×

Qualitative properties for a fourth-order rational difference equation. (English) Zbl 1082.39004

A fourth order difference equation in a rational form is considered. By bringing out a periodicity property of the nontrivial semicyclic solutions, it is shown that the positive fixed point of the equation is globally asymptotically stable.

MSC:

39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Agarwal, R.P., Difference equations and inequalities, (1992), Dekker New York, 2000 · Zbl 0784.33008
[2] Kocic, V.L.; Ladas, G., Global behavior of nonlinear difference equations of higher order with applications, (1993), Kluwer Academic Dordrecht · Zbl 0787.39001
[3] Kulenović, M.R.S.; Ladas, G.; Martins, L.F.; Rodrigues, I.W., The dynamics of \(x_{n + 1} = \frac{\alpha + \beta x_n}{A + B x_n + C x_{n - 1}}\): facts and conjectures, Comput. math. appl., 45, 1087-1099, (2003) · Zbl 1077.39004
[4] Stević, S., More on a rational recurrence relation, Appl. math. E-notes, 4, 80-84, (2004) · Zbl 1069.39024
[5] Nesemann, T., Positive nonlinear difference equations: some results and applications, Nonlinear anal., 47, 4707-4717, (2001) · Zbl 1042.39510
[6] Amleh, A.M.; Kruse, N.; Ladas, G., On a class of difference equations with strong negative feedback, J. difference equ. appl., 5, 497-515, (1999) · Zbl 0951.39002
[7] Ladas, G., Progress report on \(x_{n = 1} = (\alpha + \beta x_n + \gamma x_{n - 1}) /(A + B x_n + C x_{n - 1})\), J. difference equ. appl., 1, 211-215, (1995) · Zbl 0855.39006
[8] Stević, S., On the recursive sequence \(x_{n + 1} = g(x_n, x_{n - 1}) /(A + x_n)\), Appl. math. lett., 15, 305-308, (2002) · Zbl 1029.39007
[9] Amleh, A.M.; Grove, E.A.; Georgiou, D.A.; Ladas, G., On the recursive sequence \(x_{n + 1} = \alpha + x_{n - 1} / x_n\), J. math. anal. appl., 233, 790-798, (1999) · Zbl 0962.39004
[10] Gibbons, C.; Kulenović, M.R.S.; Ladas, G., On the recursive sequence \(x_{n + 1} = (\alpha + \beta x_{n - 1}) /(\gamma + x_n)\), Math. sci. res. hot-line, 4, 1-11, (2000) · Zbl 1039.39004
[11] Li, X.; Zhu, D., Global asymptotic stability in a rational equation, J. difference equ. appl., 9, 833-839, (2003) · Zbl 1055.39014
[12] Li, X.; Zhu, D., Global asymptotic stability for two recursive difference equations, Appl. math. comput., 150, 481-492, (2004) · Zbl 1044.39006
[13] Li, X.; Zhu, D., Global asymptotic stability of a nonlinear recursive sequence, Appl. math. lett., 17, 833-838, (2004) · Zbl 1068.39014
[14] Li, X.; Zhu, D., Two rational recursive sequences, Comput. math. appl., 47, 10-11, 1487-1494, (2004) · Zbl 1072.39008
[15] Li, X.; Zhu, D., Global asymptotic stability for a nonlinear delay difference equation, Appl. math. J. Chinese univ. ser. B, 17, 183-188, (2002) · Zbl 1013.39003
[16] Li, X.; Xiao, G., A conjecture by G. ladas, Appl. math. J. Chinese univ. ser. B, 13, 39-44, (1998) · Zbl 0902.39003
[17] Li, X., Boundedness and persistence and global asymptotic stability for a kind of delay difference equations with higher order, Appl. math. mech. (English ed.), 23, 1331-1338, (2002) · Zbl 1034.34082
[18] Li, X.; Xiao, G., Periodicity and strict oscillation for generalized lyness equations, Appl. math. mech. (English ed.), 21, 455-460, (2000) · Zbl 0965.39015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.