Picard iteration is the fixed point iteration
 |
(1)
|
also called the method of successive approximations. In fixed point theory, the name refers generally to repeated application of a map 
in order to approach one of its fixed
points (Berinde 2007).
For the initial value problem consisting of the ordinary differential equation 
and 
, Picard iteration refers more specifically to first
writing the problem as the equivalent integral equation
 |
(2)
|
and then defining the successive approximations
 |
(3)
|
Under the continuity and local Lipschitz conditions in Picard's existence theorem, the sequence of functions converges on a sufficiently small interval to the unique solution. The two usages are instances of the same construction: the integral on the right defines a map on a function space, and the ordinary differential equation iteration is its fixed point iteration (Coddington and Levinson 1955).
