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On the rational recursive sequence x n+1 =ax n-1 b+cx n x n-1 . (English) Zbl 1212.39008

Consider the difference equation

x n+1 =ax n-1 b+cx n x n-1 ,n=0,1,,

with a,b,c positive real numbers, and nonnegative initial conditions (x -1 ,x 0 ).

By a change of variables, it is reduced to

y n+1 =y n-1 p+y n y n-1 ,n=0,1,,(1)

where p=b/a, x n =(a/c) 1/2 y n .

Since this equation is semiconjugate to a Möbius transformation [cf. A. Andruch-Sobiło, M. Małgorzata, Opusc. Math. 26, No. 3 387–394 (2006; Zbl 1131.39003)], a formula for the solutions is available in terms of the parameter p and the initial data. The authors use this formula to prove that every positive solution of (1) converges to zero if p1, and converges to a periodic solution of period two if 0<p<1. [For recent results concerning the same equation, see H. Sedaghat, J. Difference Equ. Appl. 15, No. 3, 215–224 (2009; Zbl 1169.39006)].

MSC:
39A20Generalized difference equations
39A22Growth, boundedness, comparison of solutions (difference equations)