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Mat. Zametki, 2008, Volume 84, Issue 5, Pages 772–780 (Mi mz6360)  

This article is cited in 37 scientific papers (total in 37 papers)

Nontrivial Solutions of a Higher-Order Rational Difference Equation

S. Stević

Mathematical Institute, Serbian Academy of Sciences and Arts

Abstract: We prove that, for every $k\in\mathbb N$, the following generalization of the Putnam difference equation
$$ x_{n+1}=\frac{x_n+x_{n-1}+…+x_{n-(k-1)}+x_{n-k}x_{n-(k+1)}} {x_nx_{n-1}+x_{n-2}+…+x_{n-(k+1)}} ,\qquad n\in\mathbb N_0, $$
has a positive solution with the following asymptotics
$$ x_n=1+(k+1)e^{-\lambda^n}+(k+1)e^{-c\lambda^n}+o(e^{-c\lambda^n}) $$
for some $c>1$ depending on $k$, and where $\lambda$ is the root of the polynomial $P(\lambda)=\lambda^{k+2}-\lambda-1$ belonging to the interval $(1,2)$. Using this result, we prove that the equation has a positive solution which is not eventually equal to $1$. Also, for the case $k=1$, we find all positive eventually equal to unity solutions to the equation.

Keywords: difference equation, nonlinear solution, asymptotic, Putnam difference equation

DOI: https://doi.org/10.4213/mzm6360

Full text: PDF file (470 kB)
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English version:
Mathematical Notes, 2008, 84:5, 718–724

Bibliographic databases:

UDC: 512.628.4
Received: 29.10.2006

Citation: S. Stević, “Nontrivial Solutions of a Higher-Order Rational Difference Equation”, Mat. Zametki, 84:5 (2008), 772–780; Math. Notes, 84:5 (2008), 718–724

Citation in format AMSBIB
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\paper Nontrivial Solutions of a Higher-Order Rational Difference Equation
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\yr 2008
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\issue 5
\pages 772--780
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\jour Math. Notes
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\pages 718--724
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Liao Maoxin, Tang Xianhua, Xu Changjin, “On a conjecture for a higher-order rational difference equation”, Adv. Difference Equ., 2009, 394635, 9 pp.  crossref  mathscinet  zmath  isi  scopus
    2. Iricanin B., Stević S., “Eventually constant solutions of a rational difference equation”, Appl. Math. Comput., 215:2 (2009), 854–856  crossref  mathscinet  zmath  isi  scopus
    3. Iričanin B.D., Elsayed E.M., “On the max-type difference equation $x_{n+1}=\max\{A/x_n,x_{n-3}\}$”, Discrete Dyn. Nat. Soc., 2010, 675413, 13 pp.  mathscinet  zmath  isi
    4. Iričanin B.D., “Global stability of some classes of higher-order nonlinear difference equations”, Appl. Math. Comput., 216:4 (2010), 1325–1328  crossref  mathscinet  zmath  isi  scopus
    5. Stević S., “Global stability of some symmetric difference equations”, Appl. Math. Comput., 216:1 (2010), 179–186  crossref  mathscinet  zmath  isi  scopus
    6. Liao Maoxin, Tang Xianhua, Xu Changjin, “General form of some rational recursive sequences”, Comput. Math. Appl., 59:1 (2010), 360–364  crossref  mathscinet  zmath  adsnasa  isi  scopus
    7. Kent C.M., Kosmala W., Stevic S., “Long-term behavior of solutions of the difference equation $x_{n+1}=x_{n-1}x_{n-2}-1$”, Abstr. Appl. Anal., 2010, 152378, 17 pp.  mathscinet  zmath  isi
    8. Stefanidou G., Papaschinopoulos G., Schinas C.J., “On an exponential-type fuzzy difference equation”, Adv. Difference Equ., 2010, 196920, 19 pp.  crossref  mathscinet  zmath  isi
    9. Liu Wanping, Yang Xiaofan, Yang Luxing, “Global behavior of two families of nonlinear symmetric difference equations”, Discrete Dyn. Nat. Soc., 2010, 367492, 15 pp.  mathscinet  zmath  isi
    10. Iričanin B.D., Liu Wanping, “On a higher-order difference equation”, Discrete Dyn. Nat. Soc., 2010, 891564, 6 pp.  mathscinet  zmath  isi
    11. Kent C.M., Kosmala W., Radin M.A., Stević S., “Solutions of the difference equation $x_{n+1}=x_nx_{n-1}-1$”, Abstr. Appl. Anal., 2010, 469683, 13 pp.  mathscinet  zmath  isi
    12. Liu Wanping, Yang Xiaofan, Iričanin B.D., “On some $k$-dimensional cyclic systems of difference equations”, Abstr. Appl. Anal., 2010, 528648, 11 pp.  mathscinet  zmath  isi  elib
    13. Sun Taixiang, Xi Hongjian, Wu Hui, Han Caihong, “Global behavior of the difference equation $x_{n+1}=(p+x_{n-1})/(qx_n+x_{n-1})$”, Abstr. Appl. Anal., 2010, 237129, 6 pp.  mathscinet  zmath  isi
    14. Rachůnek L., Rachůnková I., “Strictly increasing solutions of nonautonomous difference equations arising in hydrodynamics”, Adv. Difference Equ., 2010, 714891  mathscinet  zmath  isi
    15. Berg L., Stević S., “On the asymptotics of the difference equation $y_n(1+y_{n-1}…y_{n-k+1})=y_{n-k}$”, J. Differ. Equ. Appl., 17:4 (2011), 577–586  crossref  mathscinet  zmath  isi  scopus
    16. Liao Maoxin, Tang Xianhua, Ouyang Zigen, Xu Changjin, “Dynamical properties of a class of higher-order nonlinear difference equations”, Appl. Math. Comput., 217:12 (2011), 5476–5479  crossref  mathscinet  zmath  isi  scopus
    17. Stević S., “On a system of difference equations with period two coefficients”, Appl. Math. Comput., 218:8 (2011), 4317–4324  crossref  mathscinet  zmath  isi  scopus
    18. Stević S., “On a system of difference equations”, Appl. Math. Comput., 218:7 (2011), 3372–3378  crossref  mathscinet  zmath  isi  scopus
    19. Papaschinopoulos G., Radin M.A., Schinas C.J., “On the system of two difference equations of exponential form: $x_{n+1}=a+bx_{n-1}e^{-y_n}, y_{n+1}=c+dy_{n-1}e^{-x_n}$”, Math. Comput. Model., 54:11-12 (2011), 2969–2977  crossref  mathscinet  zmath  isi  scopus
    20. Berg L., Stević S., “On some systems of difference equations”, Appl. Math. Comput., 218:5 (2011), 1713–1718  crossref  mathscinet  zmath  isi  scopus
    21. Stević S., “Periodicity of a class of nonautonomous max-type difference equations”, Appl. Math. Comput., 217:23 (2011), 9562–9566  crossref  mathscinet  zmath  isi  scopus
    22. Stević S., Iričanin B., “Unbounded solutions of the difference equation $x_n=x_{n-l}x_{n-k}-1$”, Abstr. Appl. Anal., 2011 (2011), 561682, 8 pp.  crossref  mathscinet  zmath  isi  scopus
    23. Berg L., Stević S., “On the asymptotics of some systems of difference equations”, J. Difference Equ. Appl., 17:9 (2011), 1291–1301  crossref  mathscinet  zmath  isi  scopus
    24. Kent C.M., Kosmala W., Stević S., “On the difference equation $x_{n+1}=x_nx_{n-2}-1$”, Abstr. Appl. Anal., 2011 (2011), 815285, 15 pp.  crossref  mathscinet  isi  scopus
    25. Liu W., Yang X., Stević S., Iričanin B., “Part-metric and its applications to cyclic discrete dynamic systems”, Abstr. Appl. Anal., 2011 (2011), 534974, 16 pp.  crossref  mathscinet  zmath  isi  scopus
    26. Liu W., Yang X., Stević S., “On a class of nonautonomous max-type difference equations”, Abstr. Appl. Anal., 2011 (2011), 436852, 15 pp.  crossref  mathscinet  zmath  isi  scopus
    27. Diblík J., Hlavičková I., “Asymptotic upper and lower estimates of a class of positive solutions of a discrete linear equation with a single delay”, Abstr. Appl. Anal., 2012 (2012), 764351, 12 pp.  crossref  mathscinet  zmath  isi  scopus
    28. Berg L., Stević S., “On the asymptotics of some difference equations”, J. Difference Equ. Appl., 18:5 (2012), 785–797  crossref  mathscinet  zmath  isi  scopus
    29. Stević S., “On some solvable systems of difference equations”, Appl. Math. Comput., 218:9 (2012), 5010–5018  crossref  mathscinet  zmath  isi  scopus
    30. Papaschinopoulos G., Radin M., Schinas C.J., “Study of the asymptotic behavior of the solutions of three systems of difference equations of exponential form”, Appl. Math. Comput., 218:9 (2012), 5310–5318  crossref  mathscinet  zmath  isi  scopus
    31. Diblik J., Iricanin B., Stevic S., Smarda Z., “On Some Symmetric Systems of Difference Equations”, Abstract Appl. Anal., 2013, 246723  crossref  mathscinet  zmath  isi  scopus
    32. Liu W., Yang X., “Global Behavior of Two Higher-Order Symmetric Difference Equations”, Util. Math., 92 (2013), 89–96  mathscinet  zmath  isi
    33. Stevic S., Alghamdi M.A., Alotaibi A., Shahzad N., “Eventual Periodicity of Some Systems of Max-Type Difference Equations”, Appl. Math. Comput., 236 (2014), 635–641  crossref  mathscinet  zmath  isi  scopus
    34. Stevic S., “On Positive Solutions of Some Classes of Max-Type Systems of Difference Equations”, Appl. Math. Comput., 232 (2014), 445–452  crossref  mathscinet  zmath  isi  scopus
    35. Stevic S., “On Families of Bounded Continuous Solutions of Some Systems of Functional-Difference Equations on Half-Intervals”, Appl. Math. Comput., 231 (2014), 169–178  crossref  mathscinet  zmath  isi  scopus
    36. Liu W., Yang X., Liu X., Stevic S., “Part-Metric and Its Applications in Discrete Systems”, Appl. Math. Comput., 228 (2014), 320–328  crossref  mathscinet  zmath  isi  scopus
    37. Tang Sh., Tang H., Xu Zh., “Global Stability of a Family of Asymmetrical Difference Equations”, Int. J. Appl. Math. Stat., 54:2 (2016), 76–82  mathscinet  isi
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