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:

    “If we do take statements to be the primary bearers of truth, there seems to be a very simple answer to the question, what is it for them to be true: for a statement to be true is for things to be as they are stated to be.”
    —J.L. (John Langshaw)

    “The parent is the strongest statement that the child hears regarding what it means to be alive and real. More than what we say or do, the way we are expresses what we think it means to be alive. So the articulate parent is less a telling than a listening individual.”
    —Polly Berrien Berends (20th century)

    “The most distinct and beautiful statement of any truth must take at last the mathematical form.”
    —Henry David Thoreau (1817–1862)