In mathematics, the Caristi fixed-point theorem (also known as the Caristi–Kirk fixed-point theorem) generalizes the Banach fixed-point theorem for maps of a complete metric space into itself.
Caristi's fixed-point theorem modifies the
[1][2] The conclusion of Caristi's theorem is equivalent to metric completeness, as proved by Weston (1977).
[3] The original result is due to the mathematicians James Caristi and William Arthur Kirk.
[4] Caristi fixed-point theorem can be applied to derive other classical fixed-point results, and also to prove the existence of bounded solutions of a functional equation.
be a complete metric space.
be a lower semicontinuous function from
into the non-negative real numbers.
has a fixed point in
The proof of this result utilizes Zorn's lemma to guarantee the existence of a minimal element which turns out to be a desired fixed point.