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Course : Mathematical Economics
Course code : OIK231
OIK231 - STYLIANOS ARVANITIS
Synopses: Lectures 12-13 (Ac. Year 2025-26)
We proceeded with the formulation and derivation of the Banach Fixed Point Theorem and some corollaries involving localization of the fixed point via the restriction into closed and invariant subspaces, as well as regarding the approximation error of the fixed point by elements of the sequence of iterations. We examined an application involving the issue of solutions existence and uniqueness (and approximations) for the Bellman equation via the BFPT. Our preparation for the derivation of this main result involved among others the derivation of Blackwell's Lemma (or Contraction Mapping Theorem) that provides sufficient conditions in terms of monotonicity and discounting for contractivity.
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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