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Global Analysis of a $(1,2)$-Type System of Non-Linear Difference Equations

Year 2021, Volume: 4 Issue: 1, 26 - 38, 29.03.2021
https://doi.org/10.33434/cams.837197

Abstract

This paper deals with the study of global analysis of following $(1,2)-$type system of non-linear difference equations:\[ u_{n+1}=\frac{\alpha v_{n-1}}{\beta +\gamma v_{n}^{p}v_{n-2}^{q}},\ \ \ \ \ \ v_{n+1}=\frac{\alpha _{1}u_{n-1}}{\beta _{1}+\gamma _{1}u_{n}^{p}u_{n-2}^{q}},\ \ \ \ n=0,1,... \] where the parameters $\alpha ,\beta ,\gamma ,\alpha _{1},\beta _{1},\gamma _{1,}p,q$ and the initial conditions $u_{i},v_{i},$ $i=-2,-1,0$ are non negative real numbers.

References

  • [1] R. Agarwal, Difference equations and inequalities, theory, methods and applications, Marcel Dekker Inc., New York 1992.
  • [2] R. Abo-Zeid and H. Kamal, Global behavior of two rational third order difference equations, Univers. J. Math. Appl., 2(4) (2019), 212-217.
  • [3] S. Abualrub and M. Aloqeili, Dynamics of the System of Difference Equations $x_{n+1}=A+y_{n-k}/y_{n},$\textit{\ }$% y_{n+1}=B+x_{n-k}/x_{n}$, Qual. Theory Dyn. Syst., 19(2) (2020), 1-19.
  • [4] A. M. Ahmed, On the Dynamics of Higher-Order Rational Difference Equation, Discrete Dyn. Nat. Soc., 2011, Article ID: 419789, 8 pages.
  • [5] A. M. Ahmed, The Dynamics of the Recursive Sequences $x_{n+1}=\frac{bx_{n-1}}{A+Bx_{n}^{p}x_{n-2}^{p}}$, J.Pure Appl. Math.: Adv. Applic., 1(2) (2009), 215-223.
  • [6] M. M. Alzubaidi and E. M. Elsayed, Analytical and Solutions of Fourth Order Difference Equations, Commun. Adv. Math. Sci., 2(1) (2019), 9-21.
  • [7] Q. Din, T. F. Ibrahim and A.Q. Khan, Behavior of a competitive system of second order difference equations, Sci. World J., Article ID: 283982.
  • [8] Q. Din, Asymptotic behavior of an anti-competitive system of second-order difference equations, J. Egyptian Math. Soc., 24(1) (2016), 37-43.
  • [9] H. M. El-Owaidy, A. M. Ahmed and A. M. Youssef, The Dynamics of the Recursive Sequence $x_{n+1}=\frac{\alpha x_{n-1}% }{\beta +\gamma x_{n-2}^{p}}$, Appl. Math. Lett., 18 (2005), 1013-1018.
  • [10] M. M. El-Dessoky, E. M. Elsayed, E. M. Elabbasy and A. Asiri, Expressions of the solutions of some systems of difference equations, J. Comput. Anal. Appl., 27(7) (2019), 1161-1172.
  • [11] S. Elaydi, An Introduction to Difference Equations, 3rd ed., Springer-Verlag, New York, 2005.
  • [12] M. Gocen and M. Guneysu, The Global Attractivity of some ratinal difference equations, J. Comp. Anal. Appl., 25 (7) (2018), 1233-1243.
  • [13] M. Gümüş and Y. Soykan, Global Character of a Six-Dimensional Nonlinear System of Difference Equations, Discrete Dyn. Nat. Soc., 2016, Article ID 6842521.
  • [14] M. Gumus and O. Ocalan, The Qualitative Analysis of a Rational System of Difference Equations, J. Fract. Calc. Appl., 9(2) (2018), 113-126.
  • [15] M. Gumus and Y. Soykan, Dynamics of Positive Solutions of a Higher Order Fractional Difference Equation with Arbitrary Powers , J. Appl. Math. Inf., 35(3-4) (2017), 267-276.
  • [16] V. Kocic and G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, Kluwer Academic Publishers, Dordrecht, (1993).
  • [17] F. H. Gumus, Yuksek Mertebeden Fark Denklemlerinin Global Davranışları Uzerine, Master Thesis, Afyon Kocatepe Universitesi Fen Bilimleri Enstitu¨su¨ Afyon (2015).
  • [18] A. Khelifa, Y. Halim and M. Berkal, Solutions of a system of two higher-order difference equations in terms of lucas sequence, Univers. J. Math. Appl., 2(4) (2019), 202-211.
  • [19] M. A. Kerker, E. Hadidi, and A. Salmi, Qualitative behavior of a higher-order nonautonomous rational difference equation, J. Appl. Math. Comput., 64 (2020), 399–409.
  • [20] M. R. S. Kulenovic and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica. Chapman and Hall/CRC, Boca Raton, London. (2002).
  • [21] M. R. S. Kulenovic and G. Ladas, Dynamics of second order rational difference equations, Chapman and Hall/CRC, (2001).
  • [22] M. Kara, N. Touafek and Y. Yazlık, Well-Defined Solutions of a Three-Dimensional System of Difference Equations, Gazi Univ. J.Sci., 33(3), 767-778.
  • [23] W. X. Ma, Global Behavior of an Arbitrary-Order Nonlinear Difference Equation with a Nonnegative Function, Mathematics, 8(5) (2020), 825.
  • [24] M. Merdan and S. Sisman, Investigation of linear difference equations with random effects, Adv. Difference Equ., 2020 (1) (2020), 1-19.
  • [25] M. Pituk, More on Poincares and Perrons Theorems for Difference Equations, J. Difference Equ. Applic., 8(3) (2002), 201-216.
  • [26] A. Sanbo, E. M. Elsayed and F. Alzahrani, Dynamics of the nonlinear rational difference equation $x_{n+1}=(Ax_{n-\alpha }x_{n-\beta }+Bx_{n-\gamma })/(Cx_{n-\alpha }x_{n-\beta }+Dx_{n-\gamma }),$; Indian J. Pure Appl. Math., 50(2) (2019), 385-401.
  • [27] M. N. Qureshi and A. Q. Khan, Global dynamics of $(1,2)$-type systems of difference equations, Malaya J. Matematik, 6(2), 408-416.
  • [28] M. N. Qureshi, Q. Din and A. Q. Khan, Asymptotic behavior of an anti-competitive system of rational difference equations, Life Sci. J., 11(2014): 1-7.
  • [29] M. N. Qureshi and Q. Din, Oualitative behavior of an anti-competitive system of third-order rational difference equations, Comput. Ecology Software, 4(2) (2014), 104-115.
  • [30] H. Sedaghat, Nonlinear difference equations theory with applications to social science models, 15. Springer Science and Business Media, (2003).
  • [31] D. T. Tollu and I. Yalc¸ınkaya, Global behavior of a three dimensional system of difference equations of order three, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68(1) (2019), 1-16.
  • [32] I. Yalcınkaya, On the Global Asymptotic Stability of a Second-Order System of Difference Equations, Discrete Dyn. Nat. Soc., 2008, Article ID 860152.
  • [33] D. Zhang, W. Ji, L. Wang and X. Li, On the symmetrical system of rational difference equation $x_{n+1}=A+y_{n-k}/y_{n}$% \textit{, }$y_{n+1}=A+x_{n-k}/x_{n}$, Appl. Math., 4 (2013), 834-837.
Year 2021, Volume: 4 Issue: 1, 26 - 38, 29.03.2021
https://doi.org/10.33434/cams.837197

Abstract

References

  • [1] R. Agarwal, Difference equations and inequalities, theory, methods and applications, Marcel Dekker Inc., New York 1992.
  • [2] R. Abo-Zeid and H. Kamal, Global behavior of two rational third order difference equations, Univers. J. Math. Appl., 2(4) (2019), 212-217.
  • [3] S. Abualrub and M. Aloqeili, Dynamics of the System of Difference Equations $x_{n+1}=A+y_{n-k}/y_{n},$\textit{\ }$% y_{n+1}=B+x_{n-k}/x_{n}$, Qual. Theory Dyn. Syst., 19(2) (2020), 1-19.
  • [4] A. M. Ahmed, On the Dynamics of Higher-Order Rational Difference Equation, Discrete Dyn. Nat. Soc., 2011, Article ID: 419789, 8 pages.
  • [5] A. M. Ahmed, The Dynamics of the Recursive Sequences $x_{n+1}=\frac{bx_{n-1}}{A+Bx_{n}^{p}x_{n-2}^{p}}$, J.Pure Appl. Math.: Adv. Applic., 1(2) (2009), 215-223.
  • [6] M. M. Alzubaidi and E. M. Elsayed, Analytical and Solutions of Fourth Order Difference Equations, Commun. Adv. Math. Sci., 2(1) (2019), 9-21.
  • [7] Q. Din, T. F. Ibrahim and A.Q. Khan, Behavior of a competitive system of second order difference equations, Sci. World J., Article ID: 283982.
  • [8] Q. Din, Asymptotic behavior of an anti-competitive system of second-order difference equations, J. Egyptian Math. Soc., 24(1) (2016), 37-43.
  • [9] H. M. El-Owaidy, A. M. Ahmed and A. M. Youssef, The Dynamics of the Recursive Sequence $x_{n+1}=\frac{\alpha x_{n-1}% }{\beta +\gamma x_{n-2}^{p}}$, Appl. Math. Lett., 18 (2005), 1013-1018.
  • [10] M. M. El-Dessoky, E. M. Elsayed, E. M. Elabbasy and A. Asiri, Expressions of the solutions of some systems of difference equations, J. Comput. Anal. Appl., 27(7) (2019), 1161-1172.
  • [11] S. Elaydi, An Introduction to Difference Equations, 3rd ed., Springer-Verlag, New York, 2005.
  • [12] M. Gocen and M. Guneysu, The Global Attractivity of some ratinal difference equations, J. Comp. Anal. Appl., 25 (7) (2018), 1233-1243.
  • [13] M. Gümüş and Y. Soykan, Global Character of a Six-Dimensional Nonlinear System of Difference Equations, Discrete Dyn. Nat. Soc., 2016, Article ID 6842521.
  • [14] M. Gumus and O. Ocalan, The Qualitative Analysis of a Rational System of Difference Equations, J. Fract. Calc. Appl., 9(2) (2018), 113-126.
  • [15] M. Gumus and Y. Soykan, Dynamics of Positive Solutions of a Higher Order Fractional Difference Equation with Arbitrary Powers , J. Appl. Math. Inf., 35(3-4) (2017), 267-276.
  • [16] V. Kocic and G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, Kluwer Academic Publishers, Dordrecht, (1993).
  • [17] F. H. Gumus, Yuksek Mertebeden Fark Denklemlerinin Global Davranışları Uzerine, Master Thesis, Afyon Kocatepe Universitesi Fen Bilimleri Enstitu¨su¨ Afyon (2015).
  • [18] A. Khelifa, Y. Halim and M. Berkal, Solutions of a system of two higher-order difference equations in terms of lucas sequence, Univers. J. Math. Appl., 2(4) (2019), 202-211.
  • [19] M. A. Kerker, E. Hadidi, and A. Salmi, Qualitative behavior of a higher-order nonautonomous rational difference equation, J. Appl. Math. Comput., 64 (2020), 399–409.
  • [20] M. R. S. Kulenovic and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica. Chapman and Hall/CRC, Boca Raton, London. (2002).
  • [21] M. R. S. Kulenovic and G. Ladas, Dynamics of second order rational difference equations, Chapman and Hall/CRC, (2001).
  • [22] M. Kara, N. Touafek and Y. Yazlık, Well-Defined Solutions of a Three-Dimensional System of Difference Equations, Gazi Univ. J.Sci., 33(3), 767-778.
  • [23] W. X. Ma, Global Behavior of an Arbitrary-Order Nonlinear Difference Equation with a Nonnegative Function, Mathematics, 8(5) (2020), 825.
  • [24] M. Merdan and S. Sisman, Investigation of linear difference equations with random effects, Adv. Difference Equ., 2020 (1) (2020), 1-19.
  • [25] M. Pituk, More on Poincares and Perrons Theorems for Difference Equations, J. Difference Equ. Applic., 8(3) (2002), 201-216.
  • [26] A. Sanbo, E. M. Elsayed and F. Alzahrani, Dynamics of the nonlinear rational difference equation $x_{n+1}=(Ax_{n-\alpha }x_{n-\beta }+Bx_{n-\gamma })/(Cx_{n-\alpha }x_{n-\beta }+Dx_{n-\gamma }),$; Indian J. Pure Appl. Math., 50(2) (2019), 385-401.
  • [27] M. N. Qureshi and A. Q. Khan, Global dynamics of $(1,2)$-type systems of difference equations, Malaya J. Matematik, 6(2), 408-416.
  • [28] M. N. Qureshi, Q. Din and A. Q. Khan, Asymptotic behavior of an anti-competitive system of rational difference equations, Life Sci. J., 11(2014): 1-7.
  • [29] M. N. Qureshi and Q. Din, Oualitative behavior of an anti-competitive system of third-order rational difference equations, Comput. Ecology Software, 4(2) (2014), 104-115.
  • [30] H. Sedaghat, Nonlinear difference equations theory with applications to social science models, 15. Springer Science and Business Media, (2003).
  • [31] D. T. Tollu and I. Yalc¸ınkaya, Global behavior of a three dimensional system of difference equations of order three, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68(1) (2019), 1-16.
  • [32] I. Yalcınkaya, On the Global Asymptotic Stability of a Second-Order System of Difference Equations, Discrete Dyn. Nat. Soc., 2008, Article ID 860152.
  • [33] D. Zhang, W. Ji, L. Wang and X. Li, On the symmetrical system of rational difference equation $x_{n+1}=A+y_{n-k}/y_{n}$% \textit{, }$y_{n+1}=A+x_{n-k}/x_{n}$, Appl. Math., 4 (2013), 834-837.
There are 33 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Emine Yener

Mehmet Gümüş

Publication Date March 29, 2021
Submission Date December 7, 2020
Acceptance Date December 28, 2020
Published in Issue Year 2021 Volume: 4 Issue: 1

Cite

APA Yener, E., & Gümüş, M. (2021). Global Analysis of a $(1,2)$-Type System of Non-Linear Difference Equations. Communications in Advanced Mathematical Sciences, 4(1), 26-38. https://doi.org/10.33434/cams.837197
AMA Yener E, Gümüş M. Global Analysis of a $(1,2)$-Type System of Non-Linear Difference Equations. Communications in Advanced Mathematical Sciences. March 2021;4(1):26-38. doi:10.33434/cams.837197
Chicago Yener, Emine, and Mehmet Gümüş. “Global Analysis of a $(1,2)$-Type System of Non-Linear Difference Equations”. Communications in Advanced Mathematical Sciences 4, no. 1 (March 2021): 26-38. https://doi.org/10.33434/cams.837197.
EndNote Yener E, Gümüş M (March 1, 2021) Global Analysis of a $(1,2)$-Type System of Non-Linear Difference Equations. Communications in Advanced Mathematical Sciences 4 1 26–38.
IEEE E. Yener and M. Gümüş, “Global Analysis of a $(1,2)$-Type System of Non-Linear Difference Equations”, Communications in Advanced Mathematical Sciences, vol. 4, no. 1, pp. 26–38, 2021, doi: 10.33434/cams.837197.
ISNAD Yener, Emine - Gümüş, Mehmet. “Global Analysis of a $(1,2)$-Type System of Non-Linear Difference Equations”. Communications in Advanced Mathematical Sciences 4/1 (March 2021), 26-38. https://doi.org/10.33434/cams.837197.
JAMA Yener E, Gümüş M. Global Analysis of a $(1,2)$-Type System of Non-Linear Difference Equations. Communications in Advanced Mathematical Sciences. 2021;4:26–38.
MLA Yener, Emine and Mehmet Gümüş. “Global Analysis of a $(1,2)$-Type System of Non-Linear Difference Equations”. Communications in Advanced Mathematical Sciences, vol. 4, no. 1, 2021, pp. 26-38, doi:10.33434/cams.837197.
Vancouver Yener E, Gümüş M. Global Analysis of a $(1,2)$-Type System of Non-Linear Difference Equations. Communications in Advanced Mathematical Sciences. 2021;4(1):26-38.

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