I'm revising for my upcoming test, and this problem dated back some years ago. I've been working on this problem for almost a day, but I don't even know how to start it correctly.
Problem
Given the open set $U$ in $\mathbb{R}^p$, $A$ closed and bounded in $U$; $V$ is an open set in $\mathbb{R}^k$; $B$ is closed, bounded, Jordan measurable set contained in $V$; and a continuous function $f:U \times V \times \mathbb{R}^n \times \mathbb{R}^{nk} \to \mathbb{R}$.
Let:
$E = C^1(B; \mathbb{R}^n)$, i.e the space of continuous functions from $B$ to $\mathbb{R}^n$ that have the first derivative continuous. We define the norm $||x||_E = \max\limits_{s \in B} \left\{ |x(s)|_2 + ||x'(s)|| \right\}$.
$F = C(A; \color{red}{\mathbb{R}^n})$, i.e the space of continuous functions from $A$ to $\mathbb{R}^n$, with the norm $||y||_F = \max\limits_{t \in A} \{ |y(t)|_2 \}$
Where $|\bullet|_2$ is the normal Euclidean norm.
Define $T: E \to F$ as follow: $$T(x)(t) = \int\limits_B f(t;s;x(s);x'(s)) ds$$
Question
Prove that $T$ is compact. $\color{green}{\mathbf{(done) :)}}$
Assume that all partial derivative of $f$ wrt every variable is continuous. Prove that:
- For all $x \in E$, $T(x)$ has continuous partial derivatives. $\color{green}{\mathbf{(this \ is \ also \ done) :)}}$
- For all $x \in E$, prove that $T$ is differentiable at $x$; and that, for every $h \in E$; $t \in A$, we'll have:
$$T'(x)(h)(t) = \int\limits_B [ \left< f_x(t;s;x(s);x'(s);h(s)) \right> + \left< f_{x'}(t;s;x(s);x'(s);h'(s)) \right>] ds$$
Where $$\left< f_x(t;s;x(s);x'(s));h(s) \right> = \sum\limits_{i = 1}^{n} \frac{\partial f}{\partial x_i}(t;s;x(s);x'(s)) h_i(s)$$
My difficulties
Should the $\color{red}{\mathbb{R}^n}$ at the beginning of the problem be changed to $\mathbb{R}$? Since I don't think that integral would return something in $\mathbb{R}^n$ at all. Or, am I missing something here?
I don't really know where that monstrous $T'(x)(h)(t)$ comes from. I would be glad if someone can give me a small push. I know that we're supposed to use the uniform continuous property of $f$ on some compact set; and the mean value theorem, but I haven't come across any functions that have $x'(s)$ as its perimeter.
Thanks guys a lot,
And have a good day,
P.S: I do have a small easy example on this kind of problem from our professor. So if you guys think that it's needed, please tell me, and I'll append it to the end of this post.