I am considering the following abstract Cauchy problem on Banach space $X$:
\begin{cases}
u'(t)=Au(t)+\big(f(t)+Bu(t)\big),&t\in[0,T],\\
u(0)=x\_0,
\end{cases}
Suppose $A$ generates an analyitc semi-group $S(t)$ on $X$, then under certain assumptions, we can study the wellposeness of the following integral equation for the original Cauchy problem:
\begin{equation}
u(t)=S(t)x\_0+\int\_0^t S(t-s)\big(f(s)+Bu(s)\big)\, ds.\tag{1}\label{eq:noper}
\end{equation}


However, we can also view $C=A+B$ as a perturbation of $A$, hence suppose $A+B$ also generates an analytic semi-group $T(t)$ on $X$, then we can also study
\begin{equation}
u(t)=T(t)x\_0+\int\_0^t T(t-s)f(s)\, ds.\tag{2}\label{eq:per}
\end{equation}


I hope to show the wellposedness of a mild solutions $u\in C([0,T];(D(A),\|\cdot\|\_A)$ to \eqref{eq:noper} and \eqref{eq:per} should be equivalent, in the sense that a solution to \eqref{eq:noper} should also be a solution to \eqref{eq:per}, also conversely. Here $\|\cdot\|\_A$ refers to the graph norm. I think it may be correct(at least with good assumptions for $x\_0$ and $B$), since intuitively, they come from two different ways to interpret the same Cauchy problem. 


For your information, the specific problem I am considering is $A$ be an elliptic 2nd order operator and $B$ be a first order operator. 


I have read Engel-Nagel' book, which presents the following form of a variation of parameters formula for perturbed semigroups, for some operator $B$,
$$T(t)x=S(t)x+\int\_0^t S(t-s)BT(s)x\,ds,$$
but I failed to use it to show the desired result. 


Is there any reference with rigorous arguments you can point out?



Equivalent ways to study a semilinear parabolic equation as a perturbed abstract Cauchy problem
You can derive that the mild solution $u$ as given in (2) satisfies (1) from the connection between the semigroups $S$ and $T$ as you cited from the Engel-Nagel book (as a minor remark, there is an $x$ missing right before the $ds$): 
Just insert the cited formula into (2), re-arrange and interchange order of integration. A function satisfying (1) must be unique (use Gronwall's Lemma), so the mild solution as given in (2) is the unique function satisfying (1). 


Please also recall that $B$ must be a bounded operator for the cited formula to be true!


