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.

Statement 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,

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

Then T has a fixed point in X, i.e. a point x₀ such that T(x₀) = x₀.

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,

$d{\big (}x,T(x){\big )}\leq \max \left\{c(f(x)),c(f(T(x)))\right\}\left(f(x)-f(T(x))\right).$

Then T has a fixed point in X.

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

$d{\big (}x,T(x){\big )}\leq \max c(f(x))\left(f(x)-f(T(x))\right){\mbox{ for all }}x\in X$

or

$d{\big (}x,T(x){\big )}\leq \max c(f(T(x)))\left(f(x)-f(T(x))\right){\mbox{ for all }}x\in X.$

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

$d{\big (}x,T(x){\big )}\leq c{\big (}d(x,T(x)){\big )}{\big (}f(x)-f(T(x)){\big )}.$

Then T has a fixed point in X.

• (Khamsi (2009)) Let $\eta$ : [0, +∞) → [0, +∞) be such that $\eta$'(0) > 0. Suppose that, for all x in X, we have

$\eta {\big (}d{\big (}x,T(x){\big )}{\big )}\leq f(x)-f(T(x)).$

Then T has a fixed point in X.

References

• Bae, J. S. and Cho, E. W. and Yeom, S. H. (1994). "A generalization of the Caristi-Kirk fixed point theorem and its applications to mapping theorems". J. Korean Math. Soc. 31 (1): 29–48. ISSN 0304-9914.{{cite journal}}: CS1 maint: multiple names: authors list (link)
• Bae, Jong Sook (2003). "Fixed point theorems for weakly contractive multivalued maps". J. Math. Anal. Appl. 284 (2): 690–697. doi:10.1016/S0022-247X(03)00387-1. ISSN 0022-247X.
• Caristi, James (1976). "Fixed point theorems for mappings satisfying inwardness conditions". Trans. Amer. Math. Soc. 215: 241–251. doi:10.2307/1999724. ISSN 0002-9947.
• Ekeland, Ivar (1974). "On the variational principle". J. Math. Anal. Appl. 47: 324–353. ISSN 0022-247x. {{cite journal}}: Check |issn= value (help)
• Ekeland, Ivar (1979). "Nonconvex minimization problems". Bull. Amer. Math. Soc. (N.S.). 1 (3): 443–474. doi:10.1090/S0273-0979-1979-14595-6. ISSN 0002-9904.
• Khamsi, M A (2009). "Remarks on Caristi's fixed point theorem". Nonlinear Analysis, Theory Methods and Applications. doi:10.1016/j.na.2008.10.042.
• Weston, J. D. (1977). "A characterization of metric completeness". Proc. Amer. Math. Soc. 64 (1): 186–188. doi:10.2307/2041008. ISSN 0002-9939.