Research Article
BibTex RIS Cite
Year 2022, , 1 - 15, 01.03.2022
https://doi.org/10.36753/mathenot.992987

Abstract

References

  • [1] Abo-Zeid, R., Kamal, H.: Global behavior of two rational third order difference equations. Universal Journal of Mathematics and Applications. 2 (4), 212-217 (2019). https://doi.org/10.32323/ujma.626465
  • [2] Abo-Zeid, R.: Behavior of solutions of a second order rational difference equation. Mathematica Moravica. 23 (1), 11-25 (2019). https://doi.org/10.5937/MatMor1901011A
  • [3] Dekkar, I., Touafek, N., Yazlik, Y.: Global stability of a third-order nonlinear system of difference equations with period- two coefficients. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas. 111 (2), 325-347 (2017). https://doi.org/10.1007/s13398-016-0297-z
  • [4] Elabbasy, E. M., El-Metwally, H. A., Elsayed, E. M.: Global behavior of the solutions of some difference equations. Advances in Difference Equations. 2011 (1), 1-16 (2011).
  • [5] Elsayed, E. M.: Qualitative behavior of a rational recursive sequence. Indagationes Mathematicae. 19 (2), 189-201 (2008). https://doi.org/10.1016/S0019-3577(09)00004-4
  • [6] Elsayed,E.M.:Qualitativepropertiesforafourthorderrationaldifferenceequation.ActaApplicandaeMathematicae. 110 (2), 589-604 (2010). https://doi.org/10.1007/s10440-009-9463-z
  • [7] Elsayed, E. M.: Dynamics of recursive sequence of order two. Kyungpook Mathematical Journal. 50 (4), 483-497 (2010). https://doi.org/10.5666/KMJ.2010.50.4.483
  • [8] Elsayed, E. M.: Solution for systems of difference equations of rational form of order two. Computational and Applied Mathematics. 33 (3), 751-765 (2014). https://doi.org/10.1007/s40314-013-0092-9
  • [9] Elsayed, E. M.: Expression and behavior of the solutions of some rational recursive sequences. Mathematical Methods in the Applied Sciences. 18 (39), 5682-5694 (2016). https://doi.org/10.1002/mma.3953
  • [10] Halim, Y., Touafek, N., Yazlik, Y.: Dynamic behavior of a second-order nonlinear rational difference equation. Turkish Journal of Mathematics. 39 (6), 1004-1018 (2015). https://doi.org/10.3906/mat-1503-80
  • [11] Halim,Y.,Bayram,M.:Onthesolutionsofahigher-orderdifferenceequationintermsofgeneralizedFibonaccisequence. Mathematical Methods in the Applied Science. 39, 2974-2982 (2016). https://doi.org/10.1002/mma.3745
  • [12] Halim, Y., Rabago, J. F. T.: On the solutions of a second-order difference equation in terms of generalized Padovan sequences. Mathematica Slovaca. 68 (3), 625-638 (2018). https://doi.org/10.1515/ms-2017-0130
  • [13] Kara, M., Yazlik, Y.: Solvability of a system of nonlinear difference equations of higher order. Turkish Journal of Mathematics. 43 (3), 1533-1565 (2019). https://dx.doi.org/10.3906/mat-1902-24
  • [14] Kara, M., Touafek, N., Yazlik, Y.: Well-defined solutions of a three-dimensional system of difference equations. Gazi University Journal of Science. 33 (3), 676-778 (2020). https://dx.doi.org/10.35378/GUJS.641441
  • [15] Kara, M., Yazlik, Y., Tollu, D. T.: Solvability of a system of higher order nonlinear difference equations. Hacettepe Journal of Mathematics & Statistics. 49 (5), 1566-1593 (2020). https://dx.doi.org/10.15672/hujms.474649
  • [16] Kara, M., Yazlik, Y.: On a solvable three-dimensional system of difference equations. Filomat. 34 (4), 1167-1186 (2020). https://dx.doi.org/10.2298/FIL2004167K
  • [17] Kara, M., Yazlik, Y.: Representation of solutions of eight systems of difference equations via generalized Padovan sequences. International Journal of Nonlinear Analysis and Applications. 12, 447-471 (2021). https://dx.doi.org/10.22075/IJNAA.2021.22477.2368
  • [18] Kara, M., Yazlik, Y.: Solvability of a (k + l)-order nonlinear difference equation. Tbilisi Mathematical Journal, 14 (2), 271-297 (2021). https://dx.doi.org/10.32513/tmj/19322008138
  • [19] Nyirenda, D., Folly-Gbetoula, M.: The invariance and formulas for solutions of some fifth-order difference equations. https://arxiv.org/pdf/1902.06482 (2019).
  • [20] Öcalan, Ö.: Oscillation of nonlinear difference equations with several coefficients. Communications in Mathematical Analysis. 4 (1), 35-44 (2008).
  • [21] Papaschinopoulos, G., Stefanidou, G.: Asymptotic behavior of the solutions of a class of rational difference equations. International Journal of Difference Equations. 5 (2), 233-249 (2010).
  • [22] Rabago, J. F. T., Bacani, J. B.: On a nonlinear difference equations. Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis. 24, 375-394 (2017).
  • [23] Stevic ́,S.,Diblik,J.,Iricˇanin,B.,Šmarda,Z.:Onasolvablesystemofrationaldifferenceequations.JournalofDiffer- ence Equations and Applications. 20 (5-6), 811-825 (2014). https://dx.doi.org/10.1080/10236198.2013.817573
  • [24] Stevic, S., Iricanin, B., Kosmala, W., Šmarda, Z.: Solvability of a nonlinear fifth-order difference equation. Mathemati- cal Methods in the Applied Science. 42, 1687-1701 (2019). https://doi.org/10.1002/mma.5467
  • [25] Tasdemir,E.,Soykan,Y.:Stabilityofnegativeequilibriumofanon-lineardifferenceequation.JournalofMathematical Sciences: Advances and Applications. 49 (1), 51-57 (2018). https://dx.doi.org/10.18642/jmsaa_7100121927
  • [26] Taskara, N., Tollu, D. T., Yazlik, Y.: Solutions of rational difference system of order three in terms of Padovan numbers. Journal of Advanced Research in Applied Mathematics. 7 (3), 18-29 (2015).
  • [27] Tollu,D.T.,Yazlik,Y.,Taskara,N.:ThesolutionsoffourRiccatidifferenceequationsassociatedwithFibonaccinumbers. Balkan Journal of Mathematics. 2 (1), 163-172 (2014).
  • [28] Tollu, D. T., Yazlik, Y., Taskara, N.: Behavior of positive solutions of a difference equation. Journal of Applied Mathematics & Informatics. 35, 217-230 (2017). https://dx.doi.org/10.14317/jami.2017.217
  • [29] Tollu, D. T., Yazlik, Y., Taskara, N.: On a solvable nonlinear difference equation of higher order. Turkish Journal of Mathematics. 42 (4), 1765-1778 (2018). https://dx.doi.org/10.3906/mat-1705-33
  • [30] Tollu, D. T., Yalcinkaya, I., Ahmad, H., Yao, S-W.: A detailed study on a solvable system related to the linear fractional difference equation. Mathematical Biosciences and Engineering. 18 (5), 5392-5408 (2021). https://dx.doi.org/10.3934/mbe.2021273
  • [31] Touafek, N.: On a second order rational difference equation. Hacettepe Journal of Mathematics and Statistics. 41 (6), 867-874 (2012).
  • [32] Touafek, N., Elsayed, E. M.: On a second order rational systems of difference equations. Hokkaido Mathematical Journal. 44 (1), 29-45 (2015).
  • [33] Yalcinkaya, I., Cinar, C.: Global asymptotic stability of a system of two nonlinear difference equations. Fasciculi Mathematici. 43, 171-180 ( 2010).
  • [34] Yalcinkaya, I., Tollu, D. T.: Global behavior of a second order system of difference equations. Advanced Studies in Contemporary Mathematics. 26 (4), 653-667 (2016).
  • [35] Yazlik, Y., Tollu, D. T., Taskara, N.: On the solutions of difference equation systems with Padovan numbers. Applied Mathematics. 4 (12A), 1-15 (2013). http://dx.doi.org/10.4236/am.2013.412A1002
  • [36] Yazlik, Y.: On the solutions and behavior of rational difference equations. Journal of Computational Analysis & Applications. 17 (3), 584-594 (2014).
  • [37] Yazlik, Y., Tollu, D. T., Tas ̧kara, N.: On the solutions of a three-dimensional system of difference equations. Kuwait Journal of Science. 43 (1), 95-111 (2016).
  • [38] Yazlik, Y., Kara, M.: On a solvable system of difference equations of higher-order with period two coefficients. Com- munications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics. 68 (2), 1675-1693 (2019). https://dx.doi.org/10.31801/cfsuasmas.548262
  • [39] Yazlik,Y.,Gungor, M.:On the solvable of nonlinear difference equation of sixth-order. Journal of Science and Arts.19 (2), 399-414 (2019).
  • [40] Yazlik, Y., Kara, M.: On a solvable system of difference equations of fifth-order. Eskisehir Tech- nical University Journal of Science and Technology B- Theoretical Sciences. 7 (1), 29-45 (2019). https://dx.doi.org/10.20290/aubtdb.422910

Solvability of a Three-Dimensional System of Nonlinear Difference Equations

Year 2022, , 1 - 15, 01.03.2022
https://doi.org/10.36753/mathenot.992987

Abstract

In this paper, we solve the following three-dimensional system of difference equations
xn
=yn4zn5yn1(an+bnzn2xn3yn4zn5)
,
yn
=zn4xn5zn1(αn+βnxn2yn3zn4xn5)
,
zn
=xn4yn5xn1(An+Bnyn2zn3xn4yn5)
, nN0,
xn=yn−4zn−5yn−1(an+bnzn−2xn−3yn−4zn−5),yn=zn−4xn−5zn−1(αn+βnxn−2yn−3zn−4xn−5),zn=xn−4yn−5xn−1(An+Bnyn−2zn−3xn−4yn−5), n∈N0,
where the sequences (an)nN0(an)n∈N0, (bn)nN0(bn)n∈N0, (αn)nN0(αn)n∈N0, (βn)nN0(βn)n∈N0, (An)nN0(An)n∈N0, (Bn)nN0(Bn)n∈N0 and the initial values xj,yjx−j,y−j, j=¯¯¯¯¯¯¯¯1,5j=1,5¯, are real numbers. In addition, the constant coefficient of the mentioned system is solved in closed form. Finally, we also describe the forbidden set of solutions of the system of difference equations.

References

  • [1] Abo-Zeid, R., Kamal, H.: Global behavior of two rational third order difference equations. Universal Journal of Mathematics and Applications. 2 (4), 212-217 (2019). https://doi.org/10.32323/ujma.626465
  • [2] Abo-Zeid, R.: Behavior of solutions of a second order rational difference equation. Mathematica Moravica. 23 (1), 11-25 (2019). https://doi.org/10.5937/MatMor1901011A
  • [3] Dekkar, I., Touafek, N., Yazlik, Y.: Global stability of a third-order nonlinear system of difference equations with period- two coefficients. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas. 111 (2), 325-347 (2017). https://doi.org/10.1007/s13398-016-0297-z
  • [4] Elabbasy, E. M., El-Metwally, H. A., Elsayed, E. M.: Global behavior of the solutions of some difference equations. Advances in Difference Equations. 2011 (1), 1-16 (2011).
  • [5] Elsayed, E. M.: Qualitative behavior of a rational recursive sequence. Indagationes Mathematicae. 19 (2), 189-201 (2008). https://doi.org/10.1016/S0019-3577(09)00004-4
  • [6] Elsayed,E.M.:Qualitativepropertiesforafourthorderrationaldifferenceequation.ActaApplicandaeMathematicae. 110 (2), 589-604 (2010). https://doi.org/10.1007/s10440-009-9463-z
  • [7] Elsayed, E. M.: Dynamics of recursive sequence of order two. Kyungpook Mathematical Journal. 50 (4), 483-497 (2010). https://doi.org/10.5666/KMJ.2010.50.4.483
  • [8] Elsayed, E. M.: Solution for systems of difference equations of rational form of order two. Computational and Applied Mathematics. 33 (3), 751-765 (2014). https://doi.org/10.1007/s40314-013-0092-9
  • [9] Elsayed, E. M.: Expression and behavior of the solutions of some rational recursive sequences. Mathematical Methods in the Applied Sciences. 18 (39), 5682-5694 (2016). https://doi.org/10.1002/mma.3953
  • [10] Halim, Y., Touafek, N., Yazlik, Y.: Dynamic behavior of a second-order nonlinear rational difference equation. Turkish Journal of Mathematics. 39 (6), 1004-1018 (2015). https://doi.org/10.3906/mat-1503-80
  • [11] Halim,Y.,Bayram,M.:Onthesolutionsofahigher-orderdifferenceequationintermsofgeneralizedFibonaccisequence. Mathematical Methods in the Applied Science. 39, 2974-2982 (2016). https://doi.org/10.1002/mma.3745
  • [12] Halim, Y., Rabago, J. F. T.: On the solutions of a second-order difference equation in terms of generalized Padovan sequences. Mathematica Slovaca. 68 (3), 625-638 (2018). https://doi.org/10.1515/ms-2017-0130
  • [13] Kara, M., Yazlik, Y.: Solvability of a system of nonlinear difference equations of higher order. Turkish Journal of Mathematics. 43 (3), 1533-1565 (2019). https://dx.doi.org/10.3906/mat-1902-24
  • [14] Kara, M., Touafek, N., Yazlik, Y.: Well-defined solutions of a three-dimensional system of difference equations. Gazi University Journal of Science. 33 (3), 676-778 (2020). https://dx.doi.org/10.35378/GUJS.641441
  • [15] Kara, M., Yazlik, Y., Tollu, D. T.: Solvability of a system of higher order nonlinear difference equations. Hacettepe Journal of Mathematics & Statistics. 49 (5), 1566-1593 (2020). https://dx.doi.org/10.15672/hujms.474649
  • [16] Kara, M., Yazlik, Y.: On a solvable three-dimensional system of difference equations. Filomat. 34 (4), 1167-1186 (2020). https://dx.doi.org/10.2298/FIL2004167K
  • [17] Kara, M., Yazlik, Y.: Representation of solutions of eight systems of difference equations via generalized Padovan sequences. International Journal of Nonlinear Analysis and Applications. 12, 447-471 (2021). https://dx.doi.org/10.22075/IJNAA.2021.22477.2368
  • [18] Kara, M., Yazlik, Y.: Solvability of a (k + l)-order nonlinear difference equation. Tbilisi Mathematical Journal, 14 (2), 271-297 (2021). https://dx.doi.org/10.32513/tmj/19322008138
  • [19] Nyirenda, D., Folly-Gbetoula, M.: The invariance and formulas for solutions of some fifth-order difference equations. https://arxiv.org/pdf/1902.06482 (2019).
  • [20] Öcalan, Ö.: Oscillation of nonlinear difference equations with several coefficients. Communications in Mathematical Analysis. 4 (1), 35-44 (2008).
  • [21] Papaschinopoulos, G., Stefanidou, G.: Asymptotic behavior of the solutions of a class of rational difference equations. International Journal of Difference Equations. 5 (2), 233-249 (2010).
  • [22] Rabago, J. F. T., Bacani, J. B.: On a nonlinear difference equations. Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis. 24, 375-394 (2017).
  • [23] Stevic ́,S.,Diblik,J.,Iricˇanin,B.,Šmarda,Z.:Onasolvablesystemofrationaldifferenceequations.JournalofDiffer- ence Equations and Applications. 20 (5-6), 811-825 (2014). https://dx.doi.org/10.1080/10236198.2013.817573
  • [24] Stevic, S., Iricanin, B., Kosmala, W., Šmarda, Z.: Solvability of a nonlinear fifth-order difference equation. Mathemati- cal Methods in the Applied Science. 42, 1687-1701 (2019). https://doi.org/10.1002/mma.5467
  • [25] Tasdemir,E.,Soykan,Y.:Stabilityofnegativeequilibriumofanon-lineardifferenceequation.JournalofMathematical Sciences: Advances and Applications. 49 (1), 51-57 (2018). https://dx.doi.org/10.18642/jmsaa_7100121927
  • [26] Taskara, N., Tollu, D. T., Yazlik, Y.: Solutions of rational difference system of order three in terms of Padovan numbers. Journal of Advanced Research in Applied Mathematics. 7 (3), 18-29 (2015).
  • [27] Tollu,D.T.,Yazlik,Y.,Taskara,N.:ThesolutionsoffourRiccatidifferenceequationsassociatedwithFibonaccinumbers. Balkan Journal of Mathematics. 2 (1), 163-172 (2014).
  • [28] Tollu, D. T., Yazlik, Y., Taskara, N.: Behavior of positive solutions of a difference equation. Journal of Applied Mathematics & Informatics. 35, 217-230 (2017). https://dx.doi.org/10.14317/jami.2017.217
  • [29] Tollu, D. T., Yazlik, Y., Taskara, N.: On a solvable nonlinear difference equation of higher order. Turkish Journal of Mathematics. 42 (4), 1765-1778 (2018). https://dx.doi.org/10.3906/mat-1705-33
  • [30] Tollu, D. T., Yalcinkaya, I., Ahmad, H., Yao, S-W.: A detailed study on a solvable system related to the linear fractional difference equation. Mathematical Biosciences and Engineering. 18 (5), 5392-5408 (2021). https://dx.doi.org/10.3934/mbe.2021273
  • [31] Touafek, N.: On a second order rational difference equation. Hacettepe Journal of Mathematics and Statistics. 41 (6), 867-874 (2012).
  • [32] Touafek, N., Elsayed, E. M.: On a second order rational systems of difference equations. Hokkaido Mathematical Journal. 44 (1), 29-45 (2015).
  • [33] Yalcinkaya, I., Cinar, C.: Global asymptotic stability of a system of two nonlinear difference equations. Fasciculi Mathematici. 43, 171-180 ( 2010).
  • [34] Yalcinkaya, I., Tollu, D. T.: Global behavior of a second order system of difference equations. Advanced Studies in Contemporary Mathematics. 26 (4), 653-667 (2016).
  • [35] Yazlik, Y., Tollu, D. T., Taskara, N.: On the solutions of difference equation systems with Padovan numbers. Applied Mathematics. 4 (12A), 1-15 (2013). http://dx.doi.org/10.4236/am.2013.412A1002
  • [36] Yazlik, Y.: On the solutions and behavior of rational difference equations. Journal of Computational Analysis & Applications. 17 (3), 584-594 (2014).
  • [37] Yazlik, Y., Tollu, D. T., Tas ̧kara, N.: On the solutions of a three-dimensional system of difference equations. Kuwait Journal of Science. 43 (1), 95-111 (2016).
  • [38] Yazlik, Y., Kara, M.: On a solvable system of difference equations of higher-order with period two coefficients. Com- munications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics. 68 (2), 1675-1693 (2019). https://dx.doi.org/10.31801/cfsuasmas.548262
  • [39] Yazlik,Y.,Gungor, M.:On the solvable of nonlinear difference equation of sixth-order. Journal of Science and Arts.19 (2), 399-414 (2019).
  • [40] Yazlik, Y., Kara, M.: On a solvable system of difference equations of fifth-order. Eskisehir Tech- nical University Journal of Science and Technology B- Theoretical Sciences. 7 (1), 29-45 (2019). https://dx.doi.org/10.20290/aubtdb.422910
There are 40 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Merve Kara 0000-0001-8081-0254

Publication Date March 1, 2022
Submission Date September 8, 2021
Acceptance Date December 31, 2021
Published in Issue Year 2022

Cite

APA Kara, M. (2022). Solvability of a Three-Dimensional System of Nonlinear Difference Equations. Mathematical Sciences and Applications E-Notes, 10(1), 1-15. https://doi.org/10.36753/mathenot.992987
AMA Kara M. Solvability of a Three-Dimensional System of Nonlinear Difference Equations. Math. Sci. Appl. E-Notes. March 2022;10(1):1-15. doi:10.36753/mathenot.992987
Chicago Kara, Merve. “Solvability of a Three-Dimensional System of Nonlinear Difference Equations”. Mathematical Sciences and Applications E-Notes 10, no. 1 (March 2022): 1-15. https://doi.org/10.36753/mathenot.992987.
EndNote Kara M (March 1, 2022) Solvability of a Three-Dimensional System of Nonlinear Difference Equations. Mathematical Sciences and Applications E-Notes 10 1 1–15.
IEEE M. Kara, “Solvability of a Three-Dimensional System of Nonlinear Difference Equations”, Math. Sci. Appl. E-Notes, vol. 10, no. 1, pp. 1–15, 2022, doi: 10.36753/mathenot.992987.
ISNAD Kara, Merve. “Solvability of a Three-Dimensional System of Nonlinear Difference Equations”. Mathematical Sciences and Applications E-Notes 10/1 (March 2022), 1-15. https://doi.org/10.36753/mathenot.992987.
JAMA Kara M. Solvability of a Three-Dimensional System of Nonlinear Difference Equations. Math. Sci. Appl. E-Notes. 2022;10:1–15.
MLA Kara, Merve. “Solvability of a Three-Dimensional System of Nonlinear Difference Equations”. Mathematical Sciences and Applications E-Notes, vol. 10, no. 1, 2022, pp. 1-15, doi:10.36753/mathenot.992987.
Vancouver Kara M. Solvability of a Three-Dimensional System of Nonlinear Difference Equations. Math. Sci. Appl. E-Notes. 2022;10(1):1-15.

20477

The published articles in MSAEN are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.