Wednesday September 19, 2001
in K9509, SFU
3:30 - 4:20
Dr Kazimierz Goebel
"Minimal displacement and optimal retraction problems in
Let (X,|| ||) be an infinite dimensional Banach space with unit ball B and unit sphere S. It is known that Brouwer's Fixed Point Theorem "strongly fails" in this setting. This means that:
The minimal displacement problem means finding uniform
of d(T) within various classes of mappings.
optimal retraction problem is a question of finding retractions of
S having relatively small Lipschitz constant.
- A. There are lipschitzian mappings T:B->B without
fixed points and, even more, such that inf || x-Tx|| =: d(T) > 0.
- B. The unit sphere S is a lipschitzian retract of
B meaning that there is a lipschitzian mapping (a retraction)
R:B->S such that T |_S =Id.$
- C. The unit sphere S is contractible to a point via