Let the following Cauchy Problem be $\displaystyle\cases{ y'(t)=f(t,y(t)) & \cr y(0)=\eta }$ for $t\in[0,T]$
Define the approximation $y_n$ of $y(t_n)$ as:
$y_0=\eta$,$\qquad$$y_1=y_0+hf(0,y_0)$,$\qquad$$y_{n+1}=y_{n-1}+hf(t_n,y_n)$, $\quad$$h=\frac{T}n$
Find an upper bound for:
$\epsilon_0=y(t_1)-y(0)-hf(0,y(0))$,$\qquad$$\epsilon_{n}=y(t_{n+1})-y(t_{n-1})-2hf(t_n,y(t_n))$ with respect to the functions:
$w_1(y;h)=\max\limits_{|t-s|\le h}|y(t)-y(s)|$$\qquad$$w_2=\max\limits_{|t-s|\le h}|y'(t)-y'(s)|$
Actually the approximation is irrelevant here, but maybe I oversee something, the problem is that I didn't use $w_1$ in both cases:
$\displaystyle\epsilon_0=y(t_1)-y(0)-hf(0,y(0))=\int_0^{t_1}y'(s)-y'(0)\le\int_0^{t_1}|y'(s)-y'(0)|\le hw_2(y';h)$
$\displaystyle\epsilon_{n}=y(t_{n+1})-y(t_{n-1})-2hf(t_n,y(t_n))=\int_{t_{n-1}}^{t_{n+1}}y'(s)-y'(t_n)\le2hw_2(y';h)$
Do you verify my steps ?
Can one get closer upper bound involving $w_1$ ? Thanks in advance.