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Eventually constant solutions of a rational difference equation. (English) Zbl 1178.39012
The authors consider the rational difference equation $$ x_{n+1} = {{x_n+x_{n-1}+x_{n-2}x_{n-3}}\over{x_nx_{n-1}+x_{n-2}+x_{n-3}}}. $$ The note displays the following results (with proofs): 1. Every positive solution that eventually equals 1 has one of the following forms {\parindent7mm \item{(a)} $\{a,1,c;1,1,1,\dots\}$ \item{(b)} $\{1,b,1,d;1,1,\dots\}$ \item{(c)} $\{a,1,1,d;1,1,\dots\}$ \item{(d)} $\{a,b,1,1,(2+ab)/(1+a+b);1,1,\dots\}$, $a\neq 1\neq b$ \item{(e)} $\{1,b,c,1,1,(2+bc)/(1+b+c);1,1,1,\dots\}$, $b\neq 1\neq c$ \par} 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.

39A20Generalized difference equations
Full Text: DOI
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