Feb 11, Lect - 15
Theorem
The approximation of of an IVP satisfies
Proof
We have
Cauchy-Peano Theorem
Let be continuous on the rectangle
Then a solution to IVP in the interval
where .
Proof
Let .
By the -approximation theorem, for each , an -approximation of the IVP such that
is uniformly bounded.
Moreover,
is equicontinuous on .
Then is uniformly bounded and equicontinuous.
Hence by Arzelà-Ascoli theorem, of which converges to uniformly.
Hence is continuous and,
Claim The limit is a solution of the IVP.
We define
By integrating from , we get
We know that uniformly. Then uniformly.
Again, as .
Hence for we get
is a solution of the IVP.