Can Peano's existence theorem be proved by the implicit function theorem?

Question

In The implicit function theorem:history,theory and applications written by Krantz & Parks, it's said that the implicit function theorem can prove the following existence theorem of ODE:

Theorem 4.1.1 If $F(t,x), (t,x) \in R×R^N$, is continuous in the $(N+1)$-dimensional region $(t_0−a,t_0+a)×B(x_0,r)$, then there exists a solution $x(t)$ of $$\frac{\mathrm{d} x}{\mathrm{d} t} =F(t,x),x(t_0)=x_0$$ defined over an interval $(t_0−h,t_0+h)$.

which is also called Peano existence theorem. The proof in the book uses this version of the implicit function theorem：

Theorem 3.4.10 Let $X,Y,Z$ be Banach spaces. Let $U \times V$ be an open subset of $X \times Y$. Suppose that $G:U \times V \to Z$ is continuous and has the property that $d_2 G$ exists and is continuous at each point of $U \times V$.
Assume that the point $(x, y) \in X \times Y$ has the property that $G(x,y)=0$ and that $d_2G(x,y)$ is invertible.
Then there are open balls $M=B_X(x, r)$ and $N=B_Y(y,s)$ such that,for each $\zeta \in M$, there is a unique $\eta \in N$ satisfying $G(\zeta,\eta)=0$. The function $f$, thereby uniquely defined near $x$ by the condition $f(\zeta)=\eta$, is continuous.

The proof process in the book is as follows： The proof process in the book

For the proof given in the book, I am puzzled: if the implicit function given by Theorem 3.4.10 is unique, why is the function $x(t)$ finally obtained in this proof not necessarily unique? My questions are as follows:

1. Is the proof given in the book correct?
2. If the proof in the book is wrong, is there a method to prove Peano's existence theorem by using the implicit function theorem?

I have this doubt because in some papers the authors claim that some version of the implicit function theorem can be used to prove Peano's existence theorem, but I looked at their references and found nothing. The most relevant literature I can find on the Internet is this book. I also didn't find convincing results in the Q&A on MSE.

Related link:

Does the implicit function theorem imply Peano existence theorem

Using the Peano existence theorem (ODE's) to imply the implicit function theorem

About the second link: In Hale's Ordinary Differential Equations (1980 version) that I can find, Peano's existence theorem is proved by Schauder fixed point theorem instead of the implicit function theorem.

Answer 1

The crux of the issue lies in the following statements on Page 65:

...that for all small enough choices of $\alpha$, there exists an $X(\alpha,\tau)$ such that: $$D_{\tau} X(\alpha, \tau) - \alpha F(\alpha \tau, X(\alpha,\tau)) = 0$$ For such an $\alpha > 0$, we define $x(t)$ by setting: $$x(t) = X(\alpha,t/\alpha)$$

The point here is that the function $x$ may depend on the choice of $\alpha$. For different choices of $\alpha$, we will find different functions. This is why we don't get uniqueness of the solution from this proof.

Now, as far as I can see, the argument is correct. You are also very right in pointing out that the Peano Existence Theorem can be proved by a fixed point theorems. A number of existence/uniqueness theorems are a consequence of fixed point principles/contraction principles. There's also a way to prove Peano's Theorem using the Arzela-Ascoli Theorem; you can easily find references of this online.