Caristi fixed-point theorem explained

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

\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]

Statement of the theorem

Let

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

Then

T has a fixed point in X; that is, a point x _{0} such that 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]

Notes and References

1. 10.1016/0022-247X(74)90025-0. Ekeland. Ivar. On the variational principle. J. Math. Anal. Appl.. 47. 1974. 324 - 353. 0022-247X. 2. free.
2. Ekeland. Ivar. Nonconvex minimization problems. Bull. Amer. Math. Soc. (N.S.). 1. 1979. 3. 443 - 474. 0002-9904. 10.1090/S0273-0979-1979-14595-6. free.
3. Weston. J. D.. A characterization of metric completeness. Proc. Amer. Math. Soc.. 64. 1977. 1. 186 - 188. 0002-9939. 10.2307/2041008. 2041008.
4. Caristi. James. Fixed point theorems for mappings satisfying inwardness conditions. Trans. Amer. Math. Soc.. 215. 1976. 241 - 251. 0002-9947. 10.2307/1999724. 1999724. free.
5. Khojasteh. Farshid. Karapinar. Erdal. Khandani. Hassan. Some applications of Caristi’s fixed point theorem in metric spaces. Fixed Point Theory and Applications. 27 January 2016. 10.1186/s13663-016-0501-z. free.
6. Book: Dhompongsa, S. . P. . Kumam . A Remark on the Caristi’s Fixed Point Theorem and the Brouwer Fixed Point Theorem . Kreinovich . V. . Statistical and Fuzzy Approaches to Data Processing, with Applications to Econometrics and Other Areas . Berlin . Springer . 2021 . 93-99 . 978-3-030-45618-4 . 10.1007/978-3-030-45619-1_7.