On the Converse of Caristi's Fixed Point Theorem

• Autorzy: Szymon Głąb
• Języki publikacji: EN

Abstrakty

EN

Let X be a nonempty set of cardinality at most $2^{ℵ₀}$ and T be a selfmap of X. Our main theorem says that if each periodic point of T is a fixed point under T, and T has a fixed point, then there exist a metric d on X and a lower semicontinuous map ϕ :X→ ℝ ₊ such that d(x,Tx) ≤ ϕ(x) - ϕ(Tx) for all x∈ X, and (X,d) is separable. Assuming CH (the Continuum Hypothesis), we deduce that (X,d) is compact.

Kategorie tematyczne

• 03E50: Continuum hypothesis and Martin's axiom
• 47H10: Fixed-point theorems
• 54H25: Fixed-point and coincidence theorems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2004
• Tom: 52
• Numer: 4
• Strony: 411-416

Daty

wydano

2004

Twórcy

autor

Szymon Głąb

• Institute of Mathematics, Technical University of Łódź, Al. Politechniki 11, 90-924 Łódź, Poland
• nstitute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa, Poland

Identyfikatory

DOI

10.4064/ba52-4-7