Limit periodic iteration (Q1113397)

Let X be a complete metric space. For \(n=1,2,3,...\), let \(f_ n\) be a function \(X\to X\) having a unique fixed point \(\alpha_ n\). Write \(F_ n=f_ 1\circ f_ 2\circ...\circ f_ n\). Under suitable conditions, there is an \(\alpha\in X\) such that, for each \(x\in X\), we have \(F_ n(x)\to \alpha\) as \(n\to \alpha\). Thus, for example, it is proved that a sufficient set of conditions for this to hold is that \(\alpha_ n=\alpha\) (all n), and that, for all n and all \(x_ 1,x_ 2\in X\) we have \(d(f_ n(x_ 1),f_ n(x_ 2))\leq K_ nd(x_ 1,x_ 2),\) where \(K_ n<1\) and where (1) \(\prod^{\infty}_{n=1}K_ n=0\). If we omit the assumption that \(\alpha_ n=\alpha\), the conclusion still holds if we replace (1) be the stronger assumption that \(K_ n\leq K<1\). Other sufficient sets of conditions are also given. There are also a number of illustrative examples.