In Theorem 1.2 of Pazy's book called Semigroups of linear operators and applications to PDEs, a fixed point method is used to show that the nonlinear problem has a solution. The map $F:C([t_0,T];X) \to C([t_0, T];X)$ is defined by
$$(Fu)(t)=T(t-t_0)u_0 + \int_{t_0}^t T(t-s) f(s,u(s)) ds, t_0 \leq t\leq T.$$
Using the assumption that $$\|T(r)v\|_X \leq M\|v\|_X$$ for all $r \in [0,T]$ and the fact that $f$ is continuous in $t$ and uniformly Lipshitz continuous with constant $L$ on $X$, Pazy shows $$\|(Fu)(t)-(Fv)(t)\| \leq ML(t-t_0)\|u-v\|.$$ This part is understandable to me. He then says using this inequality and the definition of $F$ you can show using induction that $$||(F^nu)(t)-(F^n)v(t)||\leq \frac{(ML(t-t_0))^n}{n!}||u-v||.$$
To me this part is clear without the factorial, but I am not sure how we get the bound divided by $n!$. I'm assuming it has something to do with the properties of $T(t)$ since $T(t)$ is a $C_0$ semigroup, but I still don't see it. Does anyone have any ideas where this came from?
It is a standard kind of estimate used, for example, in the proof of Picard's existence theorem. In order to get the factorial, you have to use the integral equality:
$$\begin{aligned} \|(F^2u)(t)-(F^2v)(t)\|&\leq\int_{t_0}^t \|T(t-s) [f(s,(Fu)(s))-f(s,(Fv)(s))] \|\,ds\\ &\leq M \int_{t_0}^t \|f(s,(Fu)(s))-f(s,(Fv)(s)) \|\,ds\\ &\leq M L \int_{t_0}^t \|(Fu)(s)-(Fv)(s) \|\,ds\\ &\leq M L \int_{t_0}^t ML(s-t_0)\|u-v\|\,ds\\ &= M^2 L^2\|u-v\|\int_{t_0}^t (s-t_0)\,ds\\ &= M^2 L^2\|u-v\|\frac{(s-t_0)^2}{2}\Bigg|_{s=t_0}^{s=t}\\ &= \frac{(M L(t-t_0))^2}{2}\|u-v\|\\ \end{aligned}$$
Analogously, the general case is proved by induction.