Solvability of a Second-Order Rational System of Difference Equations

Year 2023, Volume: 6 Issue: 4, 232 - 242, 31.12.2023

Messaoud Berkal , R Abo-zeıd

https://doi.org/10.33401/fujma.1383434

Abstract

In this paper, we represent the admissible solutions of the system of second-order rational difference equations given below in terms of Lucas and Fibonacci sequences: \begin{eqnarray*} \begin{split} x_{n+1}=\dfrac{L_{m+2}+L_{m+1}y_{n-1}}{L_{m+3}+L_{m+2}y_{n-1}},\quad y_{n+1}=\dfrac{L_{m+2}+L_{m+1}z_{n-1}}{L_{m+3}+L_{m+2}z_{n-1}},\\ z_{n+1}=\dfrac{L_{m+2}+L_{m+1}w_{n-1}}{L_{m+3}+L_{m+2}w_{n-1}},\quad w_{n+1}=\dfrac{L_{m+2}+L_{m+1}x_{n-1}}{L_{m+3}+L_{m+2}x_{n-1}}. \end{split} \end{eqnarray*} where $n\in\mathbb{N}_0$, $\{L_m\}_{m=-\infty}^{+\infty}$ is Lucas sequence and the initial conditions $x_{-1}$, $x_{0}$, $y_{-1}$, $y_{0}$, $z_{-1}$, $z_{0}$, $w_{-1}$, $w_{0}$ are arbitrary real numbers such that $\displaystyle v_{-i}\neq-\frac{L_{m+3}}{L_{m+2}}$, where $v_{-i}=x_{-i},y_{-i},z_{-i},w_{-i}$, $i=0,1$ and $m\in\mathbb{Z}$.

Keywords

General solution, Lucas numbers, Fibonacci numbers, system of difference equations.

References

• [1] Y. Halim, N. Touafek and E.M. Elsayed, Closed form solution of some systems of rational difference equations in terms of Fibonacci numbers, Dyn. Contin. Discrete Impulsive Syst. Ser. A., 21 (2014), 473-486. $\href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84921376512&origin=resultslist&sort=plf-f&src=s&sid=91bba5b1c3e4f1fe08d591120fc9321c&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Closed+form+solution+of+some+systems+of+rational+difference+equations+in+terms+of+Fibonacci+numbers%22%29&sl=116&sessionSearchId=91bba5b1c3e4f1fe08d591120fc9321c&relpos=2}{\color{blue}{[\mbox{Scopus}]}}$
• [2] A. Khelifa, Y. Halim and M. Berkal, On the solutions of a system of (2p+1) difference equations of higher order, Miskolc Math. Notes, 22(1) (2021), 331-350. $\href{https://doi.org/10.18514/MMN.2021.3385}{\color{blue}{[\mbox{CrossRef}]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85108574785&origin=resultslist&sort=plf-f&src=s&sid=14cd60960479e6053cb8a55b7acc85e1&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+the+solutions+of+a+system+of+%24%282p%2B+1%29%24+difference+equations+of+higher+order%22%29&sl=95&sessionSearchId=14cd60960479e6053cb8a55b7acc85e1&relpos=0}{\color{blue}{[\mbox{Scopus}]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000661139500024}{\color{blue}{[\mbox{Web of Science}]}}$
• [3] S. Stevic, Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences, Electron. J. Qual. Theory Differ. Equ., 67 (2014), 1-15. $\href{https://doi.org/10.14232/ejqtde.2014.1.67}{\color{blue}{[\mbox{CrossRef}]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84920091929&origin=resultslist&sort=plf-f&src=s&sid=a3bfaa90d452179a22299f08db027bc5&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Representation+of+solutions+of+bilinear+difference+equations+in+terms+of+generalized+Fibonacci+sequences%22%29&sl=121&sessionSearchId=a3bfaa90d452179a22299f08db027bc5&relpos=0}{\color{blue}{[\mbox{Scopus}]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000347538300001}{\color{blue}{[\mbox{Web of Science}]}}$
• [4] A. Khelifa, Y. Halim and M. Berkal, On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers, Miskolc Math. Notes, 22 (2021), 331-350. $\href{https://doi.org/10.1515/ms-2017-0378}{\color{blue}{[\mbox{CrossRef}]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85086232146&origin=resultslist&sort=plf-f&src=s&sid=c146d85ddc5d11795610d7010f9e7845&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+a+system+of+three+difference+equations+of+higher+order+solved+in+terms+of+Lucas+and+Fibonacci+numbers%22%29&sl=121&sessionSearchId=c146d85ddc5d11795610d7010f9e7845&relpos=0}{\color{blue}{[\mbox{Scopus}]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000536293100011}{\color{blue}{[\mbox{Web of Science}]}}$
• [5] A. Khelifa, Y. Halim, A. Bouchair and M. Berkal, On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers, Math. Slovaca, 70 (2020), 641-656. $\href{https://doi.org/10.1515/ms-2017-0378}{\color{blue}{[\mbox{CrossRef}]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85086232146&origin=resultslist&sort=plf-f&src=s&sid=d46d85900161fe9acbd834a0c52925c0&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+a+system+of+three+difference+equations+of+higher+order+solved+in+terms+of+Lucas+and+Fibonacci+numbers%22%29&sl=121&sessionSearchId=d46d85900161fe9acbd834a0c52925c0&relpos=0}{\color{blue}{[\mbox{Scopus}]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000536293100011}{\color{blue}{[\mbox{Web of Science}]}}$
• [6] Y. Yazlik, D. Tollu and N. Taskara, On the solutions of difference equation systems with padovan numbers, Appl. Math., 4 (2013), 15-20. $\href{https://doi.org/10.4236/am.2013.412A002 }{\color{blue}{[\mbox{CrossRef}]}}$
• [7] T. Koshy, Fibonacci and Lucas Numbers with Applications, USA: John Wiley Sons, Inc., New York, 2001. $\href{https://doi.org/10.1002/9781118742327}{\color{blue}{[\mbox{CrossRef}]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85051312326&origin=resultslist&sort=plf-f&src=s&sid=bd1c58206711dadedec5ac95f11f9fd8&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Fibonacci+and+Lucas+Numbers+with+Applications%22%29&sl=62&sessionSearchId=bd1c58206711dadedec5ac95f11f9fd8&relpos=0}{\color{blue}{[\mbox{Scopus}]}} $
• [8] S. Vajda, Fibonacci and Lucas numbers and the golden section : Theory and applications, Ellis Horwood Limited, University of Sussex, (1989).
• [9] R. Abo-Zeid, Behavior of solutions of a rational third order difference equation, J. Appl. Math. & Informatics, 38(1-2) (2020), 1-12. $\href{https://doi.org/10.14317/jami.2020.001}{\color{blue}{[\mbox{CrossRef}]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85102250656&origin=resultslist&sort=plf-f&src=s&sid=a66c530ee08b6459781eff4949a9bcd7&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Behavior+of+solutions+of+a+rational+third+order+difference+equation%22%29&sl=84&sessionSearchId=a66c530ee08b6459781eff4949a9bcd7&relpos=0}{\color{blue}{[\mbox{Scopus}]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000513943500001}{\color{blue}{[\mbox{Web of Science}]}}$
• [10] R. Abo-Zeid, On the solutions of a higher order recursive sequence, Malaya J. Mat., 8(2) (2020), 695-701. $\href{https://doi.org/10.26637/MJM0802/0063}{\color{blue}{[\mbox{CrossRef}]}}$
• [11] Y. Akrour, M. Kara, N. Touafek and Y. Yazlık, Solutions formulas for some general systems of difference equations, Miskolc Math. Notes, 22(2) (2021), 529-555. $\href{https://doi.org/10.18514/MMN.2021.3365}{\color{blue}{[\mbox{CrossRef}]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85108254784&origin=resultslist&sort=plf-f&src=s&sid=b46059849acf321ab37c790a513ccfa7&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Solutions+formulas+for+some+general+systems+of+difference+equations%22%29&sl=84&sessionSearchId=b46059849acf321ab37c790a513ccfa7&relpos=0}{\color{blue}{[\mbox{Scopus}]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000741090800004}{\color{blue}{[\mbox{Web of Science}]}}$
• [12] M. Berkal and R. Abo-Zeid, On a rational (p+1) th order difference equation with quadratic term, Univ. J. Math. Appl., 5(4) (2022), 136-144. $\href{https://doi.org/10.32323/ujma.198471}{\color{blue}{[\mbox{CrossRef}]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85148641960&origin=resultslist&sort=plf-f&src=s&sid=c9835b28ad9f840d8618d1ce54a3634e&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+a+rational+%24%28p+%2B+1%29%24+th+order+difference+equation+with+quadratic+term%22%29&sl=89&sessionSearchId=c9835b28ad9f840d8618d1ce54a3634e&relpos=0}{\color{blue}{[\mbox{Scopus}]}}$
• [13] M. Berkal, K. Berehal and N. Rezaiki, Representation of solutions of a system of five-order nonlinear difference equations, J. Appl. Math. Inform., 40(3-4) (2022), 409-431. $\href{https://doi.org/10.14317/jami.2022.409}{\color{blue}{[\mbox{CrossRef}]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85134405847&origin=resultslist&sort=plf-f&src=s&sid=4bd3f96f56735e840456fb6487ac65ed&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Representation+of+solutions+of+a+system+of+five-order+nonlinear+difference+equations%22%29&sl=101&sessionSearchId=4bd3f96f56735e840456fb6487ac65ed&relpos=0}{\color{blue}{[\mbox{Scopus}]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000888562900002}{\color{blue}{[\mbox{Web of Science}]}}$
• [14] I. Dekkar, N. Touafek, I. Yalcinkaya and M.B. Almatrafi, Explicit formulas for the solutions of a system of difference equations of fourth order in the complex domain, Math. Eng. Sci. Aerosp., 14(3) (2023).
• [15] M. Gümüş and R. Abo-Zeid, On the solutions of a (2k+2) th order difference equation, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 25(1) (2018), 129-143. $\href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85043281748&origin=resultslist&sort=plf-f&src=s&sid=2d0747af98777738784580dd0fe4dfd6&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+the+solutions+of+a+%24%282k%2B+2%29%24+th+order+difference+equation%22%29&sl=77&sessionSearchId=2d0747af98777738784580dd0fe4dfd6&relpos=0}{\color{blue}{[\mbox{Scopus}]}}$
• [16] M. Gümüş and R. Abo-Zeid, An explicit formula and forbidden set for a higher order difference equation, J. Appl. Math. Comput., 63(1-2) (2020), 133-142. $\href{https://doi.org/10.1007/s12190-019-01311-9}{\color{blue}{[\mbox{CrossRef}]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85077607973&origin=resultslist&sort=plf-f&src=s&sid=21cdd34fa10154439a52628d91e1710a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22An+explicit+formula+and+forbidden+set+for+a+higher+order+difference+equation%22%29&sl=93&sessionSearchId=21cdd34fa10154439a52628d91e1710a&relpos=0}{\color{blue}{[\mbox{Scopus}]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000505381600001}{\color{blue}{[\mbox{Web of Science}]}}$
• [17] N. Haddad, N. Touafek and J.F.T Rabago, Solution form of a higher order system of difference equation and dynamical behavior of its special case, Math. Methods Appl. Sci., 40(10) (2017), 3599-3607. $\href{https://doi.org/10.1002/mma.4248}{\color{blue}{[\mbox{CrossRef}]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85008263098&origin=resultslist&sort=plf-f&src=s&sid=4096abfab8331d6b6b57c50a7d253054&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Solution+form+of+a+higher+order+system+of+difference+equation+and+dynamical+behavior+of+its+special+case%22%29&sl=121&sessionSearchId=4096abfab8331d6b6b57c50a7d253054&relpos=0}{\color{blue}{[\mbox{Scopus}]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000402801800014}{\color{blue}{[\mbox{Web of Science}]}}$
• [18] Y. Halim, A. Khelifa, M. Berkal and A. Bouchair, On a solvable system of p difference equations of higher order, Period. Math. Hung., 85(1) (2022), 109-127. $\href{https://doi.org/10.1007/s10998-021-00421-x}{\color{blue}{[\mbox{CrossRef}]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85113165871&origin=resultslist&sort=plf-f&src=s&sid=d07df7660e624a208ec623039cdb9aed&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+a+solvable+system+of+%24p%24+difference+equations+of+higher+order%22%29&sl=81&sessionSearchId=d07df7660e624a208ec623039cdb9aed&relpos=0}{\color{blue}{[\mbox{Scopus}]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000686866800002}{\color{blue}{[\mbox{Web of Science}]}}$
• [19] Y. Halim, M. Berkal and A. Khelifa, On a three-dimensional solvable system of difference equations, Turkish J. Math., 44 (2020), 1263-1288. $\href{https://doi.org/10.3906/MAT-2001-40}{\color{blue}{[\mbox{CrossRef}]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85089383012&origin=resultslist&sort=plf-f&src=s&sid=da3682121f85189aa0b7d0f97b41604a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+a+three-dimensional+solvable+system+of+difference+equations%22%29&sl=79&sessionSearchId=da3682121f85189aa0b7d0f97b41604a&relpos=0}{\color{blue}{[\mbox{Scopus}]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000548379100014}{\color{blue}{[\mbox{Web of Science}]}}$
• [20] A. Khelifa and Y. Halim, General solutions to systems of difference equations and some of their representations, J. Appl. Math. Comput., (2021). $\href{https://doi.org/10.1007/s12190-020-01476-8}{\color{blue}{[\mbox{CrossRef}]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85099318000&origin=resultslist&sort=plf-f&src=s&sid=86eff9471f00c4c2e5179e454f34ac6d&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22General+solutions+to+systems+of+difference+equations+and+some+of+their+representations%22%29&sl=103&sessionSearchId=86eff9471f00c4c2e5179e454f34ac6d&relpos=0}{\color{blue}{[\mbox{Scopus}]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000607061700006}{\color{blue}{[\mbox{Web of Science}]}}$
• [21] A. Khelifa, Y. Halim and M. Berkal, Solutions of a system of two higher-order difference equations in terms of Lucas sequence, Univ. J. Math. Appl., 2 (2019), 202-211. $\href{https://doi.org/10.32323/ujma.610399}{\color{blue}{[\mbox{CrossRef}]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85089401805&origin=resultslist&sort=plf-f&src=s&sid=cd72f24f47cdfc2127a8ede83f3898f4&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Solutions+of+a+system+of+two+higher-order+difference+equations+in+terms+of++Lucas+sequence%22%29&sl=107&sessionSearchId=cd72f24f47cdfc2127a8ede83f3898f4&relpos=0}{\color{blue}{[\mbox{Scopus}]}}$
• [22] N. Touafek, On a second order rational difference equation, Hacettepe J. Math. Stat., 41(6) (2012), 867-874. $\href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84876236711&origin=resultslist&sort=plf-f&src=s&sid=c5e8cff718429e5b67e4b28126ce4931&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+a+second+order+rational+difference+equation%22%29&sl=63&sessionSearchId=c5e8cff718429e5b67e4b28126ce4931&relpos=2}{\color{blue}{[\mbox{Scopus}]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000315845000009}{\color{blue}{[\mbox{Web of Science}]}}$
• [23] N. Touafek, On some fractional systems of difference equations, Iran. J. Math. Sci. Inform., 9(1) (2014), 73-86. $\href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84908553526&origin=resultslist&sort=plf-f&src=s&sid=f329b967c06e5cab6cc60233eabc27f4&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+some+fractional+systems+of+difference+equations%22%29&sl=67&sessionSearchId=f329b967c06e5cab6cc60233eabc27f4&relpos=0}{\color{blue}{[\mbox{Scopus}]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000409754300007}{\color{blue}{[\mbox{Web of Science}]}}$
• [24] N. Touafek and E.M. Elsayed, On a second order rational systems of difference equations, Hokkaido Math. J., 44(1) (2015), 29-45. $\href{https://doi.org/10.14492/hokmj/1470052352}{\color{blue}{[\mbox{CrossRef}]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84934780296&origin=resultslist&sort=plf-f&src=s&sid=e9f41da76196b083b6b8d61c4954b62d&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+a+second+order+rational+systems+of+difference+equations%22%29&sl=75&sessionSearchId=e9f41da76196b083b6b8d61c4954b62d&relpos=0}{\color{blue}{[\mbox{Scopus}]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000353436000002}{\color{blue}{[\mbox{Web of Science}]}}$
• [25] D. Tollu, Y. Yazlik, and N. Taskara, On the solutions of two special types of Riccati difference equation via Fibonacci numbers, Adv. Differ. Equ., 2013 (2013), 1-7. $\href{https://doi.org/10.1186/1687-1847-2013-174}{\color{blue}{[\mbox{CrossRef}]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84893635203&origin=resultslist&sort=plf-f&src=s&sid=cd19e19abc5f2c4e47512213b6318c99&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+the+solutions+of+two+special+types+of+Riccati+difference+equation+via+Fibonacci+numbers%22%29&sl=107&sessionSearchId=cd19e19abc5f2c4e47512213b6318c99&relpos=0}{\color{blue}{[\mbox{Scopus}]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000322541000001}{\color{blue}{[\mbox{Web of Science}]}}$