, Volume 22, Issue 3, pp 223–239 | Cite as

Contractive Mapping Theorems in Partially Ordered Sets and Applications to Ordinary Differential Equations

  • Juan J. NietoEmail author
  • Rosana Rodríguez-López


We prove the existence and uniqueness of solution for a first-order ordinary differential equation with periodic boundary conditions admitting only the existence of a lower solution. To this aim, we prove an appropriate fixed point theorem in partially ordered sets.

Key Words

fixed point partially ordered set first-order differential equation lower and upper solutions 

Mathematics Subject Classification

Primary: 47H10 34B15 


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  1. 1.
    Amann, H.: Order Structures and Fixed Points. Mimeographed Lecture Notes, Ruhr-Universität, Bochum, 1977.Google Scholar
  2. 2.
    Cousot, P. and Cousot, R.: Constructive versions of Tarski’s fixed point theorems, Pacific J. Math. 82 (1979), 43–57.MathSciNetzbMATHGoogle Scholar
  3. 3.
    Heikkilä, S. and Lakshmikantham, V.: Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Marcel Dekker, Inc., New York, 1994.zbMATHGoogle Scholar
  4. 4.
    Ladde, G.S., Lakshmikantham, V. and Vatsala, A.S.: Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman, Boston, 1985.zbMATHGoogle Scholar
  5. 5.
    Ran, A.C.M. and Reurings, M.C.B.: A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Am. Math. Soc. 132 (2004), 1435–1443.CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Tarski, A.: A lattice-theoretical fixpoint theorem and its applications, Pacific J. Math. 5 (1955), 285–309.MathSciNetzbMATHGoogle Scholar
  7. 7.
    Zeidler, E.: Nonlinear Functional Analysis and Its Applications, Vol. I: Fixed-Point Theorems, Springer, New York, 1986.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Departamento de Análisis Matemático, Facultad de MatemáticasUniversidad de Santiago de CompostelaSantiago de CompostelaSpain

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