On the Converse of Caristi's Fixed Point Theorem

Szymon Głąb. "On the Converse of Caristi's Fixed Point Theorem." Bulletin of the Polish Academy of Sciences. Mathematics 52.4 (2004): 411-416. <http://eudml.org/doc/280647>.

@article{SzymonGłąb2004,
abstract = {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.},
author = {Szymon Głąb},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {abstract dynamical system; fixed point; periodic point; continuum hypothesis},
language = {eng},
number = {4},
pages = {411-416},
title = {On the Converse of Caristi's Fixed Point Theorem},
url = {http://eudml.org/doc/280647},
volume = {52},
year = {2004},
}

TY - JOUR
AU - Szymon Głąb
TI - On the Converse of Caristi's Fixed Point Theorem
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2004
VL - 52
IS - 4
SP - 411
EP - 416
AB - 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.
LA - eng
KW - abstract dynamical system; fixed point; periodic point; continuum hypothesis
UR - http://eudml.org/doc/280647
ER -