On the Difference Equation <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mi>k</mml:mi></mml:msubsup><mml:mrow><mml:msub><mml:mi>a</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:msub><mml:mi>f</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mstyle></mml:mrow></mml:math>

Stevic, Stevo

doi:https://doi.org/10.1155/2007/75262

Discrete Dynamics in Nature and Society

On this page

Abstract References Copyright Related Articles

Research Article | Open Access

Volume 2007 | Article ID 075262 | https://doi.org/10.1155/2007/75262

On the Difference Equation xn+1=∑j=0kajfj(xn−j)

Stevo Stevic¹

Received25 Feb 2007

Accepted16 Jun 2007

Published19 Aug 2007

Abstract

This paper studies the boundedness character and the global attractivity of positive solutions of the difference equation xn+1=∑j=0kajfj(xn−j),n∈ℕ0, where aj are positive numbers and fj are continuous decreasing self-maps of the interval (0,∞) for j=0,1,…,k.

References

K. S. Berenhaut, J. D. Foley, and S. Stević, “Quantitative bounds for the recursive sequence $y_{n + 1} = A + y_{n} / y_{n - k}$ ,” Applied Mathematics Letters, vol. 19, no. 9, pp. 983–989, 2006.
View at: Publisher Site | Google Scholar | 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 | Zentralblatt MATH | MathSciNet
B. Dai and X. Zou, “Permanence for a class of nonlinear difference systems,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 78607, p. 10, 2006.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. DeVault, G. Ladas, and S. W. Schultz, “Necessary and sufficient conditions for the boundedness of $x_{n + 1} = A / x_{n}^{p} + B / x_{n - 1}^{q}$ ,” Journal of Difference Equations and Applications, vol. 3, no. 3-4, pp. 259–266, 1998.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
H. M. El-Owaidy, A. A. Ragab, and M. M. El-Afifi, “On the recursive sequence $x_{n + 1} = A / x_{n}^{p} + B / x_{n - 1}^{q} + C / x_{n - 2}^{s}$ ,” Applied Mathematics and Computation, vol. 112, no. 2-3, pp. 277–290, 2000.
View at: Publisher Site | 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. L. Karakostas, “A discrete semi-flow in the space of sequences and study of convergence of sequenses defined by difference equations,” M. E. Greek Math. Soc., vol. 30, pp. 66–74, 1989.
View at: Google Scholar
G. L. 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. L. Karakostas and S. Stević, “On the difference equation $x_{n + 1} = A f (x_{n}) + f (x_{n - 1})$ ,” Applicable Analysis, vol. 83, no. 3, pp. 309–323, 2004.
View at: Publisher Site | Google Scholar
Ch. G. Philos, I. K. Purnaras, and Y. G. Sficas, “Global attractivity in a nonlinear difference equation,” Applied Mathematics and Computation, vol. 62, no. 2-3, pp. 249–258, 1994.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Stević, “On the recursive sequence $x_{n + 1} = - 1 / x_{n} + A / x_{n - 1}$ ,” International Journal of Mathematics and Mathematical Sciences, vol. 27, no. 1, pp. 1–6, 2001.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Stević, “A note on the difference equation $x_{n + 1} = \sum_{j = 0}^{k} α_{i} / x_{n - i}^{p_{i}}$ ,” Journal of Difference Equations and Applications, vol. 8, no. 7, pp. 641–647, 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, “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, “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, “On the recursive sequence $x_{n} = 1 + \sum_{i = 1}^{k} α_{i} x_{n - p_{i}} / \sum_{j = 1}^{m} β_{j} x_{n - q_{j}}$ ,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 39404, p. 7, 2007.
View at: Publisher Site | 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 | 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

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

215

Downloads

579

Citations