General solution to a difference equation and the long-term behavior of some of its solutions

Year 2025, Volume: 54 Issue: 1, 38 - 56, 28.02.2025

Stevo Stevic

https://doi.org/10.15672/hujms.1326208

Abstract

Closed-from formulas for the general solution to a difference equation are given, generalizing some special cases in the literature. We also analyze and give some comments on the results on the long-term behaviour of some solutions of the special cases.

Keywords

difference equation, general solution, closed-form formula

References

• [1] D. Adamovic, Problem 194, Mat. Vesnik, 22 (2), 270, 1970.
• [2] D. Adamovic, Solution to problem 194, Mat. Vesnik, 23, 236-242, 1971.
• [3] A. Andruch-Sobilo and M. Migda, On the rational recursive sequence $x_{n+1}=ax_{n-1}/(b+cx_nx_{n-1})$, Tatra Mt. Math. Publ. 43, 1-9, 2009.
• [4] K. Berenhaut, J. Foley and S. Stevic, Boundedness character of positive solutions of a max difference equation, J. Difference Equ. Appl. 12 (12), 1193-1199, 2006.
• [5] K. Berenhaut, J. Foley and S. Stevic, The global attractivity of the rational difference equation $y_{n}=1+(y_{n-k}/y_{n-m})$, Proc. Amer. Math. Soc. 135 (1), 1133-1140, 2007.
• [6] K. Berenhaut and S. Stevic, The behaviour of the positive solutions of the difference equation $x_n=A+(x_{n-2}/x_{n-1})^p$, J. Difference Equ. Appl. 12 (9), 909-918, 2006.
• [7] L. Berg, On the asymptotics of nonlinear difference equations, Z. Anal. Anwendungen, 21 (4), 1061-1074,2002.
• [8] L. Berg and S. Stevic, On the asymptotics of the difference equation $y_n(1+y_{n-1}\cdots y_{n-k+1})=y_{n-k}$, J. Difference Equ. Appl. 17 (4), 577-586, 2011.
• [9] 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, Commentarii Acad. Petropol. III, 1728, 85-100, 1732.
• [10] G. Boole, A Treatsie on the Calculus of Finite Differences, Third Edition, Macmillan and Co., London, 1880.
• [11] L. Brand, A sequence defined by a difference equation, Amer. Math. Monthly, 62 (7), 489-492, 1955.
• [12] A. de Moivre, Miscellanea Analytica de Seriebus et Quadraturis, J. Tonson & J.Watts, Londini, 1730.
• [13] L. Euler, Introductio in Analysin Infinitorum, Tomus Primus, Lausannae, 1748.
• [14] B. Iricanin and S. Stevic, On some rational difference equations, Ars Combin. 92, 67-72, 2009.
• [15] C. Jordan, Calculus of Finite Differences, Chelsea Publishing Company, New York, 1965.
• [16] G. L. Karakostas, Convergence of a difference equation via the full limiting sequences method, Differ. Eqs. Dyn. Syst. 1 (4), 289-294, 1993.
• [17] G. L. Karakostas, Asymptotic behavior of the solutions of the difference equation $x_{n+1}=x_n^2f(x_{n-1})$, J. Differ. Eqs. Appl. 9 (6), 599-602, 2003.
• [18] V. A. Krechmar, A Problem Book in Algebra, Mir Publishers, Moscow, 1974.
• [19] S. F. Lacroix, Traité des Differénces et des Séries, J. B. M. Duprat, Paris, 1800. (in French)
• [20] S. F. Lacroix, An Elementary Treatise on the Differential and Integral Calculus, with an Appendix and Notes by J. Herschel, J. Smith, Cambridge, 1816.
• [21] J.-L. Lagrange, Sur l’intégration d’une équation différentielle à différences finies, qui contient la théorie des suites récurrentes, Miscellanea Taurinensia, t. I, (1759), 33-42 (Lagrange OEuvres, I, 23-36, 1867).
• [22] P. S. Laplace, Recherches sur l’intégration des équations différentielles aux différences finies et sur leur usage dans la théorie des hasards, Mémoires de l’ Académie Royale des Sciences de Paris 1773, t. VII, (1776) (Laplace OEuvres, VIII, 69-197, 1891). (in French)
• [23] H. Levy and F. Lessman, Finite Difference Equations, The Macmillan Company, New York, NY, USA, 1961.
• [24] A. A. Markoff, Differenzenrechnung, Teubner, Leipzig, 1896.
• [25] L. M. Milne-Thomson, The Calculus of Finite Differences, MacMillan and Co., London, 1933.
• [26] D. S. Mitrinovic and D. D. Adamovic, Nizovi i Redovi/Sequences and Series, Naucna Knjiga, Beograd, Serbia, 1980.
• [27] D. S. Mitrinovic, J. D. Keckic, Metodi Izracunavanja Konacnih Zbirova/Methods for Calculating Finite Sums, Naucna Knjiga, Beograd, 1984.
• [28] N. E. Nörlund, Vorlesungen über Differenzenrechnung, Berlin, Springer, 1924.
• [29] G. Papaschinopoulos and C. J. Schinas, On a system of two nonlinear difference equations, J. Math. Anal. Appl. 219 (2), 415-426, 1998.
• [30] G. Papaschinopoulos and C. J. Schinas, Invariants for systems of two nonlinear difference equations, Differ. Equ. Dyn. Syst. 7, 181–196, 1999.
• [31] G. Papaschinopoulos and C. J. Schinas, Invariants and oscillation for systems of two nonlinear difference equations, Nonlinear Anal. Theory Methods Appl. 46, 967–978, 2001.
• [32] G. Papaschinopoulos, C. J. Schinas and G. Stefanidou, On a difference equation with 3-periodic coefficient, J. Difference Equ. Appl. 11 (15), 1281-1287, 2005.
• [33] G. Papaschinopoulos, C. J. Schinas and G. Stefanidou, On a k-order system of Lynesstype difference equations, Adv. Difference Equ. 2007, 31272, 13 pages, 2007.
• [34] G. Papaschinopoulos and G. Stefanidou, Trichotomy of a system of two difference equations, J. Math. Anal. Appl. 289, 216-230, 2004.
• [35] G. Papaschinopoulos and G. Stefanidou, Asymptotic behavior of the solutions of a class of rational difference equations, Inter. J. Difference Equations, 5 (2), 233-249, 2010.
• [36] G. Papaschinopoulos, G. Stefanidou and C. J. Schinas, Boundedness, attractivity, and stability of a rational difference equation with two periodic coefficients, Discrete Dyn. Nat. Soc. 2009, 973714, 23 pages, 2009.
• [37] M. H. Rhouma, The Fibonacci sequence modulo $\pi$, chaos and some rational recursive equations, J. Math. Anal. Appl. 310, 506-517, 2005.
• [38] A. Sanbo and E. M. Elsayed, Some properties of the solutions of the difference equation $x_{n+1}=ax_n+\frac{bx_nx_{n-4}}{cx_{n-3}+dx_{n-4}}$ , Open J. Discret. Appl. Math. 2 (2), 31-47, 2019.
• [39] C. Schinas, Invariants for difference equations and systems of difference equations of rational form, J. Math. Anal. Appl. 216, 164-179, 1997.
• [40] C. Schinas, Invariants for some difference equations, J. Math. Anal. Appl. 212, 281- 291, 1997.
• [41] S. Stevic, A global convergence results with applications to periodic solutions, Indian J. Pure Appl. Math. 33 (1), 45-53, 2002.
• [42] S. Stevic, On the recursive sequence$x_{n+1}=A/\prod_{i=0}^k x_{n-i}+1/\prod_{j=k+2}^{2(k+1)}x_{n-j}$ , Taiwanese J. Math. 7 (2), 249-259, 2003.
• [43] S. Stevic, 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 (6), 911-916, 2003.
• [44] S. Stevic, Asymptotic periodicity of a higher order difference equation, Discrete Dyn. Nat. Soc. 2007, 13737, 9 pages, 2007.
• [45] S. Stevic, Boundedness character of a class of difference equations, Nonlinear Anal. TMA, 70, 839-848, 2009.
• [46] S. Stevic, Global stability of a difference equation with maximum, Appl. Math. Comput. 210, 525-529, 2009.
• [47] S. Stevic, Global stability of a max-type difference equation, Appl. Math. Comput. 216, 354-356, 2010.
• [48] S. Stevic, On some periodic systems of max-type difference equations, Appl. Math. Comput. 218, 11483-11487, 2012.
• [49] S. Stevic, Solutions of a max-type system of difference equations, Appl. Math. Comput. 218, 9825-9830, 2012.
• [50] S. Stevic, 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, 4755-4764, 2013.
• [51] S. Stevic, Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences, Electron. J. Qual. Theory Differ. Equ. 2014 (67), 15 pages, 2014.
• [52] S. Stevic, Representations of solutions to linear and bilinear difference equations and systems of bilinear difference equations, Adv. Difference Equ. 2018, 474, 21 pages, 2018.
• [53] S. Stevic, J. Diblik, B. Iricanin and Z. Šmarda, On a third-order system of difference equations with variable coefficients, Abstr. Appl. Anal. 2012, 508523, 22 pages, 2012.
• [54] S. Stevic, J. Diblik, B. Iricanin and Z. Šmarda, On a solvable system of rational difference equations, J. Difference Equ. Appl. 20 (5-6), 811-825, 2014.
• [55] S. Stevic, J. Diblik, B. Iricanin and Z. Šmarda, Solvability of nonlinear difference equations of fourth order, Electron. J. Differential Equations, 2014, 264, 14 pages, 2014.
• [56] S. Stevic, B. Iricanin, W. Kosmala and Z. Šmarda, Note on the bilinear difference equation with a delay, Math. Methods Appl. Sci. 41, 9349-9360, 2018.
• [57] S. Stevic, B. Iricanin, W. Kosmala and Z. Šmarda, On a solvable class of nonlinear difference equations of fourth order, Electron. J. Qual. Theory Differ. Equ. 2022, 37, 17 pages, 2022.
• [58] S. Stevic, B. Iricanin and Z. Šmarda, On a product-type system of difference equations of second order solvable in closed form, J. Inequal. Appl. 2015, 327, 15 pages, 2015.
• [59] S. Stevic, B. Iricanin and Z. Šmarda, Solvability of a close to symmetric system of difference equations, Electron. J. Differential Equations, 2016, 159, 13 pages, 2016.
• [60] S. Stevic, B. Iricanin and Z. Šmarda, On a symmetric bilinear system of difference equations, Appl. Math. Lett. 89, 15-21, 2019.
• [61] N. N. Vorobiev, Fibonacci Numbers, Birkhäuser, Basel, 2002.
• [62] V. A. Zorich, Mathematical Analysis I, Springer, Berlin, Heidelberg, 2004.