# 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)
Lim’s theorems for multivalued mappings in CAT(0) spaces. (English) Zbl 1086.47019

Let $\left(X,D\right)$ be a metric space, $CB\left(X\right)$ (resp., $K\left(X\right)$) the family of nonempty closed bounded (resp., the family of nonempty compact) subsets of $X$, and $H$ the Hausdorff metric on $CB\left(X\right)$ induced by $d$. Then a map

$T:X\to CB\left(X\right)$

is called multivalued nonexpansive if

$H\left(Tx,Ty\right)\le d\left(x,y\right)$

for all $x,y\in X$. The authors attempt to investigate condions under which a multivalued nonexpansive map may have a fixed point. Since such a map on a complete metric space need not have a fixed point, the authors work in a complete CAT(0) space [cf. W. A. Kirk, in: Proceedings of the international conference on fixed-point theory and its applications, Valencia, Spain, July 13–19, 2003, 113–142 (2004; Zbl 1083.53061)] and assume some “inwardness” requirement on the map. Their main result goes as follows. Let $E$ be a nonempty bounded closed convex subset of a complete CAT(0) space $X$ and $T:X\to K\left(X\right)$ a nonexpansive map. Assume that $T$ is weakly inward on $E$. Then $T$ has a fixed point.

Another main result is about the existence of a common fixed point of a single-valued nonexpansive map commuting with a multivalued nonexpansive map.

##### MSC:
 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47H09 Mappings defined by “shrinking" properties 54H25 Fixed-point and coincidence theorems in topological spaces 51K10 Synthetic differential geometry 05C05 Trees 53C70 Direct methods ($G$-spaces of Busemann, etc.) 58C30 Fixed point theorems on manifolds