Lectures 15-16 (include the 3nd complementary lecture-Ac. Year 2023-24)
Saturday, May 18, 2024 at 5:28 PM
- written by user ΑΡΒΑΝΙΤΗΣ ΣΤΥΛΙΑΝΟΣAfter a brief remark on the characterization of Lipschitz continuity for functions between Euclidean spaces (see here for the develpment of the full details), we proceeded with the introduction of some general aspects of fixed point theory, and the examination of the proof and corollaries of the Banach Fixed Point Theorem. We have started our preparations for the examination of applications involving the establishement of the existence and uniqueness of functional equations, by among others proving Blackwell's Lemma. We concluded the course with an applications of the BFPT; it involves the uniqueness of solution to the Bellman equation.
Notes for the above can be found here and here. The whiteboards from analogous 2020-21 lectures (please keep in mind that those are not necessarily identical to the current lectures but they contain some common elements) can be found here and here.
Exercise: Using the BFPT and the result in Optional Exercise 11, show that if the underlying metric space is totally bounded and complete, and the f function is contractive, then its m-fold self-composition converges uniformly (w.r.t. which metric?) to the function that is constant at the unique fixed point of f.
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