Picard Iterations: Method of Successive Approximations

\textrm{For initial value problems } y' = f(t,y), \, y(t_0) = y_0 \textrm{ where the fundamental theorem } \\ \textrm{guarantees a unique solution, the method of successive approximation to the solution via Picard Iterations } \\ \phi_0(t) = y_0, \, \phi_{n+1} = y_0 + \int_{t_0}^t f(s, \phi_n(s)) \, ds, \textrm{ for } n = 0, 1, 2, \dots \\ \textrm{yields a sequence of functions } \phi_n(t) \textrm{ which converges, at least in the vicinity of } t = t_0, \textrm{ to the true solution.}