Caristi fixed-point theorem

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 ${\displaystyle \varepsilon }$-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]

Let ${\displaystyle (X,d)}$ be a complete metric space. Let ${\displaystyle T:X\to X}$ and ${\displaystyle f:X\to [0,+\infty )}$ be a lower semicontinuous function from ${\displaystyle X}$ into the non-negative real numbers. Suppose that, for all points ${\displaystyle x}$ in ${\displaystyle X,}$ $${\displaystyle d(x,T(x))\leq f(x)-f(T(x)).}$$

Then ${\displaystyle T}$ has a fixed point in ${\displaystyle X;}$ that is, a point ${\displaystyle x_{0}}$ such that ${\displaystyle T(x_{0})=x_{0}.}$ 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.^[6]