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The Knaster–Tarski theorem versus monotone nonexpansive mappings

Volume 66 / 2018

Rafael Espínola, Andrzej Wiśnicki Bulletin Polish Acad. Sci. Math. 66 (2018), 1-7 MSC: Primary 06A45, 54F05; Secondary 34A12, 46B20. DOI: 10.4064/ba8120-1-2018 Published online: 30 March 2018

Abstract

Let $X$ be a partially ordered set with the property that each family of order intervals of the form $[a,b],[a,\rightarrow )$ with the finite intersection property has a nonempty intersection. We show that every directed subset of $X$ has a supremum. Then we apply the above result to prove that if $X$ is a topological space with a partial order $\preceq $ for which the order intervals are compact, $\mathcal {F}$ is a nonempty commutative family of monotone maps from $X$ into $X$ and there exists $c\in X$ such that $c\preceq Tc$ for every $T\in \mathcal {F}$, then the set of common fixed points of $\mathcal {F}$ is nonempty and has a maximal element. The result, specialized to the case of Banach spaces, gives a general fixed point theorem for monotone mappings that drops many assumptions from several recent results in this area. An application to the theory of integral equations of Urysohn’s type is also given.

Authors

  • Rafael EspínolaDepartamento de Análisis Matemático – IMUS
    Universidad de Sevilla
    Apdo. de Correos 1160
    41080 Sevilla, Spain
    e-mail
  • Andrzej WiśnickiDepartment of Mathematics
    Rzeszów University of Technology
    Al. Powstańców Warszawy 12
    35-959 Rzeszów, Poland
    e-mail

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