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

The authors consider the rational difference equation

${x}_{n+1}=\frac{{x}_{n}+{x}_{n-1}+{x}_{n-2}{x}_{n-3}}{{x}_{n}{x}_{n-1}+{x}_{n-2}+{x}_{n-3}}·$

The note displays the following results (with proofs):

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

$\left\{a,1,c;1,1,1,\cdots \right\}$

$\left\{1,b,1,d;1,1,\cdots \right\}$

$\left\{a,1,1,d;1,1,\cdots \right\}$

$\left\{a,b,1,1,\left(2+ab\right)/\left(1+a+b\right);1,1,\cdots \right\}$, $a\ne 1\ne b$

$\left\{1,b,c,1,1,\left(2+bc\right)/\left(1+b+c\right);1,1,1,\cdots \right\}$, $b\ne 1\ne c$

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:
 39A20 Generalized difference equations
##### References:
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