zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Extensions of Banach’s contraction principle. (English) Zbl 1200.54021
For a metric space $(X,d)$ and $A,B\subset X,$ let $d(A,B)=\inf\{d(a,b) :a\in A,B\in B\}$, $A_0=\{a\in A : \exists b\in B, d(a,b)=d(A,B)\}$, and $B_0=\{b\in B : \exists a\in A,\ d(a,b)=d(A,B)\}$. The set $B$ is called approximatively compact with respect to $A$ if for every $x\in A$, every sequence $(y_n)$ in $B$ such that $d(x,y_n)\to d(x,B)$ contains a subsequence converging to some point of $B$. A mapping $T:A\to B$ is called a proximal contraction if there exists $\alpha, \, 0\leq \alpha<1,$ such that for every $(x,y)\in A\times B$ for which there exists $(u,v)\in A\times B$ with $d(u,Tx)=d(v,Ty)=d(A,B),$ the following inequality holds $$ d(u,Tx)+d(Tx,Ty)+d(v,Ty)\leq \alpha d(x,y).$$ The author shows by an example (a mapping $T:[0,1]\to[2,3]$) that a proximal contraction need not be continuous. Let $(X,d)$ be a complete metric space, $\, A,B \subset X$ nonempty closed such that $B$ is approximatively compact with respect to $A$ and $A_0,B_0$ are nonempty. Then for every proximal contraction $T:A\to B$ with $T(A_0)\subset T(B_0),$ there is a unique point $x\in A$ such that $d(x,Tx)=d(A,B)$ (Theorem 3.1). In Theorem 3.3, a similar result is proved for two mappings $T:A\to B,\, S:B\to A,$ where $T$ is a proximal contraction and $S$ is nonexpansive.

54H25Fixed-point and coincidence theorems in topological spaces
Full Text: DOI