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 -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]

  1. ^ 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.
  2. ^ 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.
  3. ^ 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.
  4. ^ 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.
  5. ^ Khojasteh, Farshid; Karapinar, Erdal; Khandani, Hassan (27 January 2016). "Some applications of Caristi's fixed point theorem in metric spaces". Fixed Point Theory and Applications. doi:10.1186/s13663-016-0501-z.