Araştırma Makalesi
Yıl 2025,
, 498 - 515, 28.04.2025
Öz
We show that a nonlinear difference equation recently considered in this journal is a special case of a solvable class of nonlinear difference equations and that the difference equation is closely related to a difference equation previously considered in the literature. We give some detailed theoretical explanations for the closed-form formulas for the solutions to the four special cases of the difference equation considered therein without giving any theoretical explanations related to them, and also show that several statements on the long-term behaviour of positive solutions to the difference equation given therein are not true.
Anahtar Kelimeler
solvable nonlinear difference equation bilinear difference equation solutions in closed form
Kaynakça
- [1] D. Adamović, Solution to problem 194, Mat. Vesnik 23, 236-242, 1971.
- [2] I. Bajo and E. Liz, Global behaviour of a second-order nonlinear difference equation, J. Difference Equ. Appl. 17 (10), 1471-1486, 2011.
- [3] L. Berg and S. Stević, 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.
- [4] 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. (in Latin)
- [5] G. Boole, A Treatsie on the Calculus of Finite Differences, Third Edition, Macmillan and Co., London, 1880.
- [6] L. Brand, A sequence defined by a difference equation, Amer. Math. Monthly 62 (7), 489-492, 1955.
- [7] B. P. Demidovich and I. A. Maron, Computational Mathematics, Mir Publishers, Moscow, 1973.
- [8] A. de Moivre, Miscellanea Analytica de Seriebus et Quadraturis, J. Tonson & J.Watts, Londini, 1730. (in Latin)
- [9] E. M. Elsayed and M. M. El-Dessoky, Dynamics and global behavior for a fourth-order rational difference equation, Hacet. J. Math. Stat. 42 (5), 479-494, 2013.
- [10] C. Jordan, Calculus of Finite Differences, Chelsea Publishing Company, New York, 1965.
- [11] G. Karakostas, The forbidden set, solvability and stability of a circular system of complex Riccati type difference equations, AIMS Mathematics 8 (11), 28033-28050, 2023.
- [12] V. A. Krechmar, A Problem Book in Algebra, Mir Publishers, Moscow, 1974.
- [13] S. F. Lacroix, Traité des Differénces et des Séries, J. B. M. Duprat, Paris, 1800. (in French)
- [14] S. F. Lacroix, An Elementary Treatise on the Differential and Integral Calculus, with an Appendix and Notes by J. Herschel, J. Smith, Cambridge, 1816.
- [15] 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). (in French)
- [16] 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)
- [17] H. Levy and F. Lessman, Finite Difference Equations, The Macmillan Company, New York, NY, USA, 1961.
- [18] A. A. Markoff, Differenzenrechnung, Teubner, Leipzig, 1896. (in German)
- [19] D. S. Mitrinović and J. D. Kečkić, Metodi Izračunavanja Konačnih Zbirova, Naučna Knjiga, Beograd, 1984. (in Serbian)
- [20] G. Papaschinopoulos and C. J. Schinas, Invariants for systems of two nonlinear difference equations, Differ. Equ. Dyn. Syst. 7, 181–196, 1999.
- [21] 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.
- [22] 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.
- [23] M. H. Rhouma, The Fibonacci sequence modulo $\pi$, chaos and some rational recursive equations, J. Math. Anal. Appl. 310, 506-517, 2005.
- [24] J. Riordan, Combinatorial Identities, John Wiley & Sons Inc., New York-London- Sydney, 1968.
- [25] C. Schinas, Invariants for difference equations and systems of difference equations of rational form, J. Math. Anal. Appl. 216, 164-179, 1997.
- [26] C. Schinas, Invariants for some difference equations, J. Math. Anal. Appl. 212, 281- 291, 1997.
- [27] S. Stević, 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.
- [28] S. Stević, Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences, Electron. J. Qual. Theory Differ. Equ. 2014, 67, 15 pages, 2014.
- [29] S. Stević, Representations of solutions to linear and bilinear difference equations and systems of bilinear difference equations, Adv. Difference Equ. 2018, 474, 21 pages, 2018.
- [30] S. Stević, General solution to a difference equation and the long-term behavior of some of its solutions, Hacet. J. Math. Stat. (2024) (in press).
- [31] S. Stević, J. Diblik, B. Iričanin and Z. Šmarda, On some solvable difference equations and systems of difference equations, Abstr. Appl. Anal. 2012, 541761, 11 pages, 2012.
- [32] S. Stević, J. Diblik, B. Iričanin and Z. Šmarda, On a solvable system of rational difference equations, J. Difference Equ. Appl. 20 (5-6), 811-825, 2014.
- [33] S. Stević, J. Diblik, B. Iričanin and Z. Šmarda, Solvability of nonlinear difference equations of fourth order, Electron. J. Differential Equations 2014, 264, 14 pages, 2014.
- [34] S. Stević, B. Iričanin, W. Kosmala and Z. Šmarda, Note on the bilinear difference equation with a delay, Math. Methods Appl. Sci. 41, 9349-9360, 2018.
- [35] S. Stević, B. Iričanin, 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.
- [36] S. Stević, B. Iričanin and Z. Šmarda, On a close to symmetric system of difference equations of second order, Adv. Difference Equ. 2015, 264, 17 pages, 2015.
- [37] S. Stević, B. Iričanin 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.
- [38] S. Stević, B. Iričanin and Z. Šmarda, On a symmetric bilinear system of difference equations, Appl. Math. Lett. 89, (15-21), 2019.
- [39] N. N. Vorobiev, Fibonacci Numbers, Birkhäuser, Basel, 2002.
Yıl 2025,
, 498 - 515, 28.04.2025
Öz
Kaynakça
- [1] D. Adamović, Solution to problem 194, Mat. Vesnik 23, 236-242, 1971.
- [2] I. Bajo and E. Liz, Global behaviour of a second-order nonlinear difference equation, J. Difference Equ. Appl. 17 (10), 1471-1486, 2011.
- [3] L. Berg and S. Stević, 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.
- [4] 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. (in Latin)
- [5] G. Boole, A Treatsie on the Calculus of Finite Differences, Third Edition, Macmillan and Co., London, 1880.
- [6] L. Brand, A sequence defined by a difference equation, Amer. Math. Monthly 62 (7), 489-492, 1955.
- [7] B. P. Demidovich and I. A. Maron, Computational Mathematics, Mir Publishers, Moscow, 1973.
- [8] A. de Moivre, Miscellanea Analytica de Seriebus et Quadraturis, J. Tonson & J.Watts, Londini, 1730. (in Latin)
- [9] E. M. Elsayed and M. M. El-Dessoky, Dynamics and global behavior for a fourth-order rational difference equation, Hacet. J. Math. Stat. 42 (5), 479-494, 2013.
- [10] C. Jordan, Calculus of Finite Differences, Chelsea Publishing Company, New York, 1965.
- [11] G. Karakostas, The forbidden set, solvability and stability of a circular system of complex Riccati type difference equations, AIMS Mathematics 8 (11), 28033-28050, 2023.
- [12] V. A. Krechmar, A Problem Book in Algebra, Mir Publishers, Moscow, 1974.
- [13] S. F. Lacroix, Traité des Differénces et des Séries, J. B. M. Duprat, Paris, 1800. (in French)
- [14] S. F. Lacroix, An Elementary Treatise on the Differential and Integral Calculus, with an Appendix and Notes by J. Herschel, J. Smith, Cambridge, 1816.
- [15] 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). (in French)
- [16] 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)
- [17] H. Levy and F. Lessman, Finite Difference Equations, The Macmillan Company, New York, NY, USA, 1961.
- [18] A. A. Markoff, Differenzenrechnung, Teubner, Leipzig, 1896. (in German)
- [19] D. S. Mitrinović and J. D. Kečkić, Metodi Izračunavanja Konačnih Zbirova, Naučna Knjiga, Beograd, 1984. (in Serbian)
- [20] G. Papaschinopoulos and C. J. Schinas, Invariants for systems of two nonlinear difference equations, Differ. Equ. Dyn. Syst. 7, 181–196, 1999.
- [21] 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.
- [22] 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.
- [23] M. H. Rhouma, The Fibonacci sequence modulo $\pi$, chaos and some rational recursive equations, J. Math. Anal. Appl. 310, 506-517, 2005.
- [24] J. Riordan, Combinatorial Identities, John Wiley & Sons Inc., New York-London- Sydney, 1968.
- [25] C. Schinas, Invariants for difference equations and systems of difference equations of rational form, J. Math. Anal. Appl. 216, 164-179, 1997.
- [26] C. Schinas, Invariants for some difference equations, J. Math. Anal. Appl. 212, 281- 291, 1997.
- [27] S. Stević, 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.
- [28] S. Stević, Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences, Electron. J. Qual. Theory Differ. Equ. 2014, 67, 15 pages, 2014.
- [29] S. Stević, Representations of solutions to linear and bilinear difference equations and systems of bilinear difference equations, Adv. Difference Equ. 2018, 474, 21 pages, 2018.
- [30] S. Stević, General solution to a difference equation and the long-term behavior of some of its solutions, Hacet. J. Math. Stat. (2024) (in press).
- [31] S. Stević, J. Diblik, B. Iričanin and Z. Šmarda, On some solvable difference equations and systems of difference equations, Abstr. Appl. Anal. 2012, 541761, 11 pages, 2012.
- [32] S. Stević, J. Diblik, B. Iričanin and Z. Šmarda, On a solvable system of rational difference equations, J. Difference Equ. Appl. 20 (5-6), 811-825, 2014.
- [33] S. Stević, J. Diblik, B. Iričanin and Z. Šmarda, Solvability of nonlinear difference equations of fourth order, Electron. J. Differential Equations 2014, 264, 14 pages, 2014.
- [34] S. Stević, B. Iričanin, W. Kosmala and Z. Šmarda, Note on the bilinear difference equation with a delay, Math. Methods Appl. Sci. 41, 9349-9360, 2018.
- [35] S. Stević, B. Iričanin, 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.
- [36] S. Stević, B. Iričanin and Z. Šmarda, On a close to symmetric system of difference equations of second order, Adv. Difference Equ. 2015, 264, 17 pages, 2015.
- [37] S. Stević, B. Iričanin 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.
- [38] S. Stević, B. Iričanin and Z. Šmarda, On a symmetric bilinear system of difference equations, Appl. Math. Lett. 89, (15-21), 2019.
- [39] N. N. Vorobiev, Fibonacci Numbers, Birkhäuser, Basel, 2002.
Toplam 39 adet kaynakça vardır.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Adi Diferansiyel Denklemler, Fark Denklemleri ve Dinamik Sistemler |
| Bölüm | Matematik |
| Yazarlar | |
| Erken Görünüm Tarihi | 27 Ağustos 2024 |
| Yayımlanma Tarihi | 28 Nisan 2025 |
| Gönderilme Tarihi | 19 Nisan 2024 |
| Kabul Tarihi | 28 Mayıs 2024 |
| Yayımlandığı Sayı | Yıl 2025 |