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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

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

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 Articles
Authors

Mehmet Gümüş 0000-0002-7447-479X

Şeyma Irmak Eğilmez 0000-0003-1781-5399

Publication Date December 31, 2023
Submission Date August 2, 2023
Acceptance Date December 19, 2023
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 Gümüş M, Eğilmez ŞI. The Qualitative Analysis of Some Difference Equations Using Homogeneous Functions. Fundam. J. Math. Appl. December 2023;6(4):218-231. doi:10.33401/fujma.1336964
Chicago Gümüş, Mehmet, and Şeyma Irmak Eğilmez. “The Qualitative Analysis of Some Difference Equations Using Homogeneous Functions”. Fundamental Journal of Mathematics and Applications 6, no. 4 (December 2023): 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 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, 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 2023), 218-231. https://doi.org/10.33401/fujma.1336964.
JAMA 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, 2023, pp. 218-31, doi:10.33401/fujma.1336964.
Vancouver Gümüş M, Eğilmez ŞI. The Qualitative Analysis of Some Difference Equations Using Homogeneous Functions. Fundam. J. Math. Appl. 2023;6(4):218-31.
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