Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

REPRESENTATION OF SOLUTIONS OF A SYSTEM OF FIVE-ORDER NONLINEAR DIFFERENCE EQUATIONS

  • BERKAL, M. (Department of Applied Mathematics, University of Alicante) ;
  • BEREHAL, K. (Abdelhafid Boussouf University, Department of Mathematics and Computer Science) ;
  • REZAIKI, N. (Abdelhafid Boussouf University, Department of Mathematics and Computer Science)
  • Received : 2020.10.20
  • Accepted : 2022.01.28
  • Published : 2022.05.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we deal with the existence of solutions of the following system of nonlinear rational difference equations with order five $x_{n+1}=\frac{y_{n-3}x_{n-4}}{y_n(a+by_{n-3}x_{n-4})}$, $y_{n+1}=\frac{x_{n-3}y_{n-4}}{x_n(c+dx_{n-3}y_{n-4})}$, n = 0, 1, ⋯, where parameters a, b, c and d are not executed at the same time and initial conditions x-4, x-3, x-2, x-1, x0, y-4, y-3, y-2, y-1 and y0 are non zero real numbers.

Keywords

References

  1. M.M. Alzubaidi and E.M. Elsayed, analytical and solutions of fourth order difference equations. Communications in Advanced Mathematical Sciences 2 (2019), 9-21.
  2. M.B. Almatrafi, Solutions structures for some systems of fractional difference equations, Open J. Math. Anal. 3 (2019), 52-61. https://doi.org/10.30538/psrp-oma2019.0032
  3. Y. Akrour, N. Touafek, Y. Halim, On system of difference equations of second order solved in closed-form, Miskolc Mathematical Notes 20 (2019), 701-717. https://doi.org/10.18514/mmn.2019.2923
  4. M.B. Almatrafi, E.M. Elsayed, Solutions and formulae for some systems of difference equations, MathLAB Journal 1 (2018), 356-369.
  5. M.B. Almatrafi, and M.M. Alzubaidi, Analysis of the qualitative behaviour of an eighth-order fractional difference equation, Open J. Discret. Appl. Math. 2 (2019), 41-47. https://doi.org/10.30538/psrp-odam2019.0010
  6. R. Abo-Zeid and H. Kamal, Global behavior of two Rational third order difference equations, Univers. J. Math. Appl. 2 (2019), 212-217.
  7. E.M. Elsayed, Solution for systems of difference equations of rational form of order two, Comp. Appl. Mat. 33 (2014), 751-765. https://doi.org/10.1007/s40314-013-0092-9
  8. E.M. Elsayed and T.F. Ibrahim, Periodicity and solutions for some systems of nonlinear rational difference equations, Hacet. J. Math. stat. 44 (2015), 1361-1390.
  9. M.M. El-Dessoky, Solution for rational rystems of difference equations of order three, Mathematics 4 (2016), 1-12. https://doi.org/10.3390/math4010001
  10. M.M. El-Dessoky, A. Khaliq, Asim Asiri, On some rational systems of difference equations, J. Nonlinear Sci. Appl. 11 (2018), 49-72 . https://doi.org/10.22436/jnsa.011.01.05
  11. M. Gumus, On a competitive system of rational difference equations, Univers. J. Math. Appl. 2 (2019), 224-228. https://doi.org/10.32323/ujma.649122
  12. Y. Halim, M. Berkal and A. Khelifa, On a three-dimensional solvable system of difference equations, Turk. J. Math. 44 (2020), 1263-1288. https://doi.org/10.3906/mat-2001-40
  13. Y. Halim, A. Khelifa and M. Berkal, Representation of solutions of a two-dimensional system of difference equations, Miskolc Mathematical Notes 21 (2020), 203-218. https://doi.org/10.18514/mmn.2020.3204
  14. Y. Halim, Global character of systems of rational difference equations, Electron. J. Math. Analysis Appl. 3 (2015), 204-214.
  15. Y. Halim, A system of difference equations with solutions associated to Fibonacci numbers, Int. J Difference Equ. 11 (2016), 65-77.
  16. Y. Halim, N. Touafek, Y. Yazlik Dynamic behavior of a second-order nonlinear rational difference equation, Turk. J. Math. 39 (2015), 1004-1018. https://doi.org/10.3906/mat-1503-80
  17. N. Haddad, J.F.T. Rabago, Dynamics of a system of k-difference equations, Electron. J. Math. Analysis Appl. 5 (2017), 242-249 .
  18. T.F. Ibrahim and N. Touafek, Max-type system of difference equations with positive twoperiodic sequences, Math. Meth. Appl. Sci. 37 (2014), 2562-2569.
  19. T.F. Ibrahim, Periodicity and global attractivity of difference equation of higher order, J. Comput. Anal. Appl. 16 (2014), 552-564.
  20. T.F. Ibrahim, Solution and Behavior of a Rational Recursive Sequence of order Four, Australian Journal of Basic and Applied Sciences, 7 (2013), 816-830.
  21. M. Kara and Y. Yazlik, Solvability of a system of nonlinear difference equations of higher order, Turk. J. Math. 43 (2019), 1533-1565. https://doi.org/10.3906/mat-1902-24
  22. M. Kara, Y. Yazlik and D.T. Tollu, Solvability of a system of h,sgher order nonlinear difference equations, Hacettepe Journal of Mathematics and Statistics 49 (2020), 1566-1593.
  23. M. Kara and Y. Yazlik, On a solvable three-dimensional system of difference equations, Filomat 34 (2020), 1167-1186. https://doi.org/10.2298/FIL2004167K
  24. M. Kara, N. Touafek and Y. Yazlik, Well-defined solutions of a three-dimensional system of difference equations, Gazi University Journal of Science 3 (2020), 767-778.
  25. M. Kara, D.T. Tollu and Y. Yazlik, Global behavior of two-dimensional difference equations system with two periodic coefficients, Tbilisi Mathematical Journal 4 (2020), 49-64.
  26. M. Kara, Y. Yazlik, N. Touafek and Y. Akrour, On a three-dimensional system of difference equations with variable coefficients, Journal of Applied Mathematics and Informatics In press.
  27. M. Kara and Y. Yazlik, On the system of difference equations $x_n=\frac{x_{n-2}y_{n-3}}{y_{n-1}(a_n+b_nx_{n-2}y_{n-3})}$, $y_n=\frac{y_{n-2}x_{n-3}}{x_{n-1}({\alpha}_n+{\beta}_ny_{n-2}x_{n-3})}$, Journal of Mathematical Extension 14 (2020), 41-59.
  28. 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 (2019), 202-211.
  29. A. Khelifa, Y. Halim, A. Bouchair and M. Berkal, On a system of three difference equations of higher order solvedin terms of Lucas and Fibonacci numbers, Math. Slovaca 70 (2020), 641-656. https://doi.org/10.1515/ms-2017-0378
  30. A.Q. Khan, Q. Din, M.N. Qureshi and T.F. Ibrahim, Global behavior of an anticompetitive system of fourth-order rational difference equations, Computational Ecology and Software 4 (2014), 35-46.
  31. S. Stevic, Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences, Electron. J. Qual. Theory Differ. Equ. 67 (2014), 15 pages.
  32. S. Stevic, On some solvable systems of difference equations, Appl. Math. Comput. 218 (2012), 5010-5018. https://doi.org/10.1016/j.amc.2011.10.068
  33. S. Stevic, M.A. Alghamdi, A. Alotaibi, E.M. Elsayed On a Class of Solvable Higher-Order Difference Equations, Filomat 31 (2017), 461-477. https://doi.org/10.2298/FIL1702461S
  34. N. Touafek and E.M. Elsayed, On the solutions of systems of rational difference equations, Math. Comput. Modelling 55 (2012), 1987-1997. https://doi.org/10.1016/j.mcm.2011.11.058
  35. Y. Yazlik, D.T. Tollu and N. Taskara, Behaviour of solutions for a system of two higherrder difference equations, J. Sci. Arts. 45 (2018), 813-826.
  36. Y. Yazlik, M. Kara, Besinci Mertebeden Fark Denklem Sisteminin Cozulebilirligi Uzerine, Eskisehir Technical University Journal of Science and Technology B-Theoretical Sciences 7 (2019), 29-45.
  37. Y. Yazlik and M. Kara, On a solvable system of difference equations of higher order with period two coefficients, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 (2019), 1675-1693. https://doi.org/10.31801/cfsuasmas.548262