Suppose we have a mapping: $$ T: \operatorname{CDF} \rightarrow \mathbb{R}^{[0,\infty)}: F \mapsto F $$ which maps Cumulative Distribution Functions to functions on $[0,\infty)$. I am looking for a fixed point of this mapping. If I apply the normal fixed point iteration: $$ F_{n+1} = TF_n, $$ I have divergence and $F_n$ becomes some ridiculous function. However since I know we should have $F(\infty) = 1$ I scale $F_n$ by setting $$ \tilde{F}_n(s) = F_0 + (1-F_0)(F_n(s) - F_n(0))/(1-F_n(0)), $$ where $F_0$ is chosen such that $T\tilde{F}_n(\infty) = 1$. If we define: $$ H: \mathbb{R}^{[0,\infty)} \rightarrow \mathbb{R}^{[0,\infty)} $$ by setting $HF=\tilde{F}$ where $\tilde{F}$ is chosen as $\tilde{F}_n$ above we are actually considering the composition: $$ T\circ H: \operatorname{CDF} \rightarrow \operatorname{CDF} $$
This method does indeed work and yields (for my example) the fixed point of $T$. Is there some general theory to this type of operation, as it seems to me that this is a technique which can be applied in a farely general framework.