Let X be a nonempty set of cardinality at most 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba52-4-7,
author = {Szymon G\l \k ab},
title = {On the Converse of Caristi's Fixed Point Theorem},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {52},
year = {2004},
pages = {411-416},
zbl = {1114.54026},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba52-4-7}
}
Szymon Głąb. On the Converse of Caristi's Fixed Point Theorem. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 52 (2004) pp. 411-416. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba52-4-7/