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Course : Mathematical Economics
Course code : OIK231
OIK231 - STYLIANOS ARVANITIS
Synopsis: Lecture 12 (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 have started our preparations for the examination of applications involving the establishement of the existence and uniqueness of functional equations.
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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