$\qquad$First of all, this question for O.D.E. comes from an end-of-book exercise with no answer.
$\qquad$Secondly, allow me to give the definitions of the relevant contents in the question to avoid ambiguity. You can also skip or skim this section until you get to the dividing line if you are familiar with it.
- Lipschitz condition
$\qquad$Say the function $f(x,y)$ satisfies the Lipschitz condition for $y$ on the region $G$, if there is a constant $L$, such that for any $(x,y_1),(x,y_2)\in G$, there goes $|f(x,y_1)-f(x,y_2)|\le L|y_1-y_2|$. Say the constant $L$ Lipschitz constant.
- Cauchy problem
$\qquad$The initial value problem $$\frac{\mathrm{d} y}{\mathrm{d} x} =f(x,y),\qquad y(x_0)=y_0,$$$\qquad$where the function $f(x,y)$ is continuous on the rectangle closed region $D:|x-x_0|\le a,|y-y_0|\le b$.
- Picard's existence and uniqueness theorem
$\qquad$Call it Picard theorem for short. Assuming that the function $f(x,y)$ is continuous on the closed region $D$ and satisfies the Lipschitz condition for $y$, then the solution of the Cauchy problem exists and is unique on the interval $|x-x_0|\le h$, where $$h=\min{\left\{ a,\frac{b}{M}\right\} },\qquad M=\max_{(x,y)\in D}{|f(x,y)|}.$$
- Picard sequence
$$y_n(x)=y_0+\int_{x_0}^{x}f\left(s,y_{n-1}(s)\right)\mathrm{d}s,\qquad |x-x_0|\le h,$$where $h$ is defined the same as in Picard theorem.
- Tonelli sequence
Note: Only the right side of the initial condition is discussed.
$\qquad$On the interval $I=[x_0,x_0+h]$, where $h$ is defined the same as in Picard theorem, construct the sequence $\{y_n(x)\}$ as follows: For each positive integer $n$, divide the interval $I$ into $n$ parts, taking the points as $x_k=x_0+kd_n$, where $d_n=\frac{h}{n},k=1,2,...,n$, and then defining $$\begin{matrix} y_n(x)=\begin{cases} y_0, & x\in\left[x_0,x_1\right], \\ y_0+\displaystyle \int_{x_0}^{x-d_n}f\left(s,y_n(s)\right)\mathrm{d}s, & x\in\left(x_1,x_n\right], \end{cases} & n=1,2,\cdots \end{matrix}.$$ We call the sequence $\{y_n(x)\}$ the Tonelli sequence.
$\qquad$In preparation for the exchange of integrals and limits in the later steps, one step in the proof of Picard theorem is to prove the uniform convergence of Picard sequences, using the difference $$|y_n(x)-y_{n-1}(x)|\le L\left|\int_{x_0}^{x}|y_{n-1}(s)-y_{n-2}(s)|\mathrm{d}s\right|,$$ treating the function column as a series of function terms, and applying the Weierstrass M discriminance.
$\qquad$In fact, the core of my problem is how to prove that the Tonelli sequence is uniformly convergent under the conditions of Picard theorem, which is that $f$ continuous on the closed region $D$ and has Lipschitz condition on $y$.
$\qquad$I had a hard time doing the similar thing on the Tonelli sequence. When I take the difference and use the Lipschitz condition, I don't get a recursive inequality, but I get an inequality involving integration between B and B itself.
$$\begin{align}|y_n(x)-y_{n-1}(x)| & = \left | \int_{x_0}^{x-d_n}f\left ( s,y_n(s) \right )\mathrm{d}s - \int_{x_0}^{x-d_{n-1}}f\left ( s,y_{n-1}(s) \right )\mathrm{d}s \right | \\&=\left | \int_{x_0}^{x-d_{n-1}}\Big ( f\big ( s,y_n(s) \big )-f\big ( s,y_{n-1}(s) \big ) \Big )\mathrm{d}s + \int_{x-d_{n-1}}^{x-d_{n}}f\left ( s,y_n(s) \right )\mathrm{d}s \right | \\&\le \left |\int_{x_0}^{x-d_{n-1}}L\left|y_n(x)-y_{n-1}(x)\right |\mathrm{d}s\right |+M|d_{n-1}-d_n|\\&= L \left |\int_{x_0}^{x-d_{n-1}}\left|y_n(x)-y_{n-1}(x)\right |\mathrm{d}s\right |+\frac{Mh}{n(n-1)}\end{align}$$
$\qquad$In addition, I know the Arzela-Ascoli theorem, and I also learned that Tonelli sequence have uniform boundedness and equicontinuity, and it seems that equicontinuity plus point-by-point convergence can lead to uniform convergence (sorry but idk if it is right), but in the process I still face problems like the above. It seems that any attempt to apply the Lipschitz condition would create such problems making the proof could not continue.
$\qquad$I don't know if I'm thinking in the right direction. If you know how to prove it or anything related to it, please discuss below, tks!
