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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 is a variation of the -variational principle of Ekeland (1974, 1979). Moreover, the conclusion of Caristi''s theorem is equivalent to metric completeness, as proved by Weston (1977). The original result is due to the mathematicians James Caristi and William Arthur Kirk.