In all the references I checked the standard initial value problem for an ODE is stated as: \begin{equation} \begin{cases} y'=F(y,t)\\ y(t_0)=y_0 \end{cases} \end{equation} for some $F:\mathbb{R}^{n+1}\to\mathbb{R}^n$, and under hypotheses on $F$ one can conclude existence, uniqueness, etc.
My question is: is there a standard way to address the problem: \begin{equation} \begin{cases} y'=F(y,t)\\ y_i(t_i)=y_{0,i}, \quad i=1,\ldots, n \end{cases} \end{equation} where the "initial condition" is not given for the same $t_0$ for all the components of $y$, but at different $t_is$? where by address I mean, to begin with, check existence and uniqueness conditions, e.g. do some version of Peano, Osgood and Caratheodory theorems hold? any reference would be appreciated. I asked the same question here https://math.stackexchange.com/questions/3485504/system-of-initial-value-problems-with-initial-conditions-at-different-t-0 and now I am trying MO.
My thoughts:
For example, using the integral formulation of the problem, in the case $n=2$ for simplicity, we have:
\begin{align} y_1(t)&=y_1(t_{01})+\int_{t_{01}}^t F_1(y_1(s),y_2(s),s)d s\\ y_2(t)&=y_2(t_{02})+\int_{t_{02}}^t F_2(y_1(s),y_2(s),s)d s \end{align}
Now if $F_1$ and $F_2$ are Lipschitz with constant $L<1/(2|t_{01}-t_{02}|)$ on the interval $I=[t_{01}-t_{02}]$ then on $I$ there exist a unique solution via the standard argument by the contraction mapping theorem. In fact define a norm on $C^1(I, \mathbb{R}^2)$ by $\lVert y-z\rVert=\max_t{|y_1(t)-z_1(t)|}+\max_t{|y_2(t)-z_2(t)|}$. Then, if $T$ is the integral operator associated with the IVP: \begin{equation} \lVert Ty-Tz\rVert\le\max_t\int_{t01}^t|F_1(y(s))-F_1(z(s)|d s+\max_t{\int_{t02}^t|F_2(y(s))-F_2(z(s)|d s} \end{equation} \begin{equation} \le2\max_t L\int_I(|y_1(s))-z_1(s)|+|y_2(s))-z_2(s)|)d s \end{equation} if $t \in [t_{01},t_{02}]$
Is this the standard way to proceed? Is there any more general/better way?
