ. Equ., 1 2018447 (2018). $\href{https://doi.org/10.1186/s13662-018-1896-0}{\color{blue}{[\mbox{CrossRef}]}}$
  • [24] S. Stevic, B. Iricanin, W. Kosmola and Z. Smarda, Note on difference equations with the right-hand side function nonincreasing in each variable, J. Inequal. Appl., 2022 (2022), 25. $\href{https://doi.org/10.1186/s13660-022-02761-9}{\color{blue}{[\mbox{CrossRef}]}}$
  • [25] M.A.E. Abdelrahman, G.E. Chatzarakis, T. Li and O. Moaaz, On the difference equations $x_{n+1}=ax_{n-l}+bx_{n-k}+f(x_{n-l},x_{n-k})$, Adv. Differ. Equ., 1 (2018), 1-14. $\href{https://doi.org/10.1186/s13662-018-1880-8}{\color{blue}{[\mbox{CrossRef}]}}$
  • [26] M.A.E. Abdelrahman, On the difference equation $z_{m+1}=f(z_{m},z_{m-1},\ldots ,z_{m-k}).$, J. Taibah Univ. Sci., 1(13) (2019), 1014-1021. $\href{https://doi.org/10.1080/16583655.2019.1678866}{\color{blue}{[\mbox{CrossRef}]}}$
  • [27] O. Moaaz, H. Mahjoub and A. Muhib, On the periodicity of general class of difference equations, Axioms, 9(3) (2019), 75. $\href{https://doi.org/10.3390/axioms9030075}{\color{blue}{[\mbox{CrossRef}]}}$
  • [28] 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, 1(11) (2023), 56-66. $\href{https://doi.org/10.36753/mathenot.1243583}{\color{blue}{[\mbox{CrossRef}]}}$
There are 28 citations in total.

Details

Primary Language English
Subjects Ordinary Differential Equations, Difference Equations and Dynamical Systems
Journal Section Research Article
Authors

Mehmet Gümüş 0000-0002-7447-479X

Şeyma Irmak Eğilmez 0000-0003-1781-5399

Submission Date August 2, 2023
Acceptance Date December 19, 2023
Publication Date December 31, 2023
DOI https://doi.org/10.33401/fujma.1336964
IZ https://izlik.org/JA24SZ88HZ
Published in Issue Year 2023 Volume: 6 Issue: 4

Cite

APA Gümüş, M., & Eğilmez, Ş. I. (2023). The Qualitative Analysis of Some Difference Equations Using Homogeneous Functions. Fundamental Journal of Mathematics and Applications, 6(4), 218-231. https://doi.org/10.33401/fujma.1336964
AMA 1.Gümüş M, Eğilmez ŞI. The Qualitative Analysis of Some Difference Equations Using Homogeneous Functions. Fundam. J. Math. Appl. 2023;6(4):218-231. doi:10.33401/fujma.1336964
Chicago Gümüş, Mehmet, and Şeyma Irmak Eğilmez. 2023. “The Qualitative Analysis of Some Difference Equations Using Homogeneous Functions”. Fundamental Journal of Mathematics and Applications 6 (4): 218-31. https://doi.org/10.33401/fujma.1336964.
EndNote Gümüş M, Eğilmez ŞI (December 1, 2023) The Qualitative Analysis of Some Difference Equations Using Homogeneous Functions. Fundamental Journal of Mathematics and Applications 6 4 218–231.
IEEE [1]M. Gümüş and Ş. I. Eğilmez, “The Qualitative Analysis of Some Difference Equations Using Homogeneous Functions”, Fundam. J. Math. Appl., vol. 6, no. 4, pp. 218–231, Dec. 2023, doi: 10.33401/fujma.1336964.
ISNAD Gümüş, Mehmet - Eğilmez, Şeyma Irmak. “The Qualitative Analysis of Some Difference Equations Using Homogeneous Functions”. Fundamental Journal of Mathematics and Applications 6/4 (December 1, 2023): 218-231. https://doi.org/10.33401/fujma.1336964.
JAMA 1.Gümüş M, Eğilmez ŞI. The Qualitative Analysis of Some Difference Equations Using Homogeneous Functions. Fundam. J. Math. Appl. 2023;6:218–231.
MLA Gümüş, Mehmet, and Şeyma Irmak Eğilmez. “The Qualitative Analysis of Some Difference Equations Using Homogeneous Functions”. Fundamental Journal of Mathematics and Applications, vol. 6, no. 4, Dec. 2023, pp. 218-31, doi:10.33401/fujma.1336964.
Vancouver 1.Gümüş M, Eğilmez ŞI. The Qualitative Analysis of Some Difference Equations Using Homogeneous Functions. Fundam. J. Math. Appl. [Internet]. 2023 Dec. 1;6(4):218-31. Available from: https://izlik.org/JA24SZ88HZ

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