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Caristi’s fixed point theorem and selections of set-valued contractions. (English) Zbl 0916.47044

Let (X,d) be a metric space and T:XX a map which need not be continuous but satisfies d(x,Tx)φ(x)-φ(Tx) for some lower semicontinuous function φ:[0,)[0,). Caristi proved this result using transfinite induction. W. A. Kirk [Colloq. Math. 36, 81-86 (1976; Zbl 0353.53041)] defined a partial ordering on X by x φ y iff d(x,y)φ(x)-φ(y) 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 T 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 η:[0,)[0,) be a function satisfying η(0)=0. Then the right hand lower Dini derivative of η at 0 (i.e., lim inf st + [η(s)-η(t)]/[s-t]) vanishes if and only if there is a complete metric space (X,d), a continuous and asymptotically regular mapping T:XX which has no fixed points and a continuous function φ:[0,)[0,) such that η(d(x,Tx))φ(x)-φ(Tx) for all xX.

47H10Fixed point theorems for nonlinear operators on topological linear spaces
54H25Fixed-point and coincidence theorems in topological spaces
47H04Set-valued operators
03E25Axiom of choice and related propositions (logic)