Von Neumann's Intersection Lemma (16.4) Implies Kakutani's Theorem (15.3) (Nikaido [1968, p. 70])

Let γ : K →→ K satisfy the hypotheses of 15.3 and set X = Y = K, E = Gr γ and set F equal to the diagonal of X × X. The hypotheses of 16.4 are then satisfied, and E ∩ F is equal to the set of fixed points of γ.

The Fan-Browder Theorem (17.1) Implies Kakutani's Theorem (15.3)

Let γ : K →→ K be convex-valued and closed and let μ(x) = {x} for each x. Then x ∈ γ(x) if and only if γ(x) ∩ μ(x) ≠ ø. Setting λ = 1, v = x and y = u ∈ γ(x), the hypotheses of 17.1 are satisfied. Thus the set of fixed points of γ is compact and nonempty.

Remark

In the hypotheses of Theorem 17.1 if γ(x) ∩ μ(x) ≠ ø, then we can take u = v and y = x. Thus if we associate to each x the set of y's given by the hypothesis, we are looking for a fixed point of the correspondence. This correspondence cannot be closed-valued however, since λ is required to be strictly positive. Thus we cannot use the Kakutani theorem to prove Theorem 17.1 in this fashion. Note that the proof of Theorem 17.1 depends only on Fan's lemma (7.4), which depends only on the K-K-M lemma (5.4), which can be proved from Sperner's lemma (4.1).