appletmagic.com Thayer Watkins Silicon Valley & Tornado Alley USA 


Shizuo Kakutani discovered and proved in 1941 a generalization of Brouwer's Fixed Point Theorem. Brouwer's theorem applies to continuous pointtopoint functions. Kakutani dealt with setvalued function; i.e., pointtoset functions.
A z such that z∈T(z) is called a fixed point of the mapping T.
Kakutani stated and proved his theorem making use of the concept of an upper semicontinuous mapping.
Upper semicontinous setvalued mappings: A mapping Φ:X→Y is called upper semicontinuous if for a any sequence {x_{i}} such that x→x_{0} and φ(x_{i})→y_{0} then y_{0}∈φ(x_{0}).
The form of the theorem proved by Kakutani was:
The general scheme of Kakutani's proof may be seen from the one dimensional case.
In the above illustration the set φ(x) is a range of values depicted by a black line segment. The aggregatge of φ(x) for all x is the black area in the above diagram.
In Kakutani's proof the simplex S is partitioned into finer and finer subdivisions. In the above illustration the simplex is just the line interval [a,b]. Suppose the partitionings are ones involving equal subintervals of (ba)/n. For a given partition {x_{i}} some value is chosen from φ(x) for each x_{i}. A function g_{n}(x) is constructed between these points (x_{i},g(x_{i}). Such a function is continuous and therefore Brouwer's Fixed Point Theorem applies. Thus, for each n there exists at least one fixed point, say x_{i}*. For consistent construction of the functions {g_{n}} and proper selection of the fixed point there is a sequence of these fixed points {x_{i}*} converging to a point x_{∞}.
Kakutani noted that the values {g_{n}(x_{i})} may be chosen arbitrarily from within the sets {φ(x_{i}} but he neglected to stress that the scheme of choosing those values has to be consistent for different values of n and likewise for the choice among possible multiple fixed points.
A few schemes which can be used to construct {g_{n}(x_{i})} are
The sequence of functions {g_{n}(x_{i})} will converge to max[φ(x)], min[φ(x)] or midpt[φ(x)], depending upon the scheme chosen and the fixed points will converge to one of the fixed points shown in the diagram.
By the upper semicontinuity of φ(x), the limit of {φ(x_{n}*)}; i.e., the limit of {x_{n}*}; must belong to φ(x_{∞}). Thus x_{∞} is a fixed point of φ(x). This fixed point is by no means unique. Any value of x in the interval [x_{1,∞}, x_{2,∞}] is a fixed point of φ(x).
Kakutani establishes the theorem for any dimension simplex and then goes on to establish the theorem for any closed, bounded convex set in Euclidean space.
(To be continued.)
HOME PAGE OF Thayer Watkins 