Suppose we want to solve the system of first order ODEs,

Here ** Y** and ** A** are n-vectors and
** F** is a nonlinear vector-valued
function on :

To insure existence and uniqueness of
a solution to (10) and hence establish
effective numerical procedures, we require that and
be continuous in the box
, where
and are positive numbers and is the vector
Euclidean norm. If for all in and if **h** is the smaller of
and , then the IVP (10) has a unique solution for
. Weaker conditions do exist, but these will suffice
for our purposes. For a more detailed discussion of existence and uniqueness
issues, consult any good text on differential equations, such as
[1,2,3].