Are Banach's Fixed Point Theorem, Arzelà-Ascoli theorem, posets, and Zorn's lemma actually connected to differential equations, and how if yes?
2026-02-23 17:42:55.1771868575
How is the Arzelà-Ascoli theorem connected to differential equations?
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Banach Fixed Point Theorem: Used to prove existence and uniqueness by applying it to the integral operator (which has been restricted to a contraction on a strip using the Lipschitz condition). This is called the Picard-Lindelof existence and uniqueness theorem.
Arzela-Ascoli: Used to prove just existence, by passing to a uniformly convergent subsequence in order to commute a limit and and integral. This is called the Cauchy-Peano existence theorem.
I do not know how you use the other two, they were not mentioned when I took the course.