Let be a metric space and a map which need not be continuous but satisfies for some lower semicontinuous function . Caristi proved this result using transfinite induction. W. A. Kirk [Colloq. Math. 36, 81-86 (1976; Zbl 0353.53041)] defined a partial ordering on by iff in order to prove this theorem. His proof uses Zorn’s lemma. F. E. Browder [in: Fixed point theorem, Appl. Proc. Sem. Halifax 1975, 23-27 (1976; Zbl 0379.54016)] gave a constructive proof using the axiom of choice only for countable families. R. Mańka [Rep. Math. Logic 22, 15-19 (1988; Zbl 0687.04003)] then gave a constructive proof based on Zermelo’s theorem. The present author gives a simple derivation of Caristi’s theorem from Zermelo’s theorem in case is continuous. On the other hand, the author describes examples of set-valued contractions which admit (not necessarily continuous) selections which satisfy the assumptions of Caristi’s theorem. Finally, the author answers a question posed by W. A. Kirk by proving the following result:
Let be a function satisfying . Then the right hand lower Dini derivative of at 0 (i.e., ) vanishes if and only if there is a complete metric space , a continuous and asymptotically regular mapping which has no fixed points and a continuous function such that for all .