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.

Caristi fixed-point theorem can be applied to derive another classical fixed-point theorems results and also to prove the existence of bounded solutions of a functional equation.

Statement of the theorem

Let (X, d) be a complete metric space. Let T : X → X and 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.

References

• 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. JSTOR 1999724.
• Ekeland, Ivar (1974). "On the variational principle". J. Math. Anal. Appl. 47 (2): 324–353. doi:10.1016/0022-247X(74)90025-0. ISSN 0022-247X.
• 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.
• Weston, J. D. (1977). "A characterization of metric completeness". Proc. Amer. Math. Soc. 64 (1): 186–188. doi:10.2307/2041008. ISSN 0002-9939. JSTOR 2041008.