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 $\varepsilon$ -variational principle of Ekeland (1974, 1979).{{cite journal

| doi = 10.1016/0022-247X(74)90025-0

| last = Ekeland

| first = Ivar

| title = On the variational principle

| journal = J. Math. Anal. Appl.

| volume = 47

| year = 1974

| pages = 324–353

| issn = 0022-247X

| issue = 2

|doi-access = free

}}{{cite journal

| last = Ekeland

| first = Ivar

| title = Nonconvex minimization problems

| journal = Bull. Amer. Math. Soc. (N.S.)

| volume = 1

| year = 1979

| issue = 3

| pages = 443–474

| issn = 0002-9904

| doi = 10.1090/S0273-0979-1979-14595-6

|doi-access = free

|url = https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-1/issue-3/Nonconvex-minimization-problems/bams/1183544327.pdf

}} The conclusion of Caristi's theorem is equivalent to metric completeness, as proved by Weston (1977).{{cite journal

| last = Weston

| first = J. D.

| title = A characterization of metric completeness

| journal = Proc. Amer. Math. Soc.

| volume = 64

| year = 1977

| issue = 1

| pages = 186–188

| issn = 0002-9939

| doi = 10.2307/2041008

| jstor = 2041008

}}

The original result is due to the mathematicians James Caristi and William Arthur Kirk.{{cite journal

| last = Caristi

| first = James

| title = Fixed point theorems for mappings satisfying inwardness conditions

| journal = Trans. Amer. Math. Soc.

| volume = 215

| year = 1976

| pages = 241–251

| issn = 0002-9947

| doi = 10.2307/1999724

| jstor = 1999724

|doi-access = free

}}

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.{{cite journal|last1=Khojasteh|first1=Farshid|last2=Karapinar|first2=Erdal|last3=Khandani|first3=Hassan|title=Some applications of Caristi's fixed point theorem in metric spaces|journal=Fixed Point Theory and Applications|date=27 January 2016|doi=10.1186/s13663-016-0501-z|doi-access=free}}

Statement of the theorem

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

$d(x, T(x)) \leq f(x) - f(T(x)).$

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.{{cite book |first1=S. |last1=Dhompongsa |first2=P. |last2=Kumam |chapter=A Remark on the Caristi’s Fixed Point Theorem and the Brouwer Fixed Point Theorem |editor-last=Kreinovich |editor-first=V. |title=Statistical and Fuzzy Approaches to Data Processing, with Applications to Econometrics and Other Areas |location=Berlin |publisher=Springer |year=2021 |pages=93–99 |isbn=978-3-030-45618-4 |doi=10.1007/978-3-030-45619-1_7}}

References

Category:Fixed-point theorems

Category:Metric geometry

Category:Theorems in real analysis