On the recursive sequence \(x_{n+1}=x_{n-1}/g(x_n)\) (Q1860928)

Several properties of the solutions of the recurrence relation \[ x_{n+1}= {x_{n-1} \over g(x_n)},\;n=1,2,\dots, \] subject to the initial conditions \(x_{-1} >0\) and \(x_0>0\) are obtained, where \(g\in C^1(R_+)\), \(g(0)=1\) and \(g'(x)>0\) for \(x\in R_+\). Then by continuity arguments, it is shown that for any \(u\in(0, \infty)\), there is a solution \(\{x_n\}\) satisfying \(x_{-1}=u\) and \(x_0g(x_0) >u\) such that \(x_0>x_1>x_2> \dots\) and \(\lim_{n\to \infty}x_n=0\).