Asymptotics of Some Classes of Higher-Order Difference Equations

Stevic, Stevo

doi:https://doi.org/10.1155/2007/56813

Discrete Dynamics in Nature and Society

On this page

Abstract References Copyright Related Articles

Research Article | Open Access

Volume 2007 | Article ID 056813 | https://doi.org/10.1155/2007/56813

Asymptotics of Some Classes of Higher-Order Difference Equations

Stevo Stevic¹

Received09 Feb 2007

Accepted23 Apr 2007

Published07 Jun 2007

Abstract

We present some methods for finding asymptotics of some classes of nonlinear higher-order difference equations. Among others, we confirm a conjecture posed by S. Stević (2005). Monotonous solutions of the equation yn=A+(yn−k/∑j=1mβjyn−qj)p, n∈ℕ0, where p,A∈(0,∞), k,m∈ℕ, qj,j∈{1,…,m}, are natural numbers such that q1<q2<⋯<qm,βj∈(0,+∞), j∈{1,…,m}, ∑j=1mβj=1, and y−s,y−s+1,…,y−1∈(0,∞), where s=max{k,qm}, are found. A new inclusion theorem is proved. Also, some open problems and conjectures are posed.

References

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 | MathSciNet
L. Berg, Asymptotische Darstellungen und Entwicklungen, Hochschulbücher für Mathematik, Band 66, 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 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 | 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
H. M. El-Owaidy, A. M. Ahmed, and M. S. Mousa, “On asymptotic behaviour of the difference equation $x_{n + 1} = α + (x_{n - 1}^{p} / x_{n}^{p})$ ,” Journal of Applied Mathematics & Computing, vol. 12, no. 1-2, pp. 31–37, 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
V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, vol. 256 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993.
View at: Zentralblatt MATH | MathSciNet
W. Kosmala and C. Teixeira, “More on the difference equation $y_{n + 1} = (p + y_{n}) / (q y_{n} + y_{n - 1})$ ,” Applicable Analysis, vol. 81, no. 1, pp. 143–151, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
N. Kruse and T. Nesemann, “Global asymptotic stability in some discrete dynamical systems,” Journal of Mathematical Analysis and Applications, vol. 235, no. 1, pp. 151–158, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. Levy and F. Lessman, Finite Difference Equations, Dover, New York, NY, USA, 1992.
View at: MathSciNet
G. Ladas, “Open problems and conjectures,” Journal of Difference Equations and Applications, vol. 4, pp. 497–499, 1998.
View at: Google Scholar
“Putnam exam,” American Mathematical Monthly, pp. 734–736, 1965.
View at: Google Scholar
S. Stević, “On stability results for a new approximating fixed points iteration process,” Demonstratio Mathematica, vol. 34, no. 4, pp. 873–880, 2001.
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, “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 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, “Periodic character of a difference equation,” Rostocker Mathematisches Kolloquium, no. 59, pp. 3–10, 2005.
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, “On the recursive sequence $x_{n + 1} = (α + β x_{n - k} / f (x_{n}, \dots, x_{n - k + 1}))$ ,” Taiwanese Journal of Mathematics, vol. 9, no. 4, pp. 583–593, 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 monotone solutions of some classes of difference equations,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 53890, p. 9, 2006.
View at: Publisher Site | Google Scholar | 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
S. Stević, “On the recursive sequence $x_{n + 1} = A + (x_{n}^{p} / x_{n - 1}^{p})$ ,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 34517, p. 9, 2007.
View at: Publisher Site | Google Scholar
S. Stević, “Nontrivial solutions of a higher-order rational difference equation,” submitted.
View at: Google Scholar
T. Sun and H. Xi, “Global asymptotic stability of a family of difference equations,” Journal of Mathematical Analysis and Applications, vol. 309, no. 2, pp. 724–728, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | 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
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 | 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. 1-2, pp. 269–282, 2005.
View at: Google Scholar | Zentralblatt MATH | MathSciNet

Copyright

Copyright © 2007 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

251

Downloads

746

Citations