On the Behaviour of the Solutions of a Second-Order Difference Equation

Gutnik, Leonid; Stevic, Stevo

doi:https://doi.org/10.1155/2007/27562

Discrete Dynamics in Nature and Society

On this page

Abstract References Copyright Related Articles

Research Article | Open Access

Volume 2007 | Article ID 027562 | https://doi.org/10.1155/2007/27562

On the Behaviour of the Solutions of a Second-Order Difference Equation

Leonid Gutnik¹and Stevo Stevic²

Received07 Dec 2006

Accepted18 Feb 2007

Published19 Apr 2007

Abstract

We study the difference equation xn+1=α−xn/xn−1, n∈ℕ0, where α∈ℝ and where x−1 and x0 are so chosen that the corresponding solution (xn) of the equation is defined for every n∈ℕ. We prove that when α=3 the equilibrium x¯=2 of the equation is not stable, which corrects a result due to X. X. Yan, W. T. Li, and Z. Zhao. For the case α=1, we show that there is a strictly monotone solution of the equation, and we also find its asymptotics. An explicit formula for the solutions of the equation are given for the case α=0.

References

A. M. Amleh, E. A. Grove, G. Ladas, and D. A. Georgiou, “On the recursive sequence $y_{n + 1} = α + y_{n - 1} / y_{n}$ ,” Journal of Mathematical Analysis and Applications, vol. 233, no. 2, pp. 790–798, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
K. S. Berenhaut, J. D. Foley, and S. Stević, “The global attractivity of the rational difference equation $y_{n} = 1 + y_{n - k} / y_{n - m}$ ,” Proceedings of the American Mathematical Society, vol. 135, no. 4, pp. 1133–1140, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
K. S. Berenhaut and S. Stević, “A note on the difference equation $x_{n + 1} = 1 / (x_{n} x_{n - 1}) + 1 / (x_{n - 3} x_{n - 4})$ ,” Journal of Difference Equations and Applications, vol. 11, no. 14, pp. 1225–1228, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
K. S. Berenhaut and S. Stević, “The behaviour of the positive solutions of the difference equation $x_{n} = A + {(x_{n - 2} / x_{n - 1})}^{p}$ ,” Journal of Difference Equations and Applications, vol. 12, no. 9, pp. 909–918, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
K. S. Berenhaut and S. Stević, “The difference equation $x_{n + 1} = α + x_{n - k} / \sum_{i = 0}^{k - 1} c_{i} x_{n - i}$ has solutions converging to zero,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1466–1471, 2007.
View at: Publisher Site | Google Scholar
L. Berg, Asymptotische Darstellungen und Entwicklungen, vol. 66 of Hochschulbücher für Mathematik, VEB Deutscher Verlag der Wissenschaften, Berlin, Germany, 1968.
View at: Zentralblatt MATH | MathSciNet
L. Berg, “On the asymptotics of nonlinear difference equations,” Zeitschrift für Analysis und ihre Anwendungen, vol. 21, no. 4, pp. 1061–1074, 2002.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
L. Berg, “Inclusion theorems for non-linear difference equations with applications,” Journal of Difference Equations and Applications, vol. 10, no. 4, pp. 399–408, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
L. Berg, “Oscillating solutions of rational difference equations,” Rostocker Mathematisches Kolloquium, no. 58, pp. 31–35, 2004.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
L. Berg, “Corrections to: “Inclusion theorems for non-linear difference equations with applications”,” Journal of Difference Equations and Applications, vol. 11, no. 2, pp. 181–182, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
L. Berg, “Nonlinear difference equations with periodic coefficients,” Rostocker Mathematisches Kolloquium, no. 61, pp. 13–20, 2006.
View at: Google Scholar
L. Berg and L. von Wolfersdorf, “On a class of generalized autoconvolution equations of the third kind,” Zeitschrift für Analysis und ihre Anwendungen, vol. 24, no. 2, pp. 217–250, 2005.
View at: Google Scholar | MathSciNet
R. DeVault, G. Ladas, and S. W. Schultz, “On the recursive sequence $x_{n + 1} = A / x_{n} + 1 / x_{n - 2}$ ,” Proceedings of the American Mathematical Society, vol. 126, no. 11, pp. 3257–3261, 1998.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. DeVault, C. Kent, and W. Kosmala, “On the recursive sequence $x_{n + 1} = p + x_{n - k} / x_{n}$ ,” Journal of Difference Equations and Applications, vol. 9, no. 8, pp. 721–730, 2003.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
G. Karakostas, “Convergence of a difference equation via the full limiting sequences method,” Differential Equations and Dynamical Systems, vol. 1, no. 4, pp. 289–294, 1993.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
G. Karakostas, “Asymptotic 2-periodic difference equations with diagonally self-invertible responses,” Journal of Difference Equations and Applications, vol. 6, no. 3, pp. 329–335, 2000.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
C. M. Kent, “Convergence of solutions in a nonhyperbolic case,” Nonlinear Analysis, vol. 47, no. 7, pp. 4651–4665, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
W. Kosmala and C. Teixeira, “More on the difference equation $y_{n + 1} = (p + y_{n - 1}) / (q y_{n} + y_{n - 1})$ ,” Applicable Analysis, vol. 81, no. 1, pp. 143–151, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Stević, “Asymptotic behaviour of a sequence defined by iteration,” Matematički Vesnik, vol. 48, no. 3-4, pp. 99–105, 1996.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. Stević, “Behavior of the positive solutions of the generalized Beddington-Holt equation,” Panamerican Mathematical Journal, vol. 10, no. 4, pp. 77–85, 2000.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. Stević, “A global convergence results with applications to periodic solutions,” Indian Journal of Pure and Applied Mathematics, vol. 33, no. 1, pp. 45–53, 2002.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. Stević, “Asymptotic behavior of a sequence defined by iteration with applications,” Colloquium Mathematicum, vol. 93, no. 2, pp. 267–276, 2002.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. Stević, “On the recursive sequence $x_{n + 1} = x_{n - 1} / g (x_{n})$ ,” Taiwanese Journal of Mathematics, vol. 6, no. 3, pp. 405–414, 2002.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. Stević, “Asymptotic behavior of a nonlinear difference equation,” Indian Journal of Pure and Applied Mathematics, vol. 34, no. 12, pp. 1681–1687, 2003.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
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 Journal of Mathematics, vol. 7, no. 2, pp. 249–259, 2003.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. Stević, “A note on periodic character of a difference equation,” Journal of Difference Equations and Applications, vol. 10, no. 10, pp. 929–932, 2004.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. Stević, “On the recursive sequence $x_{n + 1} = α + x_{n - 1}^{p} / x_{n}^{p}$ ,” Journal of Applied Mathematics & Computing, vol. 18, no. 1-2, pp. 229–234, 2005.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. Stević, “Global stability and asymptotics of some classes of rational difference equations,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 60–68, 2006.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. Stević, “On positive solutions of a $(k + 1)$ th order difference equation,” Applied Mathematics Letters, vol. 19, no. 5, pp. 427–431, 2006.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. Stević, “Existence of nontrivial solutions of a rational difference equation,” Applied Mathematics Letters, vol. 20, no. 1, pp. 28–31, 2007.
View at: Google Scholar | MathSciNet
T. Sun, H. Xi, and H. Wu, “On boundedness of the solutions of the difference equation $x_{n + 1} = x_{n - 1} / (p + x_{n})$ ,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 20652, p. 7, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
S.-E. Takahasi, Y. Miura, and T. Miura, “On convergence of a recursive sequence $x_{n + 1} = f (x_{n - 1}, x_{n})$ ,” Taiwanese Journal of Mathematics, vol. 10, no. 3, pp. 631–638, 2006.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
H. D. Voulov, “Existence of monotone solutions of some difference equations with unstable equilibrium,” Journal of Mathematical Analysis and Applications, vol. 272, no. 2, pp. 555–564, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
X.-X. Yan, W.-T. Li, and Z. Zhao, “On the recursive sequence $x_{n + 1} = α - (x_{n} / x_{n - 1})$ ,” Journal of Applied Mathematics & Computing, vol. 17, no. 1, pp. 269–282, 2005.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
X. Yang, “Global asymptotic stability in a class of generalized Putnam equations,” Journal of Mathematical Analysis and Applications, vol. 322, no. 2, pp. 693–698, 2006.
View at: Publisher Site | Google Scholar | MathSciNet

Copyright

Copyright © 2007 Leonid Gutnik and Stevo Stević. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation Order printed copies

Views

259

Downloads

782

Citations