Compactness of a set of functions

Question

During lunch break, somebody submitted us this problem today:

Let $a$ and $b$ be real numbers and $F:\mathbb R\to\mathbb R$ a continuous function.

Let $K=\{u\in C^1([a,b],\mathbb R), u'=F\circ u\}$

Prove $K$ is compact (with respect to the sup norm)

Nobody in my class has solved it.

I've thought of setting an operator $T$such that $T:u\to u'-F\circ u$ and considering its fixed points, but $T$ is not linear.

Thanks for your help.

Answer 1

I give you a counterexample:

let $F(x)=x$. Then $$\{ u \in C^1([0,1], \mathbb{R} ) : u'=u\} \supset \{ \lambda \exp : \lambda \in \mathbb{R} \}$$ is not bounded, hence it is not compact.