In the paper "The Asymptotic Behavior of Firmly Nonexpansive Mappings" by Simeon Reich and Itai Shafrir the Theorem 1 claims the following:
Theorem1 Let $D$ be a subset of a Banach space $X$ and $T: D \rightarrow X$ a firmly nonexpansive mapping. If $T$ can be iterated at $x\in D$, then for all $k\geq 1$, $$\lim_{n\rightarrow\infty}|T^{n+1}x-T^nx|=\lim_{n\rightarrow\infty}|T^{n+1}x-T^nx|/k=\lim_{n\rightarrow\infty}|T^nx/n|$$
What does it mean in this case the condition that "If $T$ can be iterated"?
The domain of $T$ is $D$. To be able to apply $T$ on $T(x)$ for some $x\in D$ you need $T(x)\in D$. Therefore $T$ can be iterated at $x$ if and only if $T^kx\in D$ for all $k>0$.
If you have a self-mapping $T:D\to D$, then $T$ can be iterated at each $x\in D$.
Example:
Let us consider $f:(-1,\infty)\to\mathbb R$ with $f(x)=\ln(x+1)$. The map $f$ is not a self-mapping since $f(-1,\infty)=\mathbb R$.
We see that $f$ can be iterated for all $x\geq 0$ since $f[0,\infty)=[0,\infty)$.
But for $x\in(-1,0)$ you can show that there exists $k>0$ such that $f^k(x)<-1$, hence $f$ can't be iterated at $x\in(-1,0)$.