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 is a variation of the "ε"-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.

tatement of the theorem

Let ("X", "d") be a complete metric space. Let "T" : "X" → "X" be a continuous function from "X" into itself, and let "f" : "X" → [0, +∞) be a lower semicontinuous function from "X" into the non-negative real numbers. Suppose that, for all points "x" in "X",

:

Then "T" has a fixed point in "X", i.e. a point "x"₀ such that "T"("x"₀) = "x"₀.

Generalisations

There are many generalisations of Caristi's theorem; some are given below. In the following, as above, ("X", "d") is a complete metric space with a function "T" : "X" → "X" and a lower semicontinuous function "f" : "X" → [0, +∞).

* (Bae, Cho and Yeom (1994)) Let "c" : [0, +∞) → [0, +∞) be upper semicontinuous from the right, and suppose that, for all "x" in "X",

::

: Then "T" has a fixed point in "X".

* (Bae, Cho and Yeom (1994)) Let "c" : [0, +∞) → [0, +∞) be non-decreasing. Assume that either

::

: or

::

: Then "T" has a fixed point in "X".

* (Bae (2003)) Let "c" : [0, +∞) → [0, +∞) be upper semicontinuous. Suppose that, for all "x" in "X", both "d"("x", "T"("x")) ≤ "f"("x") and

::

: Then "T" has a fixed point in "X".

References

* cite journal
author = Bae, J. S. and Cho, E. W. and Yeom, S. H.
title = A generalization of the Caristi-Kirk fixed point theorem and its applications to mapping theorems
journal = J. Korean Math. Soc.
volume = 31
year = 1994
issue = 1
pages = 29–48
issn = 0304-9914
* cite journal
last = Bae
first = Jong Sook
title = Fixed point theorems for weakly contractive multivalued maps
journal = J. Math. Anal. Appl.
volume = 284
year = 2003
issue = 2
pages = 690–697
issn = 0022-247X
doi = 10.1016/S0022-247X(03)00387-1
* 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
* cite journal
last = Ekeland
first = Ivar
title = On the variational principle
journal = J. Math. Anal. Appl.
volume = 47
year = 1974
pages = 324–353
issn = 0022-247x
* 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
* 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