Existence of monotone solutions of some difference equations with unstable equilibrium (Q1851317)

Let \(I\) be an open interval, \(f(u,v)\) a continuous function on \(I^2\) with a unique equilibrium \(\overline x\in I\), let \(f(u,c)\) be decreasing and \(f(c,v)\) increasing on \(I\) for every fixed \(c\in I\). Then \((f(x, x)- x)(x-\overline x)< 0\) for some \(x\in I\setminus\{\overline x\}\) is necessary and sufficient for the existence of a strictly monotone solution of the difference equation \(x_{n+1}= f(x_n, x_{n-1})\) converging to \(\overline x\).