ON THE GLOBAL ASYMPTOTIC STABILITY OF THE DIFFERENCE EQUATION

TitleON THE GLOBAL ASYMPTOTIC STABILITY OF THE DIFFERENCE EQUATION
Publication TypeJournal Article
Year of Publication2010
AuthorsVAN KHUONG, VU, PHONG, MAINAM
Secondary TitleCommunications in Applied Analysis
Volume14
Issue4
Start Page597
Pagination606
Date Published12/2010
Type of Workscientific: mathematics
ISSN1083–2564
AMS39A10.
Abstract

We investigate the dynamical behavior of the following fourth-order rational differ-
ence equation

where a ∈ [0, ∞) and the initial values x−3 , x−2 , x−1 , x0 ∈ (0, ∞). We find that the successive lengths
of positive and negative semicycles of nontrivial solutions of the above equation occur periodically.
We also show that the positive equilibrium of the equation is globally asymptotically stable.
URLhttp://www.acadsol.eu/en/articles/14/4/12.pdf
Short TitleON THE GLOBAL ASYMPTOTIC STABILITY
Refereed DesignationRefereed
Full Text

REFERENCES

[1] R. P. Agarwal, Difference Equations and Inequalities, Second Ed. Dekker, New York, 1992, 2000.
[2] V. L. Kocic, G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, Kluwer Academic, Dordrecht, 1993.
[3] M. R. S. Kulenovi ́c, G. Ladas, L. F. Martins, I. W. Rodrignes, The dynamics of x n+1 = : Facts and conjectures, Comput. Math. Appl. 45 (2003), 1087–1099.
[4] G. Ladas, Open problems and conjectures, J. Difference Equa. Appl. 2 (1996), 449–452.
[5] T. Nesemann, Positive nonlinear difference equations: some results and applications, Nonlinear Anal. 47 (2001), 4707–4717.
[6] A. M. Amleh, N. Kruse, G. Ladas, On a class of difference equations with strong negative feedback, J. Differ. Equations Appl. 5 (1999), 497–515.
[7] G. Ladas, Progress report on , J. Difference Equa. Appl. 1 (1995), 211–215.
[8] S. Stevic, On the recursive sequence , Appl. Math. Lett. 15 (2002), 305–308.
[9] A. M. Amleh, E. A. Grove, D. A. Georgiou, G. Ladas, On the recursive sequence x n+1 =, J. Math. Anal. Appl. 233 (1999), 790–798.
[10] C. Gibbon, M. R. S. Kulenovi ́c, G. Ladas, On the recursive sequence , Math. Sci. Res. Hot-line 4 (2000), 1–11.
[11] X. Li, D. Zhu, Global asymptotic stability in a rational equation, J. Difference. Equa. Appl. 9 (2003), 833–839.
[12] X. Li, D. Zhu, Global asymptotic stability for two recursive difference equations, Appl. Math. Comput. 150 (2004), 481–492.
[13] X. Li, D. Zhu, Global asymptotic stability of a nonlinear recursive sequence, Appl. Math. Appl. 17 (2004), 833–838.
[14] X. Li, D. Zhu, Two rational recursive sequence, Comput. Math. Appl. 47 (2004), 1487–1494.
[15] X. Li, D. Zhu, Global asymptotic stability for a nonlinear delay difference equation, Appl. Math. J. Chinese Univ. Ser. B 17 (2002), 183–188.
[16] X. Li, G. Xiao, et al, A conjecture by G. Ladas, Appl. Math. J. Chinese Univ. Ser. B 13 (1998), 39–44.
[17] X. Li, Boundedness and persistence and global asymptotic stability for a kind of delay difference equations with higher order, Appl. Math. Mech. (English) 23 (2003), 1331–1338.
[18] X. Li, G. Xiao, et al, Periodicity and strict oscillation for generalized Lyness equations, Appl. Math. Mech. (English) 21 (2000), 455–460.
[19] X. Li, Qualitative properties for a fourth-order rational difference equation, J. Math. Anal. Appl. 311 (2005), 103–111.
[20] X. Li, The rule of trajectory structure and global asymptotic stability for a fourth-order rational difference equation, J. Korean Math. Soc. 44 (2007), 787–797.
[21] L. Zhang, Hai-Feng Hua, Li-Ming Miao and Hong Xiang, Dinamical behavior of a third-order rational difference equation, Appl. Math. E-Notes 6 (2006), 268–275.