There is a theory of mild solutions $u \in C^0(0,T;X)$ where $X$ is a Banach space for equations of the form
$$\frac{du}{dt} + Au = f$$
where $A$ is an accretive nonlinear operator under some additional conditions ($(I+\epsilon A)$ is one-to-one onto a subspace of the range and the inverse of $(I+\epsilon A)$ is a contraction).


This theory uses discretization of the term $\frac{du}{dt}$ and the Crandall-Liggett theorem. The above conditions on $A$ give convergence of the discretization. 


My questions:


1. Is there an existence proof for such a theorem where time discretization is not used?
2. Is there any some similar such existence result for equations
$$F(u) + Au = f$$
where $F$ is a some operator that is not $\frac{d}{dt}$? I know this is a vague question but if I can't "put $F$ into $A$" then is there a way to solve this?
3. Suppose that $A=A(t)$ depends on time. I have found literature (which is quite old now) that really requires the domain of $A(t)$ to be independent of $t$: $D(A(t)) = D(A(0))$ for the results to carry through. I ask if I am missing some recent work in which the conclusions of the Crandall-Liggett are valid with minor adjustments to the premises of the theorem for time-dependent operators $A(t)$ with domains not time-independent?


My first two questions stem from the fact that I don't have $\frac{d}{dt}$ but a different operator there. I don't understand fully the necessity of $\frac{d}{dt}$...



Existence for ODE in Banach space (accretive operators and Crandall-Liggett)
I do not think it is good to ask three questions in one, since I will only address the first now. 


I would put the question back and ask you: how do you define a "solution"? Usually you have an existence proof of some object and if they are good enough, call them generalized solutions. If everything is perfect, you can try to prove some nicer properties to get back "classical" solutions, but in many instances you really need generalized solutions (viscosity solutions, shocks, etc.).


In the Crandall-Liggett theory generalized solutions are *defined* as functions which can be approximated by discretizations. Hence I do not know of and would be surprised if there would be any proof which do not use discretizations at some point.


This may work, however, for something different than the time derivative, if it retains some structural properties.


