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

The authors consider the rational difference equation $x_{n+1} = {{x_n+x_{n-1}+x_{n-2}x_{n-3}}\over{x_nx_{n-1}+x_{n-2}+x_{n-3}}}.$ The note displays the following results (with proofs):
1. Every positive solution that eventually equals 1 has one of the following forms 7mm
(a)
$$\{a,1,c;1,1,1,\dots\}$$
(b)
$$\{1,b,1,d;1,1,\dots\}$$
(c)
$$\{a,1,1,d;1,1,\dots\}$$
(d)
$$\{a,b,1,1,(2+ab)/(1+a+b);1,1,\dots\}$$, $$a\neq 1\neq b$$
(e)
$$\{1,b,c,1,1,(2+bc)/(1+b+c);1,1,1,\dots\}$$, $$b\neq 1\neq c$$
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 Multiplicative and other generalized difference equations
Full Text:

### References:

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