Periodicity and strict oscillation for generalized Lyness equations
This paper is concerned with the following recurrence relation \[ x_{n+1}= {x_n\over (a+ bx_n)x_{n- 1}},\quad n=0,1,2,\dots;\quad a,b\geq 0;\quad a+b>0, \] under the initial conditions \(x_{-1}> 0\) and \(x_0> 0\). It is shown that every solution is 5-periodic if, and only if, \(b= a^2\), 6-periodic if \(b=0\), and 4-periodic if \(a=0\). It is also shown that every solution oscillates about the positive equilibrium point \(x^*\) defined by \((a+ bx^*) x^*= 1\). In addition, the recurrence relations \(x_{n+1}= \max\{x_n,1\}/x_{n-1}\) and \(x_{n+1}= \max\{x_n, 1\}/(x_n x_{n-1})\) under initial conditions \(x_{-1}\), \(x_0>0\) are shown to have only 5-periodic and 7-periodic solutions, respectively.
scientific article; zbMATH DE number 1496571