On a family of nonlinear difference equations of the fifth order solvable in closed form

Stevo Stević; Bratislav Iričanin; Witold Kosmala; Stevo Stević; Bratislav Iričanin; Witold Kosmala

doi:10.3934/math.20231153

AIMS Mathematics

2023, Volume 8, Issue 10: 22662-22674. doi: 10.3934/math.20231153

Previous Article Next Article

Research article Special Issues

On a family of nonlinear difference equations of the fifth order solvable in closed form

1.
Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/Ⅲ, Beograd 11000, Serbia
2.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
3.
Faculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, Beograd 11000, Serbia
4.
Faculty of Mechanical and Civil Engineering in Kraljevo, University of Kragujevac, Kraljevo, Serbia
5.
Deptartment of Mathematical Sciences, Appalachian State University, Boone, NC 28608, USA

Received: 17 May 2023 Revised: 03 July 2023 Accepted: 04 July 2023 Published: 17 July 2023
MSC : 39A20

We present some closed-form formulas for the general solution to the family of difference equations

$ x_{n+1} = \Phi^{-1}\left(\Phi(x_{n-1})\frac{{\alpha} \Phi(x_{n-2})+{\beta} \Phi(x_{n-4})}{{\gamma} \Phi(x_{n-2})+{\delta} \Phi(x_{n-4})}\right), $

for $ n\in{\mathbb N}_0 $ where the initial values $ x_{-j} $, $ j = \overline{0, 4} $ and the parameters $ {\alpha}, {\beta}, {\gamma} $ and $ {\delta} $ are real numbers satisfying the conditions $ {\alpha}^2+{\beta}^2\ne 0, $ $ {\gamma}^2+{\delta}^2\ne 0 $ and $ \Phi $ is a function which is a homeomorphism of the real line such that $ \Phi(0) = 0, $ generalizing in a natural way some closed-form formulas to the general solutions to some very special cases of the family of difference equations which have been presented recently in the literature. Besides this, we consider in detail some of the recently formulated statements in the literature on the local and global stability of the equilibria as well as on the boundedness character of positive solutions to the special cases of the difference equation and give some comments and results related to the statements.
- difference equations,
- solvable equation,
- closed-form formula for solutions,
- bilinear difference equation
Citation: Stevo Stević, Bratislav Iričanin, Witold Kosmala. On a family of nonlinear difference equations of the fifth order solvable in closed form[J]. AIMS Mathematics, 2023, 8(10): 22662-22674. doi: 10.3934/math.20231153

Related Papers:

Abstract

We present some closed-form formulas for the general solution to the family of difference equations

$ x_{n+1} = \Phi^{-1}\left(\Phi(x_{n-1})\frac{{\alpha} \Phi(x_{n-2})+{\beta} \Phi(x_{n-4})}{{\gamma} \Phi(x_{n-2})+{\delta} \Phi(x_{n-4})}\right), $

for $ n\in{\mathbb N}_0 $ where the initial values $ x_{-j} $, $ j = \overline{0, 4} $ and the parameters $ {\alpha}, {\beta}, {\gamma} $ and $ {\delta} $ are real numbers satisfying the conditions $ {\alpha}^2+{\beta}^2\ne 0, $ $ {\gamma}^2+{\delta}^2\ne 0 $ and $ \Phi $ is a function which is a homeomorphism of the real line such that $ \Phi(0) = 0, $ generalizing in a natural way some closed-form formulas to the general solutions to some very special cases of the family of difference equations which have been presented recently in the literature. Besides this, we consider in detail some of the recently formulated statements in the literature on the local and global stability of the equilibria as well as on the boundedness character of positive solutions to the special cases of the difference equation and give some comments and results related to the statements.

References

[1]	D. Adamović, Solution to problem 194, Mat. Vesnik, 23 (1971), 236–242.
[2]	K. S. Berenhaut, J. D. Foley, S. Stević, Boundedness character of positive solutions of a max difference equation, J. Differ. Equ. Appl., 12 (2006), 1193–1199. https://doi.org/10.1080/10236190600949766 doi: 10.1080/10236190600949766
[3]	L. Berg, On the asymptotics of nonlinear difference equations, Z. Anal. Anwend., 21 (2002), 1061–1074. https://doi.org/10.4171/ZAA/1127 doi: 10.4171/ZAA/1127
[4]	L. Berg, S. Stević, On the asymptotics of the difference equation $y_n(1+y_{n-1}\cdots y_{n-k+1}) = y_{n-k}$, J. Differ. Equ. Appl., 17 (2011), 577–586. https://doi.org/10.1080/10236190903203820 doi: 10.1080/10236190903203820
[5]	D. Bernoulli, Observationes de seriebus quae formantur ex additione vel substractione quacunque terminorum se mutuo consequentium, ubi praesertim earundem insignis usus pro inveniendis radicum omnium aequationum algebraicarum ostenditur (in Latin), Commentarii Acad. Petropol. Ⅲ, 1728 (1732), 85–100.
[6]	L. Brand, A sequence defined by a difference equation, Am. Math. Mon., 62 (1955), 489–492. https://doi.org/10.2307/2307362 doi: 10.2307/2307362
[7]	A. de Moivre, Miscellanea analytica de seriebus et quadraturis (in Latin), J. Tonson & J. Watts, Londini, 1730.
[8]	E. M. Elsayed, B. S. Aloufi, O. Moaaz, The behavior and structures of solution of fifth-order rational recursive sequence, Symmetry, 14 (2022), 641. https://doi.org/10.3390/sym14040641 doi: 10.3390/sym14040641
[9]	B. Iričanin, S. Stević, On a class of third-order nonlinear difference equations, Appl. Math. Comput., 213 (2009), 479–483. https://doi.org/10.1016/j.amc.2009.03.039 doi: 10.1016/j.amc.2009.03.039
[10]	B. Iričanin, S. Stević, On some rational difference equations, Ars Comb., 92 (2009), 67–72.
[11]	G. L. Karakostas, Convergence of a difference equation via the full limiting sequences method, Differ. Equ. Dyn. Syst., 1 (1993), 289–294.
[12]	G. L. Karakostas, Asymptotic 2-periodic difference equations with diagonally self-invertible responces, J. Differ. Equ. Appl., 6 (2000), 329–335. https://doi.org/10.1080/10236190008808232 doi: 10.1080/10236190008808232
[13]	G. L. Karakostas, Asymptotic behavior of the solutions of the difference equation $x_{n+1} = x_n^2f(x_{n-1})$, J. Differ. Equ. Appl., 9 (2003), 599–602. https://doi.org/10.1080/1023619021000056329 doi: 10.1080/1023619021000056329
[14]	V. A. Krechmar, A problem book in algebra, Mir Publishers, Moscow, 1974.
[15]	S. F. Lacroix, Traité des differénces et des séries (in French), J. B. M. Duprat, Paris, 1800.
[16]	S. F. Lacroix, An elementary treatise on the differential and integral calculus, J. Smith, Cambridge, 1816.
[17]	J. L. Lagrange, Sur l'intégration d'une équation différentielle à différences finies, qui contient la théorie des suites récurrentes (in French), Miscellanea Taurinensia, 1759, 33–42.
[18]	P. S. Laplace, Recherches sur l'intégration des équations différentielles aux différences finies et sur leur usage dans la théorie des hasards (in French), Mém. Acad. R. Sci. Paris, Ⅶ (1776), 69–197.
[19]	H. Levy, F. Lessman, Finite difference equations, The Macmillan Company, New York, NY, USA, 1961.
[20]	L. M. Milne-Thomson, The calculus of finite differences, MacMillan and Co., London, 1933.
[21]	D. S. Mitrinović, D. D. Adamović, Nizovi i redovi/sequences and series (in Serbian), Naučna Knjiga, Beograd, Serbia, 1980.
[22]	G. Papaschinopoulos, C. J. Schinas, On a system of two nonlinear difference equations, J. Math. Anal. Appl., 219 (1998), 415–426. https://doi.org/10.1006/jmaa.1997.5829 doi: 10.1006/jmaa.1997.5829
[23]	G. Papaschinopoulos, C. J. Schinas, Invariants for systems of two nonlinear difference equations, Differ. Equ. Dyn. Syst., 7 (1999), 181–196.
[24]	G. Papaschinopoulos, C. J. Schinas, Invariants and oscillation for systems of two nonlinear difference equations, Nonlinear Anal.: Theory Methods Appl., 7 (2001), 967–978.
[25]	G. Papaschinopoulos, C. J. Schinas, G. Stefanidou, On a difference equation with 3-periodic coefficient, J. Differ. Equ. Appl., 11 (2005), 1281–1287. https://doi.org/10.1080/10236190500386317 doi: 10.1080/10236190500386317
[26]	G. Papaschinopoulos, C. J. Schinas, G. Stefanidou, On a $k$-order system of Lyness-type difference equations, Adv. Differ. Equ., 2007 (2007), 1–13.
[27]	G. Papaschinopoulos, G. Stefanidou, Asymptotic behavior of the solutions of a class of rational difference equations, Int. J. Differ. Equ., 5 (2010), 233–249.
[28]	C. J. Schinas, Invariants for difference equations and systems of difference equations of rational form, J. Math. Anal. Appl., 216 (1997), 164–179. https://doi.org/10.1006/jmaa.1997.5667 doi: 10.1006/jmaa.1997.5667
[29]	C. J. Schinas, Invariants for some difference equations, J. Math. Anal. Appl., 212 (1997), 281–291. https://doi.org/10.1006/jmaa.1997.5499 doi: 10.1006/jmaa.1997.5499
[30]	S. Stević, A global convergence results with applications to periodic solutions, Indian J. Pure Appl. Math., 33 (2002), 45–53.
[31]	S. Stević, On the recursive sequence $x_{n+1} = {\alpha}_n+(x_{n-1}/x_n)$ Ⅱ, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 911–916.
[32]	S. Stević, Boundedness character of a class of difference equations, Nonlinear Anal.: Theory Methods Appl., 70 (2009), 839–848. https://doi.org/10.1016/j.na.2008.01.014 doi: 10.1016/j.na.2008.01.014
[33]	S. Stević, Global stability of a difference equation with maximum, Appl. Math. Comput., 210 (2009), 525–529. https://doi.org/10.1016/j.amc.2009.01.050 doi: 10.1016/j.amc.2009.01.050
[34]	S. Stević, On the system of difference equations $x_n = c_ny_{n-3}/(a_n+b_ny_{n-1}x_{n-2}y_{n-3})$, $y_n = {\gamma}_n x_{n-3}/({\alpha}_n+{\beta}_n x_{n-1}y_{n-2}x_{n-3})$, Appl. Math. Comput., 219 (2013), 4755–4764. https://doi.org/10.1016/j.amc.2012.10.092 doi: 10.1016/j.amc.2012.10.092
[35]	S. Stević, Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences, Electron. J. Qual. Theory Differ. Equ., 2014 (2014), 1–15.
[36]	S. Stević, Representations of solutions to linear and bilinear difference equations and systems of bilinear difference equations, Adv. Differ. Equ., 2018 (2018), 1–21. https://doi.org/10.1186/s13662-018-1930-2 doi: 10.1186/s13662-018-1930-2
[37]	S. Stević, J. Diblik, B. Iričanin, Z. Šmarda, On a solvable system of rational difference equations, J. Difference Equ. Appl., 20 (2014), 811–825.
[38]	S. Stević, J. Diblik, B. Iričanin, Z. Šmarda, Solvability of nonlinear difference equations of fourth order, Electron. J. Differ. Equ., 2014 (2014), 1–14. https://doi.org/10.1080/10236198.2013.817573 doi: 10.1080/10236198.2013.817573
[39]	S. Stević, B. Iričanin, W. Kosmala, Z. Šmarda, On a nonlinear second-order difference equation, J. Inequal. Appl., 2022 (2022), 1–11. https://doi.org/10.1186/s13660-022-02822-z doi: 10.1186/s13660-022-02822-z
[40]	S. Stević, B. Iričanin, Z. Šmarda, Solvability of a close to symmetric system of difference equations, Electron. J. Differ. Equ., 2016 (2016), 1–13.
[41]	S. Stević, B. Iričanin, Z. Šmarda, On a product-type system of difference equations of second order solvable in closed form, J. Inequal. Appl., 2015 (2015), 1–15. https://doi.org/10.1186/s13660-015-0835-9 doi: 10.1186/s13660-015-0835-9
[42]	S. Stević, B. Iričanin, Z. Šmarda, On a symmetric bilinear system of difference equations, Appl. Math. Lett., 89 (2019), 15–21. https://doi.org/10.1016/j.aml.2018.09.006 doi: 10.1016/j.aml.2018.09.006
[43]	V. A. Zorich, Mathematical analysis Ⅰ, Springer, Berlin, Heidelberg, 2004.

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)