Solvability of a Three-Dimensional System of Nonlinear Difference Equations

Merve Kara

doi:10.36753/mathenot.992987

EN

Solvability of a Three-Dimensional System of Nonlinear Difference Equations

Abstract

In this paper, we solve the following three-dimensional system of difference equations

\begin{matrix} x_{n} \end{matrix}

=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,

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

{(a_{n})}_{n \in N_{0}}

,

{(b_{n})}_{n \in N_{0}}

,

{(α_{n})}_{n \in N_{0}}

,

{(β_{n})}_{n \in N_{0}}

,

{(A_{n})}_{n \in N_{0}}

,

{(B_{n})}_{n \in N_{0}}

and the initial values

x_{- j}, y_{- j}

,

j = ¯ ¯¯¯¯¯¯ ¯ 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.

Keywords

system of difference equations, closed-form, forbidden set

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

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Merve Kara ^*
0000-0001-8081-0254
Türkiye

Publication Date

March 1, 2022

Submission Date

September 8, 2021

Acceptance Date

December 31, 2021

Published in Issue

Year 2022 Volume: 10 Number: 1

DOI

https://doi.org/10.36753/mathenot.992987

IZ

https://izlik.org/JA84ML26ND

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

1.Kara M. Solvability of a Three-Dimensional System of Nonlinear Difference Equations. Math. Sci. Appl. E-Notes. 2022;10(1):1-15. doi:10.36753/mathenot.992987

Chicago

Kara, Merve. 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.

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

[1]M. Kara, “Solvability of a Three-Dimensional System of Nonlinear Difference Equations”, Math. Sci. Appl. E-Notes, vol. 10, no. 1, pp. 1–15, Mar. 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 1, 2022): 1-15. https://doi.org/10.36753/mathenot.992987.

JAMA

1.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, Mar. 2022, pp. 1-15, doi:10.36753/mathenot.992987.

Vancouver

1.Merve Kara. Solvability of a Three-Dimensional System of Nonlinear Difference Equations. Math. Sci. Appl. E-Notes. 2022 Mar. 1;10(1):1-15. doi:10.36753/mathenot.992987

INDEXING & ABSTRACTING & ARCHIVING

20477 The published articles in MSAEN are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Solvability of a Three-Dimensional System of Nonlinear Difference Equations

Abstract

Solvability of a Three-Dimensional System of Nonlinear Difference Equations

Abstract

Keywords

References

Details

Primary Language

Subjects

Journal Section

Authors

Publication Date

Submission Date

Acceptance Date

Published in Issue

DOI

IZ

Cite

Cited By

On solutions of three-dimensional system of difference equations with constant coefficients

On Some Solvable Systems of Some Rational Difference Equations of Third Order

On a Class of Difference Equations System of Fifth-Order

On the Dynamics of Some Three-Dimensional Systems of Difference Equations

On a difference equation whose solution is related to Fibonacci numbers

On a solvable difference equations system

On the Balancing-bilinear system of difference equations: solutions via some number sequences and global stability

On the Solvability of Some Systems of Nonlinear Difference Equations

On the solvability in closed-form of a rational three-dimensional system of difference equations