Global attractivity in the recursive sequence \(x_{n+1}=(\alpha-\beta x_{n})/(\gamma-x_{n-1})\). (English) Zbl 1030.39024

Following on the analysis of the rational recursive sequence \[ x_{n+1}=(\alpha-\beta x_{n})/(\gamma-x_{n-1}),\quad n=0,1,2,\ldots \] with arbitrary \(x_{0}\), \(x_{-1}\), \(\alpha\geq 0\), \(\beta>0\), \(\gamma>0\), it is shown that the positive equilibrium is an attractor and an estimate is obtained for the attraction basin.


39A12 Discrete version of topics in analysis
39B05 General theory of functional equations and inequalities
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
Full Text: DOI


[1] Kocic, V.L; Ladas, G; Rodrigues, I.W, On rational recursive sequences, J. math. anal. appl., 173, 127-157, (1993) · Zbl 0777.39002
[2] Kocic, V.L; Ladas, G, Global attractivity in a second-order nonlinear difference equation, J. math. anal. appl., 180, 144-150, (1993) · Zbl 0802.39001
[3] Kocic, V.L; Ladas, G, Global behavior of nonlinear difference equations of higher order with application, (1993), Kluwer Academic Publishers Dordrecht · Zbl 0787.39001
[4] Aboutaleb, M.T; El-Sayed, M.A; Hamza, A.E, Stability of the recursive sequence xn+1=(α−βxn)/(γ+xn−1), J. math. anal. appl., 261, 126-133, (2001) · Zbl 0990.39009
[5] W.T. Li, H.R. Sun, Global attractivity in a rational recursive sequence, Appl. Math. Lett., in press · Zbl 1019.39007
[6] Darwen, C; Patula, W.T, Properties of a certain lyness equation, J. math. anal. appl., 218, 458-478, (1998) · Zbl 0896.39003
[7] Feuer, J; Janowski, E.J; Ladas, G, Lyness-type equations in the third quadrant, Nonlinear anal. TMA, 30, 1183-1189, (1997) · Zbl 0893.39004
[8] DeVault, R; Kosmala, W; Ladas, G; Schultz, S.W, Global behavior of yn+1=(p+yn−k)/(qyn+yn−k), Nonlinear anal. TMA, 47, 4743-4751, (2001) · Zbl 1042.39523
[9] El-Metwally, H; Grove, E.A; Ladas, G; Levins, R; Radin, M, On the difference equation xn+1=α+βxn−1e−xn, Nonlinear anal. TMA, 47, 4623-4634, (2001) · Zbl 1042.39506
[10] Kulenovic, M.R.S; Ladas, G; Prokup, N.R, A rational difference equation, Comput. math. applic., 41, 671-678, (2001) · Zbl 0985.39017
[11] Greaf, J.R; Qian, C; Spikes, P.W, Stability in a population model, Appl. math. comput., 89, 119-132, (1998) · Zbl 0904.39005
[12] El-Owaidy, H.M; El-Afifi, M.M, A note on the periodic cycle of xn+2=(1+xn+1)/xn, Appl. math. comput., 109, 301-306, (2000) · Zbl 1023.39010
[13] El-Owaidy, H.M; Ragab, A.A; El-Afifi, M.M, On the recursive sequence xn+1=A/xnp+B/xn−1q+C/xn−2s, Appl. math. comput., 112, 277-290, (2000) · Zbl 1023.39011
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.