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 formula_1-variational principle of Ekeland (1974, 1979). 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.
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.
Statement of the theorem.
Let formula_2 be a complete metric space. Let formula_3 and formula_4 be a lower semicontinuous function from formula_5 into the non-negative real numbers. Suppose that, for all points formula_6 in formula_7
formula_8
Then formula_9 has a fixed point in formula_10 that is, a point formula_11 such that formula_12 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.
References.
 ;templatestyles src="Reflist/styles.css" / ;