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)
Hyperconvexity and approximate fixed points. (English) Zbl 0694.54033
A metric space (S,d) is called hyperconvex if any collection of closed balls of radius $r\sb{\alpha}$ and center $s\sb{\alpha},\alpha \in A$, which satisfy $d(s\sb{\alpha},s\sb{\beta})\le r\sb{\alpha}+r\sb{\beta}$ for all $\alpha$, $\beta$ in A, has nonempty intersection. The author begins with a good review and set of references for known results. He defines the $\epsilon$-fixed point set of T: $S\to S$ to be $F\sb{\epsilon}(T)=\{s:d(s,Ts)\le \epsilon \}$, then proves an approximate fixed point theorem: if T: $H\to H$ is nonexpansive and H is hyperconvex, then $F\sb{\epsilon}(T)$ is hyperconvex and, if nonempty, is a nonexpansive retract of H. If $F\sb{\epsilon}(T)$ is also bounded, then the fixed point set $F(T)=F\sb 0(T)$ is nonempty as well. The rest of the paper gives interesting results and discussion of the hyperconvex hull, N(S), of S as defined by {\it J. R. Isbell} [Comment. Math. Helv. 39, 65- 76 (1964; Zbl 0151.302)]. In particular, if T:N(S)$\to N(S)$ is nonexpansive, then $S\subset F\sb{\epsilon}(T)$ implies $N(S)=F\sb{\epsilon}(T)$. Also, any realization of $N(c\sb 0)$ in $\ell\sb{\infty}$ is all of $\ell\sb{\infty}$. This is nice since realizations of N(S) in a hyperconvex space H containing S always exist but, though isometric, are not generally unique. The author’s first “Remark” is unfortunate in that it gives a badly flawed example to discount Remark 3.1 of the Isbell paper cited above. In fact, it is the reviewer’s opinion that minor modification of the author’s lemma proof verifies Isbell’s remark.
Reviewer: D.E.Sanderson

54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties
Full Text: DOI
[1] Aronszajn, N.; Panitchpakdi, P.: Extensions of uniformly continuous transformations and hyperconvex metric spaces. Pacif. J. Math. 6, 405-439 (1956) · Zbl 0074.17802
[2] Baillon, J. B.: Nonexpensive mapping and hyperconvex spaces. Contemp. math. 72, 11-19 (1988)
[3] Bruck, R. E.: Asymptotic behavior of nonexpensive mappings. Contemp. math. 18, 1-47 (1983) · Zbl 0528.47039
[4] Connor, J.; Loomis, I.: Linear isometries of subalgebras of l$\infty $which contain c. Abstr. am. Math. soc. 55, 92 (1988)
[5] Edelstein, M.: On nonexpensive mappings of Banach spaces. Proc. camb. Phil. soc. 60, 439-447 (1964) · Zbl 0196.44603
[6] Isbell, J. R.: Six theorems about injective metric spaces. Comment. math. Helvetici 39, 65-76 (1964) · Zbl 0151.30205
[7] Jawhari, E.; Misane, D.; Ponzet, M.: Retracts: graphs and ordered sets from the metric point of view. Contemp. math. 57 (1986) · Zbl 0597.54028
[8] Kirk, W. A.: Fixed point theory for nonexpansive mappings II. Contemp. math. 18, 121-140 (1983) · Zbl 0511.47037
[9] Lacey, H. E.: The isometric theory of classical Banach spaces. (1974) · Zbl 0285.46024
[10] Lin M. & Sine R., On the fixed point set of nonexpansive order preserving maps (to appear). · Zbl 0662.47030
[11] Mankiewicz, P.: On extensions of isometries in normed spaces. Bull. acad. Pol sci. Sér. sci. Math. astr. Phys. 20, 367-371 (1972) · Zbl 0234.46019
[12] Ray, W. O.; Sine, R. C.: Nonexpansive mappings with precompact orbits. Lecture notes in mathematics 886, 409-416 (1981)
[13] Sine, R. C.: On nonlinear contraction semigroups in sup norm spaces. Nonlinear analysis 3, 885-890 (1979) · Zbl 0423.47035
[14] Sine, R. C.: Rigidity properties of nonexpansive mappings. Nonlinear analysis 11, 777-794 (1987) · Zbl 0662.47029
[15] Soardi, P.: Existence of fixed points of nonexpansive mappings in certain Banach lattices. Proc. am. Math soc. 73, 25-29 (1979) · Zbl 0371.47048
[16] Wells, J. H.; Williams, L. R.: Embeddings and extensions in analysis. (1975) · Zbl 0324.46034