Notions of derivative and integral in Banach spaces in Brezis' book

Question

In chapter 7, page 184 of the book Functional analysis by Haim Brezis, the author gave a theorem of Cauchy, Lipchizt and Picard as follows:

Theorem 7.3 (Cauchy, Lipchitz, Picard). Let $E$ be a Banach space and let $F: E\rightarrow E$ be a Lipchitz map, i.e, there is a constant L such that $ \|Fu - Fv\| \leq L \| u -v \|, \forall u,v \in E.$ Then given any $u_0 \in E$, there exists a unique solution $u \in C^1([0,\infty), E)$ of the problem

\begin{equation} \begin{cases} \frac{du}{dt}(t) = Fu(t) \quad on \ [0,\infty), \\ u(0) = u_0. \end{cases} \end{equation}

However, he did not give any definition of what is the derivative $\dot u(t) := \frac{du}{dt}(t)$ in the Banach space. Then, in the proof, he says that this is equivalent to the integral equation

$$ u(t) = u_0 + \int^t_0 F(u(s))ds. $$

I am fine with all the reasoning he gives in the proof but I don't know how to interpret the derivative and the integral above. What I have learned in some references is that the derivative of a path can be defined as

$$ \dot u(t) := \lim_{h \rightarrow 0}\frac{u(t+h)-u(t)}{h}, $$

provided the limit exists.

Question 1: Is this the correct spirit (the definition of derivative above) in this theorem?

For the integral $\int^t_0 F(u(s))ds$, a definition that I have leaned is Bochner integral. In this definition, we have to have a measure space (which in our case is $\mathbb{R}$) to construct integral of "simple functions" (taking finite values in Banach space) and then use completeness of $E$ to define the general integral. I would like to understand the sense of integral so my next question is

Question 2: Does Brezis implicitly mean it is in the sense of Bochner integral in this theorem? Or is it a different kind of integrals? Or perhaps do other kinds of integrals (which I don't know many) coincide in Banach spaces?

Question 3: When we differentiate the integral in the sense of the derivative above, does it give $F(u(t)$? (So the two operations are inverse of each other)

I'm amateur I don't know much about these "general" integrals in Banach spaces. I would hope to get a detailed explanation (and some references to read) if possible :(...

Thank you for your help, I hope I have put the context well enough :D.

Answer 1

1. Yes this is the correct spirit, you will encounter in these books many notation for the derivative, but for $U \in C^1 (I,E)$, you have $$u'(t) = \frac{du}{dt} (t) = \dot{u}(t)$$ It only depends of the frame, for very abstract theorems it is often $u'$, when you want to specify the variable (other can be fixed) you use $\frac{d}{dt}$ and for physics and EDO, $\dot{u}$. But it also depends from the author...

For the integral, each one will, do the main interest is the fundamental analysis theorem that states if $f$ is continuous, there is an unique $F$ such that $F' = f$ and $F(0) = 0$ and it is $\int_0^x f(t) dt$. Just take Riemann's it is sufficient here since you function is very regular (they are all the same!).

2. I think it is not relevant to give more explanation.
3. Yes obviously with any integral you chose.

The Banach space condition is relevant when you study the proof, for example there is one with a fixed point theorem (Picard I think), the real version though is with $F(x,t)$ Lipschitz and $u'(t) = F(u(t),t)$.