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)

    “A sentence is made up of words, a statement is made in words.... Statements are made, words or sentences are used.”
    —J.L. (John Langshaw)

    “No statement about God is simply, literally true. God is far more than can be measured, described, defined in ordinary language, or pinned down to any particular happening.”
    —David Jenkins (b. 1925)