zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Caristi’s fixed point theorem and selections of set-valued contractions. (English) Zbl 0916.47044

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