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 -variational principle of Ekeland (1974, 1979).[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.[5]