In The implicit function theorem written by Krantz & Parks, it's said that the implicit function theorem implies the following existence theorem of ODE:
Theorem 4.1.1 If $F(t,x)$, $(t,x)\in\mathbb R\times\mathbb R^N$, is continuous in the $(N+1)$-dimensional region $(t_0-a,t_0+a)\times B(x_0,r)$, then there exists a solution $x(t)$ of $$\frac{dx}{dt}=F(t,x),\qquad x(t_0)=x_0$$ defined over an interval $(t_0-h,t_0+h)$.
It's Peano existence theorem, I think. However, there seems a gap in their proof. WLOG, suppose $t_0=0$. They constructed $\mathcal H\colon[0,1]\times\mathcal B_1\to\mathcal B_0\times\mathbb R$, where $\mathcal B_0$ is the space of bounded continuous $\mathbb R^N$-valued functions on $(-a,a)$ normed canonically, and $\mathcal B_1$ is the space of bounded continuously differentiable $\mathbb R^N$-valued functions on $(-a,a)$ that also have a bounded derivative, normed canonically by $\sup\lvert f\rvert+\sup\lvert\dot f\rvert$, as follows: $$\mathcal H[\alpha,X(\tau)]=[X'(\tau)-\alpha F(\alpha\tau,X(\tau)),X(0)-x_0]$$. Note that $\mathcal H[0,x_0]=[0,0]$, where $x_0$ on the left side denotes the constant function. Then they claim that the existence theorem follows from the implicit function theorem. However, under the only condition that $F$ is continuous, there's no evidence that $\mathcal H$ is partially differentiable with respect to $X$ for $\alpha\in(0,x_0)$.
Can we fix the preceding proof in some extent?
PS: I posted the question not only because I want to comprehend such a proof, but also want to understand the relation between ODE and implicit functions. It seems certain that such a proof cannot be alright, since the canonical implicit function theorem is also a uniqueness theorem, which implies the local uniqueness of a solution of ODE. However, I want to know how to fix it. I doubt it might rely on a more general implicit function theorem.