The Ran–Reurings fixed point theorem without partial order: A simple proof

  • Hichem Ben-El-MechaiekhEmail author


The purpose of this note is to generalize the celebrated Ran–Reurings fixed point theorem to the setting of a space with a binary relation that is only transitive (and not necessarily a partial order) and a relation-complete metric. The arguments presented here are simple and straightforward. It is also shown that extensions by Rakotch and by Hu and Kirk of Edelstein’s generalization of the Banach contraction principle to local contractions on chainable complete metric spaces are derived from the Ran–Reurings theorem.


Existence and uniqueness of a fixed point contraction local contraction transitive relation monotonic chainability monotoniccomplete metric 

Mathematics Subject Classification



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  1. 1.
    M. R. Alfuraidan and M. A. Khamsi, Caristi fixed point theorem in metric spaces with a graph. Abstr. Appl. Anal. 2014 (2014), Article ID 303484, 5 pp.Google Scholar
  2. 2.
    Edelstein M.: An extension of Banach’s contraction principle. Proc. Amer. Math. Soc. 12, 7–10 (1961)zbMATHMathSciNetGoogle Scholar
  3. 3.
    A. Granas and J. Dugundji, Fixed Point Theory. Springer, New York, 2003.Google Scholar
  4. 4.
    T. Hu and W. A. Kirk, Local contractions in metric spaces. Proc. Amer. Soc. 68 (1978), 121–124.Google Scholar
  5. 5.
    R. D. Holmes, Fixed points for local radial contractions. In: Fixed Point Theory and Its Applications (Proc. Sem., Dalhousie Univ., Halifax, N. S., 1975), S. Swaminathan, ed., Academic Press, New York, 1976, 79–89.Google Scholar
  6. 6.
    Jachymski J.: The contraction principle for mappings on a metric space with a graph. Proc. Amer. Soc. 136, 1359–1373 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    J. J. Nieto and R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22 (2005), 223–239.Google Scholar
  8. 8.
    Rakotch E.: A note on α-locally contractive mappings. Bull. Res. Council Israel Sect. F 10, 188–191 (1962)MathSciNetGoogle Scholar
  9. 9.
    A. C. M. Ran and M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Amer. Math. Soc. 132 (2004), 1435–1443.Google Scholar

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© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsBrock UniversitySaint CatharinesCanada

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