I need a reference for: Existence and Uniqueness of a general ODEs with a linear operator I'm looking for a reference of a theorem that establishes the existence and uniqueness of the following general ODE:
Let $Q_n$ is a finite dimentional Hilbert space and let the operator $A:Q_n\to Q_n$ is a self adjoint and a positive definite linear operator. Let $f\in L^2(0,T;Q_n)$ and $g\in Q_n$  We search $u(t)\in Q_n$ such that for any $t\in [0,T]$, 
$\dfrac{du(t)}{dt}+Au(t)=f(t),$
$u(0)=g.$
Note that $u\in L^2(0,T;Q_n)$
 A: Assuming $f(t)$ is continuous, the existence and uniqueness of 
$\dfrac{du}{dt} = Au + f(t), \tag{1}$
with
$u(0) = g, \tag{2}$
follows immediately as an application of the general theorems on existence and uniqueness for ODEs, which may be found, among other places, in Ordinary Differential Equations, by Jack K. Hale, published by Dover Press in 2009.  See Chapters 0 and I for a thorough discussion, especially Theorem 3.1 of section I.3.  Many other texts and treatises on ODEs also cover this material.
In the particular case at hand, in accord with the general theory (which may be found in Hale's book), existence and uniqueness follow from the Lipschitz continuity of the right-hand side in the variable $u$ and its continuity in $t$ which, since $Au$ is a linear map and $f(t)$ is continuous, are easy to establish.  Indeed for any linear map such as $A$, $\Vert Au_1 - Au_2 \Vert  \le \Vert A \Vert \Vert u_1 - u_2 \Vert$, which establishes the Lipschitz continuity of $u \to Au$.
Hope this helps.  Cheerio,
and as always,
Fiat Lux!!!
A: The solution of your initial value problem is
$$
u(t)=\mathrm{e}^{-tA}g+\int_0^t\mathrm{e}^{-(t-s)A}f(s)\,ds.
$$
All graduate textbooks on ODEs study this extensively.
See for example:


*

*Coddington & Levinson,

*Hartmann.
