Kakutani Fixed-point Theorem - Infinite Dimensional Generalizations

Infinite Dimensional Generalizations

Kakutani's fixed point theorem was extended to infinite dimensional locally convex topological vector spaces by Irving Glicksberg and Ky Fan. To state the theorem in this case, we need a few more definitions:

Upper semicontinuity

A set-valued function φ: X→2Y is upper semicontinuous if for every open set W ⊂ Y, the set {x| φ(x) ⊂ W} is open in X.

Kakutani map

Let X and Y be topological vector spaces and φ: X→2Y be a set-valued function. If Y is convex, then φ is termed a Kakutani map if it is upper semicontinuous and φ(x) is non-empty, compact and convex for all x ∈ X.

Then the Kakutani-Glicksberg-Fan theorem can be stated as:

Let S be a non-empty, compact and convex subset of a locally convex topological vector space. Let φ: S→2S be a Kakutani map. Then φ has a fixed point.

The corresponding result for single-valued functions is the Tychonoff fixed point theorem.

If the space on which the function is defined is Hausdorff in addition to being locally convex, then the statement of the theorem becomes the same as that in the Euclidean case:

Let S be a non-empty, compact and convex subset of a locally convex Hausdorff space. Let φ: S→2S be a set-valued function on S which has a closed graph and the property that φ(x) is non-empty and convex for all x ∈ S. Then the set of fixed points of φ is non-empty and compact.