On the period five trichotomy of all positive solutions of \(x_{n+1}=(p+x_{n-2})/x_{n}\).
The authors study the asymptotic behavior of positive solutions of the difference equation in the title, where \(p\) is a positive real parameter. For any \(p>0\) there exists a unique positive equilibrium \(\bar{x}=(1+\sqrt{1+4p})/2\). When \(p=1\) there exists a cycle of period five. By exploiting some special properties of the equation the authors show that if \(0<p<1\) or \(p\geq2\) every positive bounded solution converges to \(\bar{x}\). When \(0 < p < 1\) the existence of unbounded solutions is proved. When \(p\geq2\) the positive equilibrium is globally asymptotically stable. The authors make a conjecture: if \(1<p<2\), then the positive equilibrium \(\bar{x}\) is globally asymptotically stable.
scientific article; zbMATH DE number 2056494