PapersNo Access

STABILITY OF THE kTH ORDER LYNESS' EQUATION WITH A PERIOD-k COEFFICIENT

E. J. JANOWSKI

Department of Mathematics, University of Rhode Island, Kingston, RI 02881-0816, USA

Search for more papers by this author

M. R. S. KULENOVIĆ

Department of Mathematics, University of Rhode Island, Kingston, RI 02881-0816, USA

Author for correspondence.

Search for more papers by this author

, and

Z. NURKANOVIĆ

Department of Mathematics, University of Tuzla, 75000 Tuzla, Bosnia and Herzegovina

Search for more papers by this author

https://doi.org/10.1142/S0218127407017227Cited by:9 (Source: Crossref)

Abstract

We first investigate the Lyapunov stability of the period-three solution of Todd's equation with a period-three coefficient:

where

α,β, and γ positive.

Then for k = 2,3,… we extend our stability result to the k-order equation,

where p_n is a periodic coefficient of period k with positive real values and x_-k+1,…,x_-1, x₀ ∈ (0, ∞).

Keywords:

References

E. Barbeau, B. Gelbord and S. Tanny, J. Diff. Eqs. Appl. 1, 291 (1995). Crossref, Google Scholar
G. Bastien and M. Rogalski, J. Diff. Eqs. Appl. 10, 977 (2004). Crossref, Web of Science, Google Scholar
G. Bastien and M. Rogalski, J. Math. Anal. Appl. 300, 303 (2004). Crossref, Web of Science, Google Scholar
F. Beukers and R. Cushman, J. Diff. Eqs. 143, 191 (1998). Crossref, Web of Science, Google Scholar
A. Cima, A. Gasull and A. Manosa, Int. J. Bifurcation and Chaos 16, 631 (2006), DOI: 10.1142/S0218127406015027. Link, Web of Science, Google Scholar
C. A. Clark, E. J. Janowski and M. R. S. Kulenović, J. Math. Anal. Appl. 307, 292 (2005). Crossref, Web of Science, Google Scholar
S. Elaydi and R. Sacker, J. Diff. Eqs. 208, 258 (2005). Crossref, Web of Science, Google Scholar
S. Elaydi and R. Sacker, Global stability of periodic orbits of nonautonomous difference equations in population biology and the Cushing–Henson conjectures, Proc. Eighth Int. Conf. Difference Equations and Applications (Chapman & Hall/CRC, Boca Raton, FL, 2005) pp. 113–126. Google Scholar
J. Feuer, E. Janowski and G. Ladas, J. Diff. Eqs. Appl. 2, 167 (1996). Crossref, Google Scholar
J. E. Franke and J. F. Selgrade, J. Math. Anal. Appl. 286, 64 (2003). Crossref, Web of Science, Google Scholar
L. Gardini, G. I. Bischi and C. Mira, Int. J. Bifurcation and Chaos 13, 1841 (2003), DOI: 10.1142/S0218127403007679. Link, Web of Science, Google Scholar
E. A. Grove and G. Ladas , Periodicities in Nonlinear Difference Equations ( CRC Press , Boca Raton, Florida , 2004 ) . Crossref, Google Scholar
S. M. Henson and J. M. Cushing, J. Math. Biol. 36, 201 (1997). Crossref, Web of Science, Google Scholar
V. L. Kocic and G. Ladas , Global Behavior of Nonlinear Difference Equations of Higher Order with Applications ( Kluwer Academic Publishers , Dordreht/Boston/London , 1993 ) . Crossref, Google Scholar
V. L. Kocic, G. Ladas and I. W. Rodrigues, J. Math. Anal. Appl. 173, 127 (1993). Crossref, Web of Science, Google Scholar
M. R. S. Kulenović, Appl. Math. Lett. 13, 1 (2000). Crossref, Web of Science, Google Scholar
M. R. S. Kulenović and O. Merino , Discrete Dynamical Systems and Difference Equations with Mathematica ( Chapman and Hall/CRC , Boca Raton, London , 2002 ) . Crossref, Google Scholar
M. R. S. Kulenović, G. Ladas and C. B. Overdeep, J. Diff. Eqs. Appl. 10, 915 (2004). Google Scholar
H. Sedaghat , Nonlinear Difference equations. Theory with Applications to Social Science Models , Mathematical Modelling: Theory and Applications 15 ( Kluwer Academic Publishers , Dordrecht , 2003 ) . Google Scholar
Zeeman, E. C. [1996] "Geometric unfolding of a difference equation," preprint (Hertford College, Oxford) . Google Scholar