On a Class of Difference Equations System of Fifth-Order

Yıl 2024, Cilt: 7 Sayı: 3, 186 - 202, 30.09.2024

Merve Kara , Yasin Yazlik

https://doi.org/10.33401/fujma.1492703

Öz

In the current paper, we investigate the following new class of system of difference equations \begin{align} u_{n+1}=&f^{-1}\left( g\left( v_{n-1}\right) \frac{A_{1}f\left( u_{n-2}\right)+B_{1}g\left( v_{n-4}\right) }{C_{1}f\left( u_{n-2}\right)+D_{1}g\left( v_{n-4}\right)}\right), \nonumber \\ v_{n+1}=&g^{-1}\left( f\left( u_{n-1}\right) \frac{A_{2}g\left( v_{n-2}\right)+B_{2}f\left( u_{n-4}\right) }{C_{2}g\left( v_{n-2}\right)+D_{2}f\left( u_{n-4}\right)}\right) ,\ n\in \mathbb{N}_{0}, \nonumber \end{align} where the initial conditions $u_{-p}$, $v_{-p}$, for $p=\overline{0,4}$ are real numbers, the parameters $A_{r}$, $B_{r}$, $C_{r}$, $D_{r}$, for $r\in\{1,2\}$ are real numbers, $A_{r}^{2}+B_{r}^{2}\neq 0\neq C_{r}^{2}+D_{r}^{2}$, for $r\in\{1,2\}$, $f$ and $g$ are continuous and strictly monotone functions, $f\left( \mathbb{R}\right) =\mathbb{R}$, $g\left( \mathbb{R}\right) =\mathbb{R}$, $f\left( 0\right) =0$, $g\left( 0\right) =0$. In addition, we solve aforementioned general two dimensional system of difference equations of fifth-order in explicit form. Moreover, we obtain the solutions of mentioned system according to whether the parameters being zeros or not. Finally, we present an interesting application.

Anahtar Kelimeler

Difference equations systems, Solution, Stability

Kaynakça

• [1] R. Abo-Zeid and H. Kamal, On the solutions of a third order rational difference equation, Thai. J. Math., 18(4)(2020), 1865-1874. $ \href{https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1108}{\mbox{[Web]}} $
• [2] R. Abo-Zeid, Global behavior and oscillation of a third order difference equation, Quaest. Math., 44(9) (2021), 1261-1280. $\href{https://doi.org/10.2989/16073606.2020.1787537}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85087820294&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.2989%2F16073606.2020.1787537%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000276301800001}{\mbox{[Web of Science]}} $
• [3] Y. Halim, N. Touafek and Y. Yazlık, Dynamic behavior of a second-order nonlinear rational difference equation, Turkish J. Math., 39(6)(2015), 1004-1018. $ \href{https://doi.org/10.3906/mat-1503-80}{\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000366443700018}{\mbox{[Web of Science]}} $
• [4] T.F. Ibrahim, Periodicity and global attractivity of difference equation of higher order, J. Comput. Anal. Appl., 16(1)(2014), 552-564. $ \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000330602500016}{\mbox{[Web of Science]}}$
• [5] D.T. Tollu, Y. Yazlık and N. Tas¸kara, Behavior of positive solutions of a difference equation, J. Appl. Math. Inform., 35(3)(2017), 217-230. $ \href{https://doi.org/10.14317/jami.2017.217}{\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000411010100001}{\mbox{[Web of Science]}} $
• [6] A. Ghezal, Note on a rational system of (4k+4)􀀀order difference equations: periodic solution and convergence, J. Appl. Math. Comput., 69(2)(2022), 2207-2215. $ \href{https://doi.org/10.1007/s12190-02201830-y}{\mbox{[CrossRef]}} $
• [7] M. Kara, Y. Yazlık and D.T. Tollu, Solvability of a system of higher order nonlinear difference equations, Hacet. J. Math. Stat., 49(5)(2020), 1566-1593. $ \href{https://doi.org/10.15672/hujms.474649}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85092901185&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.15672%2Fhujms.474649%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000581099500002}{\mbox{[Web of Science]}} $
• [8] M. Kara and Y. Yazlık, On a solvable three-dimensional system of difference equations, Filomat, 34(4)(2020), 1167-1186. $ \href{https://doi.org/10.2298/FIL2004167K}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85097881022&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.2298%2FFIL2004167K%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000600790800010}{\mbox{[Web of Science]}}$
• [9] M. Kara, D.T. Tollu and Y. Yazlık, Global behavior of two-dimensional difference equations system with two period coefficients, Tbil. Math. J., 13(4)(2020), 49-64. $ \href{https://doi.org/10.32513/tbilisi/1608606049}{\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000603341900006}{\mbox{[Web of Science]}} $
• [10] M. Kara and Y. Yazlık, On eight solvable systems of difference equations in terms of generalized Padovan sequences, Miskolc Math. Notes, 22(2)(2021), 695-708. $ \href{https://doi.org/10.18514/MMN.2021.3234}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85123234232&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.18514%2FMMN.2021.3234%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000741090800016}{\mbox{[Web of Science]}} $
• [11] M. Kara and Y. Yazlık, Solvable three-dimensional system of higher-order nonlinear difference equations, Filomat, 36(10)(2022), 3453-3473. $ \href{https://doi.org/10.2298/FIL2210453K}{\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000916889600019}{\mbox{[Web of Science]}} $
• [12] M. Kara and Y. Yazlık, On a solvable system of rational difference equations of higher order, Turkish. J. Math., 46(2)(2022), 587-611. $\href{https://doi.org/10.3906/mat-2106-1}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85125523157&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.3906%2Fmat-2106-1%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000696638900001}{\mbox{[Web of Science]}} $
• [13] M. Kara and Y. Yazlık, On the solutions of three-dimensional system of difference equations via recursive relations of order two and Applications, J. Appl. Anal. Comput., 12(2)(2022), 736-753. $ \href{https://doi.org/10.11948/20210305}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85128170166&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.11948%2F20210305%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000784384600017}{\mbox{[Web of Science]}} $
• [14] M. Kara, Solvability of a three-dimensional system of non-liner difference equations, Math. Sci. Appl. E-Notes, 10(1)(2022), 1-15. $ \href{https://doi.org/10.36753/mathenot.992987}{\mbox{[CrossRef]}} $
• [15] N. Tas¸kara, D.T. Tollu, N. Touafek and Y. Yazlık, A solvable system of difference equations, Commun. Korean. Math. Soc., 35(1)(2020), 301-319. $ \href{https://doi.org/10.4134/CKMS.c180472}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85082331235&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.4134%2FCKMS.c180472%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000508684900022}{\mbox{[Web of Science]}}$
• [16] N. Touafek, On a general system of difference equations defined by homogeneous functions, Math. Slovaca, 71(3)(2021), 697-720. $ \href{https://doi.org/10.1515/ms-2021-0014}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85108378993&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.1515%2Fms-2021-0014%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000663038900014}{\mbox{[Web of Science]}} $
• [17] İ. Yalc¸ınkaya, On the global asymptotic behavior of a system of two nonlinear difference equations, Ars. Combin., 95(2010), 151-159. $\href{https://hdl.handle.net/20.500.12395/25123}{\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000276676500014}{\mbox{[Web of Science]}} $
• [18] İ. Yalc¸ınkaya and D. T. Tollu, Global behavior of a second order system of difference equations, Adv. Stud. Contemp. Math., 26(4) (2016), 653-667.
• [19] Y. Yazlık, D.T. Tollu and N. Tas¸kara, On the solutions of difference equation systems with Padovan numbers, Appl. Math., 4(12A)(2013), 1-15. $ \href{https://doi.org/10.4236/am.2013.412A1002}{\mbox{[CrossRef]}} $
• [20] Y. Yazlık and M. Kara, On a solvable system of difference equations of higher-order with period two coefficients, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68(2)(2019), 1675-1693. $ \href{https://doi.org/10.31801/cfsuasmas.548262}{\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000488869500039}{\mbox{[Web of Science]}} $
• [21] Y. Yazlık and M. Kara, On a solvable system of difference equations of fifth-order, Eskisehir Tech. Univ. J. Sci. Tech. B- Theor. Sci., 7(1)(2019), 29-45. $ \href{https://doi.org/10.20290/aubtdb.422910}{\mbox{[CrossRef]}} $
• [22] A. De Moivre, The Doctrine of Chances, 3nd edition, In Landmark Writings in Western Mathematics, London, (1756). $ \href{https://www.ime.usp.br/~walterfm/cursos/mac5796/DoctrineOfChances.pdf}{\mbox{[Web]}} $
• [23] D.T. Tollu, Y. Yazlık and N. Taşkara, On a solvable nonlinear difference equation of higher order, Turkish J. Math., 42(4)(2018), 1765-1778. $ \href{https://doi.org/10.3906/mat-1705-33}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85050724550&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.3906%2Fmat-1705-33%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000439579600017}{\mbox{[Web of Science]}} $
• [24] E.M. Elabbasy and E.M. Elsayed, Dynamics of a rational difference equation, Chin. Ann. Math., 30(2)(2009), 187-198. $ \href{https://doi.org/10.1007/s11401-007-0456-9}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-63049109137&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.1007%2Fs11401-007-0456-9%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000264261300008}{\mbox{[Web of Science]}} $
• [25] E.M. Elabbasy, H.A. El-Metwally and E. M. Elsayed, Global behavior of the solutions of some difference equations, Adv. Difference Equ., 2011(1)(2011), 1-16. $\href{https://doi.org/10.1186/1687-1847-2011-28}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84855197924&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.1186%2F1687-1847-2011-28%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000307015900001}{\mbox{[Web of Science]}} $
• [26] E.M. Elsayed, Qualitative behavior of a rational recursive sequence, Indag. Math., 19(2)(2008), 189-201. $\href{https://doi.org/10.1016/S0019-3577(09)00004-4}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-60849086178&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.1016%2FS0019-3577%2809%2900004-4%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000262876500003}{\mbox{[Web of Science]}} $
• [27] E.M. Elsayed, Qualitative properties for a fourth order rational difference equation, Acta. Appl. Math., 110(2)(2010), 589-604. $ \href{https://doi.org/10.1007/s10440-009-9463-z}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-78650854764&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.1007%2Fs10440-009-9463-z%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000276510500005}{\mbox{[Web of Science]}} $
• [28] S. Stevic, M.A. Alghamdi, N. Shahzad and D.A. Maturi, On a class of solvable difference equations, Abstr. Appl. Anal., 2013(2013), 1-7. $\href{https://doi.org/10.1155/2013/157943}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84893668152&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.1155%2F2013%2F157943%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000328396000001}{\mbox{[Web of Science]}} $
• [29] R.P. Agarwal and E.M. Elsayed, On the solution of fourth-order rational recursive sequence, Adv. Stud. Contemp. Math., 20(4) (2010), 525-545. $ \href{https://www.researchgate.net/profile/Elsayed-Elsayed-7/publication/267441756_On_the_solution_of_fourth-order_rational_recursive_sequence/links/547dcd9b0cf2cfe203c22479/On-the-solution-of-fourth-order-rational-recursive-sequence.pdf}{\mbox{[Web]}} $
• [30] E.M. Elsayed, Qualitative behavior of difference equation of order two, Math. Comput. Model., 50(7-8)(2009), 1130-1141. $ \href{https://doi.org/10.1016/j.mcm.2009.06.003}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-69249219001&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.1016%2Fj.mcm.2009.06.003%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000269475200018}{\mbox{[Web of Science]}} $
• [31] E.M. Elsayed, F. Alzahrani, I. Abbas and N.H. Alotaibi, Dynamical behavior and solution of nonlinear difference equation via Fibonacci sequence, J. Appl. Anal. Comput., 10(1)(2020), 282-296. $ \href{https://doi.org/10.11948/20190143}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85078863700&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.11948%2F20190143%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000503991100020}{\mbox{[Web of Science]}} $
• [32] E.M. Elsayed, B.S. Aloufi and O. Moaaz, The behavior and structures of solution of fifth-order rational recursive sequence, Symmetry, 14(4)(2022), 1-18. $ \href{https://doi.org/10.3390/sym14040641}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85127546089&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.3390%2Fsym14040641%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000785510800001}{\mbox{[Web of Science]}} $
• [33] S. Stevic, B. Iricanin and W. Kosmala, On a family of nonlinear difference equations of the fifth order solvable in closed form, AIMS Math., 8(10)(2023), 22662-22674. $ \href{https://doi.org/10.3934/math.20231153}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85165098393&origin=resultslist&sort=plf-f&src=s&sid=48a6f1c982306bd91c68d4da1fe32d0f&sot=b&sdt=b&s=DOI%2810.3934%2Fmath.20231153%29&sl=29&sessionSearchId=48a6f1c982306bd91c68d4da1fe32d0f&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001034072000006}{\mbox{[Web of Science]}} $