Let $f:R\times R^n \rightarrow R^n$ be a vector field such that the equation
$$ \dot{x} = f(t,x), \ \ x(t_0) = x_0, $$
has a unique solution $x(t)$ on the interval $[t_0,t_1]$. Note that we can rewrite the solution as the Volterra integral
$$ x(t) = x_0 + \int_{t_0}^tf(s,x(s))\ ds, $$
and can (sometimes) seek a solution by studying the limit of the sequence $$ x^{(n+1)}(t) = x_0 + \int_0^tf(s,x^n(s))\ ds, \ \ x^0(t) = x_0. $$ Now suppose we define an operator $P$ on $C[t_0,t_1]$ by $P(\cdot) = x_0 + \int_0^tf(s,\cdot)\ ds$. The sequence above can then be considered to be a dynamical system on $C[t_0,t_1]$. The fixed points of this system correspond to solutions of the original nonlinear differential equation and in some cases, such as when $P$ turns out to be a contraction, we can show that the sequence above converges to the fixed point.
To what extent has this sort of approach been investigated in the literature?