Skip to main content
SpringerLink
Log in

On the behavior of the solutions of an abstract system of difference equations

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

The aim of the present work is to study of the behavior of the solutions of the following abstract system of difference equations defined by

$$\begin{aligned} x_{n+1}=f_{1}(x_{n},x_{n-1})+f_{2}(y_{n},y_{n-1}),\,y_{n+1}=g_{1}(x_{n},x_{n-1})+g_{2}(y_{n},y_{n-1}) \end{aligned}$$

where \(n\in \mathbb {N}_{0}\), the initial values \(x_{-1}\), \(x_{0}\), \(y_{-1}\) and \(y_{0}\) are positive real numbers, and the functions \( f_{1},\;f_{2},\;g_{1},\;g_{2}:(0,+\infty )^{2}\rightarrow (0,+\infty )\) are continuous and homogeneous of degree zero.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abdelrahman, M.A.E.: On the difference equation \(z_{m+1}=f(z_m,z_{m-1},...,z_{m-k})\). J. Taibah Univ. Sci. 13(1), 1014–1021 (2019)

    Article  Google Scholar 

  2. Amleh, A.M., Grove, E.A., Ladas, G., Georgiou, D.A.: On the recursive seqience \(x_{n+1}=\alpha +\frac{x_{n-1} }{x_{n}}\). J. Math. Anal. Appl. 233, 790–798 (1999)

    Article  MathSciNet  Google Scholar 

  3. Border, K. C.: Eulers Theorem for Homogeneous Functions. https://mathproblems123.files.wordpress.com/2011/02/eulerhomogeneity.pdf

  4. Camouzis, E., DeVault, R., Papaschinopoulos, G.: On the recursive sequence. Adv. Differ. Equ. 2005, 548479 (2005)

    Article  MathSciNet  Google Scholar 

  5. Dekkar, I., Touafek, N., Yazlik,Y.: Global stability of a third-order nonlinear system of difference equations with period-two coefficients, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 111 (2017), 325–347

  6. Devault, R., Scultz, S.W.: On the dynamics of \(x_{n+1}=\frac{ax_{n}+bx_{n-1}}{cx_{n}+dx_{n-2}}\). Commun. Appl. Nonlinear Anal. 12, 35–40 (2005)

    Google Scholar 

  7. Devault, R., Kocic, V.L., Stutson, D.: Global behavior of solutions of the nonlinear difference equation \( x_{n+1}=p_{n}+\frac{x_{n-1}}{x_{n}}\). J. Differ. Equ. Appl. 11(8), 707–719 (2005)

    Article  Google Scholar 

  8. Elaydi, S.: An Introduction to Di.erence Equations, Undergraduate Texts in Mathematics, 3rd edition, Springer, New York (2005)

  9. El-Owaidy, H.M., Ahmed, A.M., Mousa, M.S.: On asymptotic behaviour of the difference equation \(x_{n+1}=\alpha +\frac{x_{n-k}}{x_{n}}\). Appl. Math. Comput. 147(1), 163–167 (2004)

    MathSciNet  MATH  Google Scholar 

  10. Elsayed, E.M.: New method to obtain periodic solutions of period two and three of a rational difference equation. Nonlinear Dyn. 79, 241–250 (2015)

    Article  MathSciNet  Google Scholar 

  11. Grove, E.A, Ladas, G.: Periodicities in nonlinear difference equations, Advances in Discrete Math-ematics and Applications, Vol. 4, CHAPMAN and HALL/CRC, Boca Raton (2005)

  12. Gumus, M.: The global asymptotic stability of a system of difference equations. J. Difference Equ. Appl. 24, 976–991 (2018)

    Article  MathSciNet  Google Scholar 

  13. Haddad, N., Touafek, N., Rabago, J.F.T.: Solution form of a higher-order system of difference equations and dynamical behavior of its special case. Math. Methods Appl. Sci. 40(10), 3599–3607 (2017)

    Article  MathSciNet  Google Scholar 

  14. Halilm, Y., Touafek, N., Yazlik, Y.: Dynamic behavior of a second-order nonlinear rational difference equation. Turk. J. Math. 39, 1004–1018 (2015)

    Article  MathSciNet  Google Scholar 

  15. Ibrahim, T.F., Touafek, N.: On a third order rational difference equation with variable coeffitients. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 20, 251–264 (2013)

    MathSciNet  MATH  Google Scholar 

  16. Kocic, V.L., Ladas,G.: Global Behavior of nonlinear di.erence equations of higher order with applications, , Vol. 256, Kluwer Academic Publisher, Dordrecht, 1993

  17. Kurbanli, A.S., Cinar, C., Yalcinkaya, I.: On the behavior of positive solutions of the system of rational difference equations \(x_{n+1}=x_{n-1}/(y_{n}x_{n+1}+1)\), \( y_{n+1}=y_{n-1}/(x_{n}y_{n+1}+1)\). Math. Comput. Model. 53, 1261–1267 (2011)

    Article  Google Scholar 

  18. 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)

    Article  Google Scholar 

  19. Moaaz, O.: Dynamics of difference equation \(x_{n+1}=f(x_{n-l},x_{n-k})\), Adv. Differ. Equ., 2018 (2018), Article 447, 14 pages

  20. Moaaz, O., Chalishajar, D., Bazighifan, O.: Some qualitative behavior of solutions of general class of difference equations, Mathematics, 7 (2019), Article 585, 12 pages

  21. Ozkan, O., Kurbanli, A. S.: On a system of difference equations, Discrete Dyn. Nat. Soc., 2013 (2013), Article ID 970316, 7 pages

  22. Papaschinopoulos, G., Schinas, C.J., Stefanidou, G.: On the nonautonomous difference equation \(x_{n+1}=A_{n}+\frac{x_{n-1}^{p}}{x_{n}^{q} }\). Appl. Math. Comput. 217(12), 5573–5580 (2011)

    MathSciNet  MATH  Google Scholar 

  23. Saleh, M., Abu-Baha, S.: Dynamics of a higher order rational difference equation. Appl. Math. Comput. 181(1), 84–102 (2006)

    MathSciNet  MATH  Google Scholar 

  24. Taskara, N., Tollu, D.T., Touafek, N., Yazlik, Y.: A solvable system of difference equations. Commun. Korean Math. Soc. 35(1), 301–319 (2020)

    MathSciNet  MATH  Google Scholar 

  25. Tollu, D.T., Yalcinkaya, I.: Global behavior of a three-dimensional system of difference equations of order three. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68(1), 1–16 (2019)

    Article  MathSciNet  Google Scholar 

  26. Touafek, N.: On a second order rational difference equation, Hacet. J. Math. Stat. 41, 867–874 (2012)

    MathSciNet  MATH  Google Scholar 

  27. Touafek, N.: On a general system of difference equations defined by homogeneous functions. Math. Slovaca 71(3), 697–720 (2021)

    Article  MathSciNet  Google Scholar 

  28. Touafek, N., Tollu, D. T., Akrour, Y.: On a general homogeneous three-dimensional system of difference equations, Electron. Res. Arch., (2021). https://www.aimsciences.org/article/doi/10.3934/era.2021017

  29. Wang, C., Jing, X., Hu, X., Li, R.: On the periodicity of a max-type rational difference equation. J. Nonlinear Sci. Appl. 10(9), 4648–4661 (2017)

    Article  MathSciNet  Google Scholar 

  30. Yalcinkaya, I., Cinar, C., Simsek, D.: Global asymptotic stability of a system of difference equations. Appl. Anal. 87(6), 677–687 (2008)

    Article  MathSciNet  Google Scholar 

  31. Yalcinkaya, I., Tollu, D.T.: Global behavior of a second-order system of difference equations. Adv. Stud. Contemp. Math. Kyungshang 26(4), 653–667 (2016)

    MATH  Google Scholar 

  32. Yazlik, Y., Elsayed, E.M., Taskara, N.: On the Behaviour of the Solutions of Difference Equation Systems. J. Comput. Anal. Appl. 16(1), 932–941 (2014)

    MathSciNet  MATH  Google Scholar 

  33. Yazlik, Y., Kara, M.: On a solvable system of difference equations of higher-order with period two coefficients. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68, 1675–1693 (2019)

    Article  MathSciNet  Google Scholar 

  34. Yazlik, Y., Tollu, D.T., Taskara, N.: On the solutions of a three-dimensional system of difference equations. Kuwait J. Sci. 43(1), 95–111 (2016)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thanks the three referees for their comments and suggestions. M. Boulouh and N. Touafek were supported by DGRSDT (MESRS-DZ).

Author information

Authors and Affiliations

  1. LMAM Laboratory, Department of Mathematics, Mohamed Seddik Ben Yahia University, Jijel, Algeria

    Mounira Boulouh & Nouressadat Touafek

  2. Department of Mathematics and Computer Sciences, Faculty of Science, Necmettin Erbakan University, 42090, Konya, Turkey

    Durhasan Turgut Tollu

Authors
  1. Mounira Boulouh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nouressadat Touafek
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Durhasan Turgut Tollu
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Boulouh, M., Touafek, N. & Tollu, D.T. On the behavior of the solutions of an abstract system of difference equations. J. Appl. Math. Comput. 68, 2937–2969 (2022). https://doi.org/10.1007/s12190-021-01641-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-021-01641-7

Keywords

Mathematics Subject Classification

Search

Navigation