A trichotomy result for non-autonomous rational difference equations

Frank Palladino ; Michael Radin

Open Mathematics, Tome 9 (2011), p. 1135-1142 / Harvested from The Polish Digital Mathematics Library

Résumé

We study non-autonomous rational difference equations. Under the assumption of a periodic non-autonomous parameter, we show that a well known trichotomy result in the autonomous case is preserved in a certain sense which is made precise in the body of the text. In addition we discuss some questions regarding whether periodicity preserves or destroys boundedness.

Détail

Publié le : 2011-01-01

Zbl 1236.39016

EUDML-ID : urn:eudml:doc:269377

Bibliographie

[1] Camouzis E., Ladas G., When does periodicity destroy boundedness in rational equations?, J. Difference Equ. Appl., 2006, 12(9), 961–979 http://dx.doi.org/10.1080/10236190600822369 | Zbl 1112.39003

[2] Camouzis E., Ladas G., Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Adv. Discrete Math. Appl., 5, Chapman & Hall/CRC Press, Boca Raton, 2007 http://dx.doi.org/10.1201/9781584887669 | Zbl 1129.39002

[3] Cushing J.M., Henson S.M., A periodically forced Beverton-Holt equation, J. Difference Equ. Appl., 2002, 8(12), 1119–1120 http://dx.doi.org/10.1080/1023619021000053980 | Zbl 1023.39013

[4] Elaydi S., Sacker R.J., Nonautonomous Beverton-Holt equations and the Cushing-Henson conjectures, J. Difference Equ. Appl., 2005, 11(4–5), 337–346 http://dx.doi.org/10.1080/10236190412331335418 | Zbl 1084.39005

[5] El-Metwally H.A., Grove E.A., Ladas G., A global convergence result with applications to periodic solutions, J. Math. Anal. Appl., 2000, 245(1), 161–170 http://dx.doi.org/10.1006/jmaa.2000.6747 | Zbl 0971.39004

[6] Grove E.A., Kostrov Y., Ladas G., Schultz S.W., Riccati difference equations with real period-2 coefficients, Comm. Appl. Nonlinear Anal., 2007, 14(2), 33–56 | Zbl 1126.39001

[7] Grove E.A., Ladas G., Periodicities in Nonlinear Difference Equations, Adv. Discrete Math. Appl., 4, Chapman & Hall/CRC Press, Boca Raton, 2005 | Zbl 1078.39009

[8] Grove E.A., Ladas G., Predescu M., Radin M., On the global character of the difference equation $x_{n + 1} = \frac{{\alpha + \gamma x_n - (2k + 1) + \delta x_{n - 2l} }}{{A + x_{n - 2l} }} $ , J. Difference Equ. Appl., 2003, 9(2), 171–199 http://dx.doi.org/10.1080/1023619021000054015 | Zbl 1038.39004

[9] Karakostas G.L., Stević S., On the recursive sequence $x_{n + 1} = B + \frac{{x_{n - k} }}{{a_0 x_n + \cdots + a_{k - 1} x_{n - k + 1} + \gamma }} $ , J. Difference Equ. Appl., 2004, 10(9), 809–815 http://dx.doi.org/10.1080/10236190410001659732 | Zbl 1068.39012

[10] Kocić V.L., Ladas G., Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Math. Appl., 256, Kluwer, Dordrecht, 1993 | Zbl 0787.39001

[11] Kulenovic M.R.S., Merino O., Stability analysis of Pielou’s equation with period-two coefficient, J. Difference Equ. Appl., 2007, 13(5), 383–406 http://dx.doi.org/10.1080/10236190601045929 | Zbl 1123.39004

[12] Palladino F.J., Difference inequalities, comparison tests, and some consequences, Involve, 2008, 1(1), 91–100 http://dx.doi.org/10.2140/involve.2008.1.91 | Zbl 1154.39012

[13] Palladino F.J., On the characterization of rational difference equations, J. Difference Equ. Appl., 2009, 15(3), 253–260 http://dx.doi.org/10.1080/10236190802119903 | Zbl 1169.39005

[14] Palladino F.J., On periodic trichotomies, J. Difference Equ. Appl., 2009, 15(6), 605–620 http://dx.doi.org/10.1080/10236190802258677 | Zbl 1207.39018

[15] Stevic S., Behavior of the positive solutions of the generalized Beddington-Holt equation, Panamer. Math. J., 2000, 10(4), 77–85 | Zbl 1039.39005

[16] Stevic S., On the recursive sequence x n+1 = α n + x n−1/x n. II, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 2003, 10(6), 911–916

[17] Stevic S., A short proof of the Cushing-Henson conjecture, Discrete Dyn. Nat. Soc., 2006, ID 37264 | Zbl 1149.39300

[18] Stevic S., On positive solutions of a (k + 1)th order difference equation, Appl. Math. Lett., 2006, 19(5), 427–431 http://dx.doi.org/10.1016/j.aml.2005.05.014 | Zbl 1095.39010