zbMATH — the first resource for mathematics

The dynamics of \(x_{n+1}= \frac {\alpha + \beta x_n}{A+Bx_n+Cx_{n-1}}\) facts and conjectures. (English) Zbl 1077.39004
The authors investigate the global character of the solutions of the equation in the title with nonnegative parameters and initial conditions such as the global attractivity, local and global asymptotic stability of the unique positive equilibrium, the boundedness and periodic nature of the solutions.

39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
Full Text: DOI
[1] Amleh, A.M.; Grove, E.A.; Ladas, G.; Georgiou, D.A., On the recursive sequence \(yn+1 = α + yn−1yn\), J. math. anal. appl., 233, 790-798, (1999) · Zbl 0962.39004
[2] Cunningham, K.; Kulenović, M.R.S.; Ladas, G.; Valicenti, S., On the recursive sequence \(χn+1=α+βχnA+Bχn+Cxn-1\), (), 4603-4614 · Zbl 1042.39522
[3] Gibbons, C.; Kulenović, M.R.S.; Ladas, G., On the recursive sequence., Math. sci. res. hot-line, 4, 2, 1-11, (2000) · Zbl 1039.39004
[4] Jaroma, J.H., On the global asymptotic stability of \(yn=1=a+bynA+yn-1\), () · Zbl 0860.39016
[5] Kocic, V.L.; Ladas, G., Global behaviour of nonlinear difference equations of higher order with applications, (1993), Kluwer Academic Dordrecht · Zbl 0787.39001
[6] Kocic, V.L.; Ladas, G.; Rodrigues, I.W., On the rational recursive sequences, J. math. anal. appl., 173, 127-157, (1993) · Zbl 0777.39002
[7] Kocic, V.L.; Ladas, G.; Tzanetopoulos, G.; Thomas, E., On the stability of lyness’ equation, Discrete and impulsive systems, 1, 245-254, (1995) · Zbl 0869.39004
[8] Kulenović, M.R.S.; Ladas, G.; Prokup, N.R., On the recursive sequence \(χn=1=αχn+βχn-11+xn\), J. differ. equations appl., 6, 5, 563-576, (2000) · Zbl 0966.39003
[9] Kulenović, M.R.S.; Ladas, G.; Prokup, N.R., A rational difference equation, Computers math. applic., 41, 5/6, 671-678, (2001) · Zbl 0985.39017
[10] Kulenović, M.R.S.; Ladas, G.; Sizer, W., On the recursive sequence \(yn+1=αyn+βyn-1γYn+Cyn-1\), Math. sci. res. hot-line, 2, 5, 1-16, (1998) · Zbl 0960.39502
[11] Kuruklis, S.; Ladas, G., Oscillation and global attractivity in a discrete delay logistic model, Quart. appl. math., 50, 227-233, (1992) · Zbl 0799.39004
[12] Krause, U., Stability trichotomy, path stability, and relative stability for positive nonlinear difference equations of higher order, J. diff. equa. appl., 1, 323-346, (1995) · Zbl 0855.39005
[13] Agarwal, R., Difference equations and inequalities. theory, methods and applications, (1992), Marcel Dekker New York
[14] Elaydi, S., An introduction to difference equations, (1999), Springer-Verlag New York · Zbl 0954.39011
[15] Grove, E.A.; Ladas, G.; McGrath, L.C.; Teixeira, C.T., Existence and behavior of solutions of a rational system, Comm. appl. nonlinear anal., 8, 1-25, (2001) · Zbl 1035.39013
[16] Kelley, W.G.; Peterson, A.C., Difference equations, (1991), Academic Press
[17] Philos, Ch.G.; Purnaras, I.K.; Sficas, Y.G., Global attractivity in a nonlinear difference equation, Appl. math. comput., 62, 249-258, (1994) · Zbl 0817.39005
[18] Kulenović, M.R.S., Invariants and related Liapunov functions for difference equations, Appl. math. lett., 13, 7, 1-8, (2000) · Zbl 0958.39021
[19] Kulenovic, M.R.S.; Ladas, G., Dynamics of second order rational difference equations, (2001), Chapman & Hall/CRC Press Boca Raton, FL · Zbl 0985.39017
[20] Ladas, G., Open problems and conjectures, J. diff. equa. appl., 1, 3, 317-321, (1995)
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.