existence and uniqueness of solution to Cauchy problem
be a Cauchy problem, where is
Lipschitz continuous with respect to the first variables (i.e. with respect to ).
Then there exists a unique solution of the Cauchy problem, defined in a neighborhood of .
Solving the Cauchy problem is equivalent to solving the following integral equation
Let be the set of continuous functions . We’ll assume to be chosen such that the 11 denotes the closed ball . In this ball, therefore, is Lipschitz continuous with respect to the first variable, in other words, there exists a real number such that
for all points sufficiently near to .
Now let’s define the mapping as follows
We make the following observations about .
The Lipschitz continuity of yields
If we choose these conditions ensure that
, i.e. doesn’t send us outside of .
In particular, the second point allows us to apply Banach’s theorem and define
and which therefore locally solves the Cauchy problem.
|Title||existence and uniqueness of solution to Cauchy problem|
|Date of creation||2013-03-22 16:54:45|
|Last modified on||2013-03-22 16:54:45|
|Last modified by||ehremo (15714)|