Sufficient Conditions for Wellposedness
We shall now give a sufficient condition for wellposedness of an IVP and this is given in the following theorem:
Theorem: If satisfies a Lipschitz condition on
then is well-posed on with respect to all initial data.
Proof: We can show this by considering the perturbed problem
where and are the small perturbations. Let be the norm defined as .
Let . Here is the changed solution and is the true solution. Now subtracting (1.7) from (1.10), we get
and .
satisfies a Lipschitz condition on 
is well-posed on 
with respect to all initial data. 
and 
are the small perturbations. Let 
be the norm 
defined as 
. 
. Here 
is the changed solution and 
is the true solution. Now subtracting (1.7) from (1.10), we get 
.
