What are the most important theorems in fixed point theory and why are they so important?
I know some: Banach's contraction principle, Brouwers fixed point theorem, caristi fixed point theore...
I also know that Banach's contraction principle has a crucial role in existence and uniqueness of the solution of the Cauchy problem in ode-s.
You should add to your list the Kakutani's fixed point theorem. It's application in Game Theory by John F. Nash, has provided the single most prominent result in the contemporary Game Theory, that is the existance of an equilibrium - which is known as the Nash Equilibrium - in every finite game.
The importance can be seen, that by applying the fixed point theorem, Nash's original paper, where he initiated the concept of the Nash Equilibrium and which is (again) the single most cited paper in Game Theory, has a length of less than a page.
