Shizuo Kakutani's Fixed Point Theorem

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Shizuo Kakutani discovered and proved in 1941 a generalization of Brouwer's Fixed Point Theorem. Brouwer's theorem applies to continuous point-to-point functions. Kakutani dealt with set-valued function; i.e., point-to-set functions.

Let M be a compact, convex subset of Euclidean n-space. Let T be a continuous set-valued function on M; i.e., a mapping from M to the set of all subsets of M, T→P(M). If T is such that T(x) is convex for all x belonging to M then there exists a z such that T(z) contains z.

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 semi-continuous mapping.
Upper semi-continous set-valued mappings: A mapping Φ:X→Y is called upper semi-continuous if for a any sequence {x_i} such that x→x₀ and φ(x_i)→y₀ then y₀∈φ(x₀).
The form of the theorem proved by Kakutani was:
If x→φ(x) is an upper semi-continuous point-to-set mapping of an r-dimensional closed simplex S into its power set P(S), then there exists x₀∈S such that x₀∈φ(x₀).

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 (b-a)/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
g_n(x_i) = max[φ(x_i)]
g_n(x_i) = min[φ(x_i)]
g_n(x_i) = midpt[φ(x_i)] = ½{max[φ(x_i)] + min[φ(x_i)]}

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 semi-continuity 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.)

Shizuo Kakutani, "A Generalization of Brouwer's Fixed Point Theorem," Duke Mathematical Journal, vol. 8 (1941), pp.457-459.

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Let M be a compact, convex subset of Euclidean n-space. Let T be a continuous set-valued function on M; i.e., a mapping from M to the set of all subsets of M, T→P(M). If T is such that T(x) is convex for all x belonging to M then there exists a z such that T(z) contains z.

If x→φ(x) is an upper semi-continuous point-to-set mapping of an r-dimensional closed simplex S into its power set P(S), then there exists x0∈S such that x0∈φ(x0).

If x→φ(x) is an upper semi-continuous point-to-set mapping of an r-dimensional closed simplex S into its power set P(S), then there exists x₀∈S such that x₀∈φ(x₀).