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On the converse of caristi's fixed point theorem

Głąb S.

Bulletin of the Polish Academy of Sciences. Mathematics

2004

Vol. 52, nr 4

411-416

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.