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)
A selection theorem for quasi-lower semicontinuous mappings in hyperconvex spaces. (English) Zbl 1101.54025
Let $X$ be a paracompact topological space, $Y$ a hyperconvex metric space and $F:X\to Y$ a multifunction with sub-admissible values. A subset $B\subset Y$ is called sub-admissible if $coC\subset B$ for each finite $C\subset B$, where $coC$ denotes the intersection of all closed balls in $Y$ containing $C$. It is proved that if $F$ is quasi-lsc (i.e., for each $x\in X$ and $\varepsilon>0$ there is a point $y\in F(x)$ and a neighbourhood $U(x)$ of $x$ such that for each $t\in U(x)$, $F(t)\cap B(y,\varepsilon)\ne\emptyset)$, then $F$ admits a continuous selection. Two fixed-point theorems are deduced. Remark: It would be interesting to locate the generalized convexity used in the paper under review in the framework of Bielawski’s simplicial convexity, cf. {\it R. Bielawski} [J. Math. Anal. Appl, 127, 155--171 (1987; Zbl 0638.52002)].

54C65Continuous selections
26E25Set-valued real functions
52A01Axiomatic and generalized geometric convexity
Full Text: DOI
[1] Aronszajn, N.; Panitchpakdi, P.: Extensions of uniformly continuous transformations and hyperconvex metric spaces. Pacific J. Math. 6, 405-439 (1956) · Zbl 0074.17802
[2] Bardaro, C.; Ceppitelli, R.: Some further generalizations of knaster -- Kuratowski -- mazurkiewicz theorem and minimax inequalities. J. math. Anal. appl. 132, 484-490 (1988) · Zbl 0667.49016
[3] Ben-El-Mechaiekh, H.; Oudadess, M.: Some selection theorems without convexity. J. math. Anal. appl. 195, 614-618 (1995) · Zbl 0845.54012
[4] Horvath, C. D.: Some results on multivalued mappings and inequalities without convexity. Nonlinear and convex analysis, 99-106 (1987)
[5] Horvath, C. D.: Contractibility and generalized convexity. J. math. Anal. appl. 156, 341-357 (1991) · Zbl 0733.54011
[6] Horvath, C. D.: Extension and selection theorems in topological spaces with a generalized convexity structure. Ann. fac. Sci. Toulouse 2, 253-269 (1993) · Zbl 0799.54013
[7] Hou, J. -C.: Michaels selection theorem under an assumption weaker than lower semicontinuous in H-spaces. J. math. Anal. appl. 259, 501-508 (2001) · Zbl 0987.54022
[8] Khamsi, M. A.: KKM and Ky Fan theorems in hyperconvex metric spaces. J. math. Anal. appl. 204, 298-306 (1996) · Zbl 0869.54045
[9] Khamsi, M. A.; Kirk, W. A.; Martinez, C. M.: Fixed point and selection theorems in hyperconvex spaces. Proc. amer. Math. soc. 128, 3275-3283 (2000) · Zbl 0959.47032
[10] Michael, E.: Continuous selections I. Ann. of math. 63, 361-381 (1956) · Zbl 0071.15902
[11] Park, S.: Fixed point theorems in hyperconvex metric spaces. Nonlinear anal. 37, 467-472 (1999) · Zbl 0930.47023
[12] Park, S.; Sims, B.: Remarks on fixed point theorems in hyperconvex spaces. Nonlinear funct. Anal. appl. 5, 51-64 (2000) · Zbl 0968.47021
[13] Sine, R. C.: Hyperconvexity and nonexpansive multifunctions. Trans. amer. Math. soc. 315, 755-767 (1989) · Zbl 0682.47029
[14] Wu, X.; Li, F.: Approximate selection theorems in H-spaces with applications. J. math. Anal. appl. 231, 118-132 (1999) · Zbl 0985.54019
[15] Wu, X.; Yuan, X.: Approximate selections, fixed points, almost fixed points of multivalued mappings and generalized quasi-variational inequalities in H-spaces. Nonlinear anal. 38, 249-258 (1999) · Zbl 0940.54025
[16] Wu, X.; Thompson, B.; Yuan, X.: On continuous selection problems for multivalued mappings with the local intersection property in hyperconvex metric spaces. J. appl. Anal. 9, 249-260 (2003) · Zbl 1061.47045
[17] Zheng, X.: Approximate selection theorems and their applications. J. math. Anal. appl. 212, 88-97 (1997) · Zbl 0918.54018