Taking a linear operator inside an integral I am currently reading up on the Newton iteration, and have come across a step in a proof that I don't understand, and am having difficulty finding in a numerical analysis textbook (as I don't know if it has a specific name).
I think I need an inequality that states
$A\int_{\Omega}B(x) \text{d}x \geq \int_{\Omega}AB(x) \text{d}x$
For $B$ a generic operator, and $A$ a linear operator on a convex domain $D$. Is anyone aware of a theorem like this?
If there isn't a general theorem of the above form I am including more detailed information about the specific instance with which I am having difficulties below.
I start with a nonlinear system
$F(x) = 0$
It is assumed that $F: D \rightarrow \mathbb{R}^{n}$ is continuously Frechet differentiable on convex domain $D$, and that the inverse of the derivative $F'(x)$ exists for $x \in D$.
The Newton iteration is written as
$x^{k+1} = x^{k} + \Delta x^{k}$
where $\Delta x^{k} = -F'(x^{k})^{-1}F(x^{k})$.
Now we take two Newton sequences $\{x^{k}\}$ and $\{y^{k}\}$ starting from $x^{0}$ and $y^{0}$. Using the above definitions it can be shown that
$x^{k+1} - y^{k+1} = x^{k} + \Delta x^{k} - y^{k} - \Delta y^{k} \\ = F'(x^{k})^{-1}\left( F'(x^{k})(x^{k} - y^{k}) - \int_{0}^{1}F'(y^{k} + t(x^{k}-y^{k}))(x^{k}-y^{k})\text{d}t \right) + \\ \quad \quad F'(x^{k})^{-1}(F'(y^{k}) - F'(x^{k}))\Delta{y}^{k}.$
From this point the next step in the book is to state that
$||x^{k+1} - y^{k+1}|| \leq \int_{t=0}^{1}||F'(x^{k})^{-1}\left( F'(x^{k}) - F'(y^{k} + t(x^{k} - y^{k})) \right) (x^{k} - y^{k}) \text{d}t|| + \\ \quad \quad \quad ||F'(x^{k})^{-1}(F'(y^{k}) - F'(x^{k}))\Delta y^{k}||$,
which is where I get lost. I get the use of the triangle inequality, and I know that $||\int_{\Omega}x|| \leq \int_{\Omega}||x||$, but I haven't come accross anything that would allow me to take the linear operator $F'(x^{k})^{-1}$ inside the integral sign, such that (as far as I can tell) I would need an inequality of the form
$F'(x^{k})^{-1}\int_{\Omega}G(x^{k})\text{d}x \geq \int_{\Omega}F'(x^{k})^{-1}G(x)\text{d}t$
for the previous inequality to hold. Is this correct? Can someone point me in the right direction?
Cheers,
Keeran

I think what you have to use is just the fact that integration is linear.
By looking at the $i$th coordinate you get
$\begin{align} \left(A \int f(x) dx\right)_i 
&= \sum_{j} A_{i,j} \left(\int f(x) dx\right)_j\\
&= \sum_{j} A_{i,j} \int f(x)_j dx\\ 
&= \int \sum_{j} A_{i,j}f(x)_j dx\\ 
&= \int (A f(x))_i dx \\
&= \left(\int A f(x) dx \right)_i
\end{align}$
and this gives you $A \int f(x) dx = \int A f(x) dx$.
