Kakutani Fixed-point Theorem

Non-example

The requirement that φ(x) be convex for all x is essential for the theorem to hold.

Consider the following function defined on :

$f(x)= \begin{cases} 3/4 & 0 \le x < 0.5 \\ \{ 3/4, 1/4 \} & x = 0.5 \\ 1/4 & 0.5 < x \le 1 \\ \end{cases}$

The function has no fixed point. Though it satisfies all other requirements of Kakutani's theorem, its value fails to be convex at x = 0.5.

Kakutani Fixed-point Theorem - Non-example