International Journal of Nonlinear Analysis and Applications
On a solvable system of difference equations via some number sequences
Document Type : Research Paper
Authors
1 Department of Mathematics, Kamil Ozdag Science Faculty, Karamanoglu Mehmetbey University, Karaman, Turkey
2 Department of Mathematics, Faculty of Science and Art, Nevsehir Haci Bektacs Veli University, Nevsehir, Turkey
Abstract
In this paper, we show that the following three-dimensional rational system of difference equations
\begin{equation*}
x_{n}=\frac{z_{n-1}z_{n-3}}{bx_{n-2}+az_{n-3}}, \ y_{n}=\frac{x_{n-1}x_{n-3}}{dy_{n-2}+cx_{n-3}}, \ z_{n}=\frac{y_{n-1}y_{n-3}}{fz_{n-2}+ey_{n-3}}, \ n\in \mathbb{N}_{0},
\end{equation*}
where the parameters $a, b, c, d, e, f$\ and the initial values $x_{-i},y_{-i},z_{-i}$, $i \in \{1,2,3\}$, are real numbers, can be solved in explicit form. In addition, the solutions of aforementioned systems according to the special cases of the parameters are given in closed form. Later, the forbidden set of the initial values for aforementioned system is described. Finally, an application and numerical examples to support our results are given.
Keywords
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Appl. 2 (2019), no. 4, 212–217.
[2] R. Abo-Zeid, Behavior of solutions of a second order rational difference equation, Math. Morav. 23 (2019), no.
3, 11–25.
[3] R. Abo-Zeid and H. Kamal, On the solutions of a third order rational difference equation, Thai J. Math. 18
(2020), no. 4, 1865–1874.
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difference equations, Discrete Dyn. Nat. Soc. 2018 (2018).
[5] I. Dekkar, N. Touafek and Y. Yazlik, Global stability of a third-order nonlinear system of difference equations with
period-two coefficients, RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 111 (2017), no. 2, 325–347.
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[7] E.M. Elabbasy, H.A. El-Metwally and E.M. Elsayed, Global behavior of the solutions of some difference equations,
Adv. Difference Equ. 2011 (2011), no. 1, 1–6.
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(2009), no. 2, 187–198.
[9] E.M. Elsayed, Qualitative properties for a fourth order rational difference equation, Acta Appl. Math. 110 (2010),
no. 2, 589–604.
[10] E.M. Elsayed, Solution for systems of difference equations of rational form of order two, Comput. Appl. Math.
33 (2014), no. 3, 751–765.
[11] E.M. Elsayed, Expression and behavior of the solutions of some rational recursive sequences, Math. Methods Appl.
Sci. 18 (2016), no. 39, 5682–5694.
[12] N. Haddad, N. Touafek and J.F.T. Rabago, Well-defined solutions of a system of difference equations, J. Appl.
Math. Comput. 56 (2018), no. 1-2, 439–458.
[13] Y. Halim, N. Touafek and Y. Yazlik, Dynamic behavior of a second-order nonlinear rational difference equation,
Turkish J. Math. 39 (2015), no. 6, 1004–1018.[14] Y. Halim and M. Bayram, On the solutions of a higher-order difference equation in terms of generalized Fibonacci
sequence, Math. Methods Appl. Sci. 39 (2016), no. 11, 2974–2982.
[15] Y. Halim and J.F.T. Rabago, On the solutions of a second-order difference equation in terms of generalized
Padovan sequences, Math. Slovaca 68 (2018), no. 3, 625–638.
[16] M. Kara and Y. Yazlik, On a solvable three-dimensional system of difference equations, Filomat 34 (2020), no. 4,
1167–1186.
[17] M. Kara and Y. Yazlik, Representation of solutions of eight systems of difference equations via generalized Padovan
sequences, Int. J. Nonlinear Anal. Appl. 12 (2021) 447–471.
[18] M. Kara and Y. Yazlik, Solvability of a (k + l)-order nonlinear difference equation, Tbil. Math. J. 14 (2021),
271–297.
[19] M. Kara and Y. Yazlik, Solvability of a nonlinear three-dimensional system of difference equations with constant
coefficients, Math. Slovaca 71 (2021), no. 5, 1133–1148.
[20] M. Kara and Y. Yazlik, On eight solvable systems of difference equations in terms of generalized Padovan sequences, Miskolc Math. Notes. 22 (2021), no. 2, 695–708.
[21] M. Kara, Solvability of a three-dimensional system of non-liner difference equations, Math. Sci. Appl. E-Notes.
10 (2022), no. 1, 1–15.
[22] M.R.S. Kulenovic and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica, New
York, NY, USA, 2002.
[23] H. Levy and F. Lessman, Finite Difference Equations, Macmillan, New York, 1961.
[24] S. Stevi´c, On a two-dimensional solvable system of difference equations, Electron. J. Qual. Theory Differ. Equ.
104 (2018), 1–18.
[25] S. Stevi´c, AE. Ahmed, W. Kosmala and Z. Smarda, ˇ On a class of difference equations with interlacing indices,
Adv. Difference Equ. 297 (2021), 1–16.
[26] N. Taskara, K. Uslu and DT. Tollu, The periodicity and solutions of the rational difference equation with periodic
coefficients, Comput. Math. Appl. 62 (2011), no. 4, 1807–1813.
[27] N. Taskara, D.T. Tollu and Y. Yazlik, Solutions of rational difference system of order three in terms of Padovan
numbers, J. Adv. Res. Appl. Math. 7 (2015), no. 3, 18–29.
[28] N. Taskara, D.T. Tollu, N. Touafek and Y. Yazlik, A solvable system of difference equations, Comm. Korean
Math. Soc. 35 (2020), no. 1, 301–319.
[29] D.T. Tollu, Y. Yazlik and N. Taskara, On fourteen solvable systems of difference equations, Appl. Math. Comput.
233 (2014), 310–319.
[30] D.T. Tollu, Y. Yazlik and N. Taskara, The solutions of four Riccati difference equations associated with Fibonacci
numbers, Balkan J. Math. 2 (2014), no. 1, 163–172.
[31] D.T. Tollu, Y. Yazlik and N. Taskara, On a solvable nonlinear difference equation of higher order, Turkish J.
Math. 42 (2018), 1765–1778.
[32] D.T. Tollu, Y. Yalcinkaya, H. Ahmad and S.-W. Yao, A detailed study on a solvable system related to the linear
fractional difference equation, Math. Biosci. Eng. 18 (2021), no. 5, 5392–5408.
[33] N. Touafek, On a second order rational difference equation, Hacet. J. Math. Stat. 41 (2012), no. 6, 867–874.
[34] N. Touafek and E.M. Elsayed, On a second order rational systems of difference equations, Hokkaido Math. J. 44
(2015), 29–45.
[35] I. Yalcinkaya, C. Cinar and D. Simsek, Global asymptotic stability of a system of difference equations, Appl. Anal.
87 (2008), no. 6, 677–687.
[36] I. Yalcinkaya, On the global asymptotic behavior of a system of two nonlinear difference equations, Ars Combin.
95 (2010), 151–159.[37] I. Yalcinkaya and D.T. Tollu, Global behavior of a second order system of difference equations, Adv. Stud. Contemp. Math. 26 (2016), no. 4, 653–667.
[38] Y. Yazlik, D.T. Tollu and N. Taskara, On the solutions of difference equation systems with Padovan numbers,
Appl. Math. 4 (2013), no. 12A, 15–20.
[39] Y. Yazlik, On the solutions and behavior of rational difference equations, J. Comput. Anal. Appl. 17 (2014), no.
3, 584–594.
[40] Y. Yazlik 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 (2019), no. 2, 1675–1693.
[41] Y. Yazlik and M. Kara, On a solvable system of difference equations of fifth-order, Eski¸sehir Tech. Univ. J. Sci.
Tech. B-Theoret. Sci. 7 (2019), no. 1, 29–45.
Appl. 2 (2019), no. 4, 212–217.
[2] R. Abo-Zeid, Behavior of solutions of a second order rational difference equation, Math. Morav. 23 (2019), no.
3, 11–25.
[3] R. Abo-Zeid and H. Kamal, On the solutions of a third order rational difference equation, Thai J. Math. 18
(2020), no. 4, 1865–1874.
[4] A.M. Alotaibi, M.S.M. Noorani and M.A. El-Moneam, On the solutions of a system of third order rational
difference equations, Discrete Dyn. Nat. Soc. 2018 (2018).
[5] I. Dekkar, N. Touafek and Y. Yazlik, Global stability of a third-order nonlinear system of difference equations with
period-two coefficients, RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 111 (2017), no. 2, 325–347.
[6] A. De Moivre, The Doctrine of Chances. In Landmark Writings in Western Mathematics, 3rd edn. London, 1756.
[7] E.M. Elabbasy, H.A. El-Metwally and E.M. Elsayed, Global behavior of the solutions of some difference equations,
Adv. Difference Equ. 2011 (2011), no. 1, 1–6.
[8] M.E. Elmetwally and E.M. Elsayed, Dynamics of a rational difference equation, Chin. Ann. Math. Ser. B. 30B
(2009), no. 2, 187–198.
[9] E.M. Elsayed, Qualitative properties for a fourth order rational difference equation, Acta Appl. Math. 110 (2010),
no. 2, 589–604.
[10] E.M. Elsayed, Solution for systems of difference equations of rational form of order two, Comput. Appl. Math.
33 (2014), no. 3, 751–765.
[11] E.M. Elsayed, Expression and behavior of the solutions of some rational recursive sequences, Math. Methods Appl.
Sci. 18 (2016), no. 39, 5682–5694.
[12] N. Haddad, N. Touafek and J.F.T. Rabago, Well-defined solutions of a system of difference equations, J. Appl.
Math. Comput. 56 (2018), no. 1-2, 439–458.
[13] Y. Halim, N. Touafek and Y. Yazlik, Dynamic behavior of a second-order nonlinear rational difference equation,
Turkish J. Math. 39 (2015), no. 6, 1004–1018.[14] Y. Halim and M. Bayram, On the solutions of a higher-order difference equation in terms of generalized Fibonacci
sequence, Math. Methods Appl. Sci. 39 (2016), no. 11, 2974–2982.
[15] Y. Halim and J.F.T. Rabago, On the solutions of a second-order difference equation in terms of generalized
Padovan sequences, Math. Slovaca 68 (2018), no. 3, 625–638.
[16] M. Kara and Y. Yazlik, On a solvable three-dimensional system of difference equations, Filomat 34 (2020), no. 4,
1167–1186.
[17] M. Kara and Y. Yazlik, Representation of solutions of eight systems of difference equations via generalized Padovan
sequences, Int. J. Nonlinear Anal. Appl. 12 (2021) 447–471.
[18] M. Kara and Y. Yazlik, Solvability of a (k + l)-order nonlinear difference equation, Tbil. Math. J. 14 (2021),
271–297.
[19] M. Kara and Y. Yazlik, Solvability of a nonlinear three-dimensional system of difference equations with constant
coefficients, Math. Slovaca 71 (2021), no. 5, 1133–1148.
[20] M. Kara and Y. Yazlik, On eight solvable systems of difference equations in terms of generalized Padovan sequences, Miskolc Math. Notes. 22 (2021), no. 2, 695–708.
[21] M. Kara, Solvability of a three-dimensional system of non-liner difference equations, Math. Sci. Appl. E-Notes.
10 (2022), no. 1, 1–15.
[22] M.R.S. Kulenovic and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica, New
York, NY, USA, 2002.
[23] H. Levy and F. Lessman, Finite Difference Equations, Macmillan, New York, 1961.
[24] S. Stevi´c, On a two-dimensional solvable system of difference equations, Electron. J. Qual. Theory Differ. Equ.
104 (2018), 1–18.
[25] S. Stevi´c, AE. Ahmed, W. Kosmala and Z. Smarda, ˇ On a class of difference equations with interlacing indices,
Adv. Difference Equ. 297 (2021), 1–16.
[26] N. Taskara, K. Uslu and DT. Tollu, The periodicity and solutions of the rational difference equation with periodic
coefficients, Comput. Math. Appl. 62 (2011), no. 4, 1807–1813.
[27] N. Taskara, D.T. Tollu and Y. Yazlik, Solutions of rational difference system of order three in terms of Padovan
numbers, J. Adv. Res. Appl. Math. 7 (2015), no. 3, 18–29.
[28] N. Taskara, D.T. Tollu, N. Touafek and Y. Yazlik, A solvable system of difference equations, Comm. Korean
Math. Soc. 35 (2020), no. 1, 301–319.
[29] D.T. Tollu, Y. Yazlik and N. Taskara, On fourteen solvable systems of difference equations, Appl. Math. Comput.
233 (2014), 310–319.
[30] D.T. Tollu, Y. Yazlik and N. Taskara, The solutions of four Riccati difference equations associated with Fibonacci
numbers, Balkan J. Math. 2 (2014), no. 1, 163–172.
[31] D.T. Tollu, Y. Yazlik and N. Taskara, On a solvable nonlinear difference equation of higher order, Turkish J.
Math. 42 (2018), 1765–1778.
[32] D.T. Tollu, Y. Yalcinkaya, H. Ahmad and S.-W. Yao, A detailed study on a solvable system related to the linear
fractional difference equation, Math. Biosci. Eng. 18 (2021), no. 5, 5392–5408.
[33] N. Touafek, On a second order rational difference equation, Hacet. J. Math. Stat. 41 (2012), no. 6, 867–874.
[34] N. Touafek and E.M. Elsayed, On a second order rational systems of difference equations, Hokkaido Math. J. 44
(2015), 29–45.
[35] I. Yalcinkaya, C. Cinar and D. Simsek, Global asymptotic stability of a system of difference equations, Appl. Anal.
87 (2008), no. 6, 677–687.
[36] I. Yalcinkaya, On the global asymptotic behavior of a system of two nonlinear difference equations, Ars Combin.
95 (2010), 151–159.[37] I. Yalcinkaya and D.T. Tollu, Global behavior of a second order system of difference equations, Adv. Stud. Contemp. Math. 26 (2016), no. 4, 653–667.
[38] Y. Yazlik, D.T. Tollu and N. Taskara, On the solutions of difference equation systems with Padovan numbers,
Appl. Math. 4 (2013), no. 12A, 15–20.
[39] Y. Yazlik, On the solutions and behavior of rational difference equations, J. Comput. Anal. Appl. 17 (2014), no.
3, 584–594.
[40] Y. Yazlik 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 (2019), no. 2, 1675–1693.
[41] Y. Yazlik and M. Kara, On a solvable system of difference equations of fifth-order, Eski¸sehir Tech. Univ. J. Sci.
Tech. B-Theoret. Sci. 7 (2019), no. 1, 29–45.
)
Volume 13, Issue 2
July 2022Pages 2611-2637
- Receive Date: 19 April 2022
- Revise Date: 20 June 2022
- Accept Date: 29 June 2022