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Global asymptotic stability in a rational recursive sequence. (English) Zbl 1071.39018
The global stability of the difference equation \[ x_{n}=\frac{a+bx_{n-1}+cx_{n-1}^2}{d-x_{n-2}}, \quad n=1,2,\dots \] where \(a,b\geq 0\) and \(c,d>0\) is studied. It is shown that one nonnegative equillibrium point of the equation is a global attractor with a basin that is determined by the parameters, and every positive solution of the equation in the basin exponentially converges to the attractor.

MSC:
39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
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[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] 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
[3] Li, W.; Sun, H., Global attractivity in a rational recursive sequence, Dyn. syst. appl, 11, 339-346, (2002) · Zbl 1019.39007
[4] Yan, X.; Li, W., Global attractivity in the recursive sequence xn+1=(α−βxn)/(γ−xn−1), Appl. math. comput, 138, 415-423, (2003)
[5] Yan, X.; Li, W.; Sun, H., Global attractivity in a higher order nonlinear difference equation, Appl. math. E-notes, 2, 51-58, (2002) · Zbl 1004.39010
[6] Yan, X.; Li, W., Global attractivity in a rational recursive sequence, Appl. math. comput, 145, 1-12, (2003) · Zbl 1044.39013
[7] Amleh, A.M.; Grove, E.A.; Ladas, G., On the recursive sequence xn+1=α+xn−1/xn, J. math. anal. appl, 233, 790-798, (1999) · Zbl 0962.39004
[8] Gibbons, C.; Kulenović, M.R.S.; Ladas, G., On the recursive sequence \(xn+1=α+βxn−1γ+xn\), Math. sci. res. hotline, 4, 2, 1-11, (2000)
[9] 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
[10] Kosmala, W.A.; Kulenović, M.R.S.; Ladas, G.; Teixeira, C.T., On the recursive sequence yn+1=(p+yn−1)/(qyn+yn−1), J. math. anal. appl, 251, 571-586, (2000) · Zbl 0967.39004
[11] Cunningham, K.; Kulenović, M.R.S.; Ladas, G.; Valicenti, S.V., On the recursive sequence \(xn+1=α+βxnBxn+Cxn−1\), Nonlinear anal, 47, 4603-4614, (2001) · Zbl 1042.39522
[12] Kulenović, M.R.S.; Ladas, G.; Prokup, N.R., A rational difference equation, Comput. math. appl, 41, 671-678, (2001) · Zbl 0985.39017
[13] Abu-Saris, R.M.; DeVault, R., Global stability of \(yn+1=A+ynyn−k\), Appl. math. lett, 16, 173-178, (2003) · Zbl 1049.39002
[14] El-Owaidy, H.M.; Ahmed, A.M.; Mousa, M.S., On the recursive sequences \(xn+1=−αxn−1β±xn\), Appl. math. comput, 145, 747-753, (2003) · Zbl 1034.39004
[15] El-Afifi, M.M., On the recursive sequence \(xn+1=α+βxn+γxn−1Bxn+Cxn−1\), Appl. math. comput, 147, 617-628, (2004) · Zbl 1041.39001
[16] El-Owaidy, H.M.; Ahmed, A.M.; Mousa, M.S., On asymptotic behaviour of the difference equation \(xn+1=α+xn−kxn\), Appl. math. comput, 147, 163-167, (2004) · Zbl 1042.39001
[17] Li, L., Stability properties of nonlinear difference equations and conditions for boundedness, Comput. math. appl, 38, 29-35, (1999) · Zbl 0936.39004
[18] Zhang, D.; Shi, B.; Gai, M., A rational recursive sequence, Comput. math. appl, 41, 301-306, (2001) · Zbl 0985.39016
[19] Kocic, V.L.; Ladas, G., Global behavior of nonlinear difference equations of higher order with applications, (1993), Kluwer Academic Publishers Dordrecht · Zbl 0787.39001
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