The Knaster-Tarski theorem versus monotonne nonexpansive mappings

• Autorzy: Espínola R. , Wiśnicki A.

Identyfikatory

DOI

10.4064/ba8120-1-2018

• Języki publikacji: EN

Abstrakty

EN

Let X be a partially ordered set with the property that each family of order intervals of the form [a, b], [a,→) 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 ⪯ for which the order intervals are compact, F is a nonempty commutative family of monotone maps from X into X and there exists c ∈ X such that c ⪯ Tc for every T ∈ F, then the set of common fixed points of 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.

Słowa kluczowe

EN

monotone mapping; nonexpansive mapping; fixed point; partially ordered set; directed set; Banach space; Urysohn-type equation

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2018
• Tom: Vol. 66, no. 1
• Strony: 1--7
• Opis fizyczny: Bibliogr. 13 poz.

Twórcy

autor

Espínola R.

• Departamento de Análisis Matemático – IMUS, Universidad de Sevilla, Apdo. de Correos 1160, 41080 Sevilla, Spain

autor

Wiśnicki A.

• Department of Mathematics, Rzeszów University of Technology, Al. Powstańców Warszawy 12, 35-959 Rzeszów, Poland

Bibliografia

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• [12] 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.
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Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).