On the converse of caristi's fixed point theorem

• Autorzy: Głąb S.
• Języki publikacji: EN

Abstrakty

EN

Let X be a nonempty set of cardinality at most 2^[aleph]0 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 [phi] : X --> R+ such that d(x,Tx] is less than or equal phi[x] - phi(Tx) for all x belongs to X, and (X, d) is separable. Assuming CH (the Continuum Hypothesis), we deduce that (X,d) is compact.

Słowa kluczowe

EN

abstract dynamical system; fixed point; periodic point; continuum hypothesis

PL

układ dynamiczny abstrakcyjny; punkt stały; punkt okresowy; hipoteza continuum

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2004
• Tom: Vol. 52, nr 4
• Strony: 411--416
• Opis fizyczny: Bibliogr. 7 poz.

Twórcy

autor

Głąb S.

• Institute of Mathematics Technical University of Łódź 90-924 Łódź, Poland and Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa, Poland,

Bibliografia

• [1] C. Bessaga, On the converse of the Banach fixed point principle, Colloq. Math. 7 (1959), 41-43.
• [2] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241-251.
• [3] J. Dugundji and A. Granas, Fixed Point Theory I, Polish Sci. Publ.,Warszawa, 1982.
• [4] J. Jachymski, Converses to fixed point theorems of Zermelo and Caristi, Nonlinear Anal. 52 (2003), 1455-1463.
• [5] L. Janoš, An application of combinatorial techniques to a topological problem, Bull. Austral. Math. Soc. 9 (1973), 439-443.
• [6] -, Compactification and linearization of abstract dynamical systems, Acta Univ. Carolin. Math. Phys. 38 (1997), 63-70.
• [7] A. S. Kechris, Classical Descriptive Set Theory, Springer, New York, 1998.