Caristi’s Fixed Point Theorem

  • P. V. SubrahmanyamEmail author
Part of the Forum for Interdisciplinary Mathematics book series (FFIM)


In 1976, Caristi [4] published a novel generalization of the contraction principle. Using transfinite arguments which was also later simplified by Wong [15]. Brondstedt [3] provided an alternative proof by introducing an interesting partial order. On the other hand, Ekeland [6] established a variational principle whence deducing Caristi’s theorem. Brezis and Browder [2] proved an ordering principle also leading to this fixed point theorem. Subsequently Altman [1], Turinici [14, 15] and others have extended this principle. In this chapter, we discuss some of these as well as proofs of Caristi’s theorem by Kirk [8], Penot [10] and Seigel [11]. That both Ekcland’s principle and Caristi’s theorem characterize completeness is also brought out.


  1. 1.
    Altman, M.: A generalization of the Brezis-Browder principle on ordered sets. Nonlinear Anal. 6, 157–165 (1982)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Brezis, H., Browder, F.E.: A general principle on ordered sets in nonlinear functional analysis. Adv. Math. 21, 355–364 (1976)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brondstedt, A.: On a lemma of Bishop and Phelps. Pac. J. Math. 55, 335–341 (1974)Google Scholar
  4. 4.
    Caristi, J.: Fixed point theorems for mappings satisfying inwardness conditions. Trans. Am. Math. Soc. 125, 241–251 (1976)MathSciNetCrossRefGoogle Scholar
  5. 5.
    De Figuiredo, D.G.: Lectures on the Ekeland Variational Principle with Applications and Detours. T.I.F.R (1972)Google Scholar
  6. 6.
    Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Jachymski, J.R.: Caristi’s fixed point theorem and selections of set-valued contractions. J. Math. Anal. Appl. 229, 55–67 (1998)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kirk, W.A.: Caristi’s fixed point theorem and metric convexity. Colloq. Math. 36, 81–86 (1976)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kirk, W.A.: Contraction mappings and extensions. In: Kirk, W.A., Sims, B. (eds.) Handbook of Metric Fixed Point Theory. Springer, Berlin (2001)CrossRefGoogle Scholar
  10. 10.
    Penot, J.P.: A short constructive proof of the Caristi fixed point theorem. Publ. Math. Univ. Paris 10, 1–3 (1976)Google Scholar
  11. 11.
    Siegel, J.: A new proof of Caristi’s fixed point theorem. Proc. Am. Math. Soc. 66, 54–56 (1977)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Sullivan, F.: A characterization of complete metric spaces. Proc. Am. Math. Soc. 83, 345–346 (1981)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Takahashi, W.: Existence theorems generalizing fixed point theorems for multivalued mappings. In: Baillon, J.B., Thera, M. (eds.) Fixed Point Theory and Applications, vol. 252, pp. 387–406. Longman Scientific & Technical, Harlow (1991)Google Scholar
  14. 14.
    Turinici, M.: A generalization of Altman’s ordering principle. Proc. Am. Math. Soc. 90, 128–132 (1984)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Turinici, M.: Remarks about a Brezis-Browder principle. Fixed Point Theory 4, 109–117 (2003)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Wong, C.S.: On a fixed point theorem of contractive type. Proc. Am. Math. Soc. 57, 282–284 (1976)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology MadrasChennaiIndia

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