Existence and Uniqueness of Fixed Point in Partially Ordered Sets and Applications to Ordinary Differential Equations

Abstract

We prove some fixed point theorems in partially ordered sets, providing an extension of the Banach contractive mapping theorem. Having studied previously the nondecreasing case, we consider in this paper nonincreasing mappings as well as non monotone mappings. We also present some applications to first–order ordinary differential equations with periodic boundary conditions, proving the existence of a unique solution admitting the existence of a lower solution.

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Correspondence to Juan J. Nieto or Rosana Rodríguez-López.

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Research partially supported by Ministerio de Educación y Ciencia and FEDER, Project MTM2004-06652-C03-01, and by Xunta de Galicia and FEDER, Projects PGIDIT02PXIC20703PN and PGIDIT05PXIC20702PN

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Nieto, J.J., Rodríguez-López, R. Existence and Uniqueness of Fixed Point in Partially Ordered Sets and Applications to Ordinary Differential Equations. Acta. Math. Sin.-English Ser. 23, 2205–2212 (2007). https://doi.org/10.1007/s10114-005-0769-0

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Keywords

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

MR (2000) Subject Classification

  • 47H10
  • 34B15