Kakutani Fixed-point Theorem - Statement

Statement

Kakutani's theorem states:

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

When we say that the graph of is closed, we mean that for all sequences and such that, and for all, we have .

Read more about this topic:  Kakutani Fixed-point Theorem

Famous quotes containing the word statement:

    “I think, therefore I am is the statement of an intellectual who underrates toothaches.”
    —Milan Kundera (b. 1929)

    “It is commonplace that a problem stated is well on its way to solution, for statement of the nature of a problem signifies that the underlying quality is being transformed into determinate distinctions of terms and relations or has become an object of articulate thought.”
    —John Dewey (1859–1952)

    “Eroticism has its own moral justification because it says that pleasure is enough for me; it is a statement of the individual’s sovereignty.”
    —Mario Vargas Llosa (b. 1936)