Consider the following autonomous differential equation:
$$y'=f(y)$$
where $y=y(t)$ with $y_0 = y(0)$.
Let us suppose that $y^{[0]}(t)$ is a good approximation for $y(t)$. Using this function in the right part of the differential equation, we obtain
$$y(t) \approx y_0 + \int\limits_{0}^t f(y^{[0]}(\tau)) {\rm d}\tau \equiv y^{[1]}(t)$$
That is, we arrived at the following iteration formula:
$$y^{[k+1]}(t) = y_0 + \int\limits_{0}^t f(y^{[k]}(\tau)) {\rm d}\tau$$
Should this kind of iteration converge to $y(t)$? If so, then is this an existing method to solve differential equations?