Multiagent Systems

Prof. Dr. Jürgen Dix · Department of Informatics, TUC Multiagent Systems, WS 06/07 217/731
3. Decision Making (1) 2. Non-coop Games
Let f : D → D be continuous. Let F : R n → D be the
extension of f to R n , for which we have
F (x) = f( x
||x|| ) on Rn \ D.
For k ∈ N, k ≠ 0, put f k (x) := ∫ R n k n φ(ky)F (x − y)dy.
Show that the restriction of f k to D maps D into D. Show
that the mappings F k are continuously differentiable and
approximate in the topology of uniform convergence the
mapping F . Show that if there exists a continuous
mapping f : D → D without fixed points, then there will
also exist a continuously differential mapping without
fixed points. It follows, that it suffices to proof the
Brouwer Fixpoint Theorem only for continuously
differentiable mappings.