Feb 12, Lect - 16
Banach Fixed Point Theorem
Suppose is a complete metric space and is a contraction. Then has a unique fixed point .
Corollary
Let be such that is a contraction for some . Then has a unique fixed point.
Existence and Uniqueness of IVP by Fixed Point Theorem
Under the same hypothesis of Picard’s theorem, the IVP has a unique solution.
Proof
Let .
is a closed ball in a Banach space, with supremum norm,
Hence, is a complete metric space.
Define by
For , , we have . It is clear that fixed points of are solutions of the IVP.
Claim The solution is unique.
Let and consider,
Thereby,
If we choose large enough so that,
is a contraction.
Hence by Banach Fixed Point Theorem, has a unique fixed point. Hence, the IVP has a unique solution.
Theorem
Let,
(i)
(ii)
be two IVPs for with the solutions and .
Let and for all in .
Then, given any , such that
whenever and .
Proof
Let , and we have and .
Given that and .
Then by Picard’s theorem, and such that
Since , we have
and .
Therefore,
By Gronwall’s inequality,
Feb 16, Lect - 15
Generalized Gronwall’s Inequality
If , , and are non-negative continuous functions defined for , then the inequality
Proof
Given that,
when
Since is non-negative, from (1),
Using the Integrating Factor (IF), , we have
From (1), we have