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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 $T:X\to X$ a map which need not be continuous but satisfies $d(x,Tx)\le\varphi(x)-\varphi(Tx)$ for some lower semicontinuous function $\varphi:[0,\infty)\to[0,\infty)$. Caristi proved this result using transfinite induction. {\it W. A. Kirk} [Colloq. Math. 36, 81-86 (1976; Zbl 0353.53041)] defined a partial ordering on $X$ by $x\le_\varphi y$ iff $d(x,y)\le\varphi(x)-\varphi(y)$ in order to prove this theorem. His proof uses Zorn’s lemma. {\it 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. {\it 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 $\eta(0)=0$. Then the right hand lower Dini derivative of $\eta$ at $0$ (i.e., $\liminf_{s\to t^+}[\eta(s)-\eta(t)]/[s-t]$) 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 $\varphi:[0,\infty)\to[0,\infty)$ such that $\eta(d(x,Tx))\le\varphi(x)-\varphi(Tx)$ for all $x\in X$.

MSC:
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)
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Full Text: DOI
References:
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