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Eventually constant solutions of a rational difference equation. (English) Zbl 1178.39012

The authors consider the rational difference equation

x n+1 =x n +x n-1 +x n-2 x n-3 x n x n-1 +x n-2 +x n-3 ·

The note displays the following results (with proofs):

1. Every positive solution that eventually equals 1 has one of the following forms

{a,1,c;1,1,1,}

{1,b,1,d;1,1,}

{a,1,1,d;1,1,}

{a,b,1,1,(2+ab)/(1+a+b);1,1,}, a1b

{1,b,c,1,1,(2+bc)/(1+b+c);1,1,1,}, b1c

where x -3 =a, x -2 =b, x -1 =c, x 0 =d for convenience.

2. For almost all initial values, positive solutions are not eventually equal to 1.

MSC:
39A20Generalized difference equations
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