*(English)*Zbl 0916.47044

Let $(X,d)$ be a metric space and $T:X\to X$ a map which need not be continuous but satisfies $d(x,Tx)\le \phi \left(x\right)-\phi \left(Tx\right)$ for some lower semicontinuous function $\phi :[0,\infty )\to [0,\infty )$. 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 $x{\le}_{\phi}y$ iff $d(x,y)\le \phi \left(x\right)-\phi \left(y\right)$ 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 $\eta :[0,\infty )\to [0,\infty )$ be a function satisfying $\eta \left(0\right)=0$. Then the right hand lower Dini derivative of $\eta $ at 0 (i.e., ${lim\; inf}_{s\to {t}^{+}}[\eta \left(s\right)-\eta \left(t\right)]/[s-t]$) vanishes if and only if there is a complete metric space $(X,d)$, a continuous and asymptotically regular mapping $T:X\to X$ which has no fixed points and a continuous function $\phi :[0,\infty )\to [0,\infty )$ such that $\eta \left(d\right(x,Tx\left)\right)\le \phi \left(x\right)-\phi \left(Tx\right)$ for all $x\in X$.

##### MSC:

47H10 | Fixed point theorems for nonlinear operators on topological linear spaces |

54H25 | Fixed-point and coincidence theorems in topological spaces |

47H04 | Set-valued operators |

03E25 | Axiom of choice and related propositions (logic) |