A MISCELLANY OF SELECTION THEOREMS WITHOUT CONVEXITY

• Journal title : Honam Mathematical Journal
• Volume 35, Issue 4,  2013, pp.757-764
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2013.35.4.757
Title & Authors
A MISCELLANY OF SELECTION THEOREMS WITHOUT CONVEXITY
Kim, Hoonjoo;

Abstract
In this paper, we give sufficient conditions for a map with nonconvex values to have a continuous selection and the selection extension property in LC-metric spaces under the one-point extension property. And we apply it to weakly lower semicontinuous maps and generalize previous results. We also get a continuous selection theorem for almost lower semicontinuous maps with closed sub-admissible values in $\small{\mathbb{R}}$-trees.
Keywords
selection;LC-metric space;weakly lower semicontinuous;quasi-lower semicontinuous;almost lower semicontinuous;one-point extension property;selection extension;hyperconvex;R-tree;
Language
English
Cited by
References
1.
C. Bardaro and R. Ceppitelli, Some further generalizations of Knaster-Kuratowski-Mazurkiewicz theorem and minimax inequalities, J. Math. Anal. Appl. 132 (1988), 484-490.

2.
H. Ben-El-Mechaiekh and M. Oudadess, Some selection theorems without convexity, J. Math. Anal. Appl. 195 (1995), 614-618.

3.
L. -J. Chu and C. -H. Huang, Generalized selection theorems without convexity, Nonlinear Anal. 73 (2010), 3224-3231.

4.
F. Deutsch and V. Indumathi and K. Schnatz, Lower semicontinuity, almost lower semicontinuity, and continuous selections for set-valued mappings, J. approx. theory 53 (1988), 266-294.

5.
V. G. Gutev, Unified selection and factorization theorems, Compt. Rend. Acad. Bulgur. Sci. 40 (1987), 13-15.

6.
V. G. Gutev, Selection theorems under an assumption weaker than lower semicontinuity, Topol. Appl. 50 (1993), 129-138.

7.
C. D. Horvath, Contractibility and generalized convexity, J. Math. Anal. Appl. 156 (1991), 341-357.

8.
C. D. Horvath, Extension and selection theorems in topological spaces with a generalized convexity structure, Ann. Fac. Sci. Toulouse Math. (6) 2 (1993), 253-269.

9.
H. Kim and S. Lee, Approximate selections of almost lower semicontinuous multimaps in C-spaces, Nonlinear Anal. 64 (2006), 401-408.

10.
W. A. Kirk, Hyperconvexity of R-trees, Fund. Math. 156 (1998), 67-72.

11.
J.T. Markin, A selection theorem for quasi-lower semicontinuous mappings in hyperconvex spaces, J. Math. Anal. Appl. 321 (2006), 862-866.

12.
J.T. Markin, Fixed points, selections and best approximation for multivalued mappings in R-trees, Nonlinear Anal. 67 (2007), 2712-2716.

13.
E. Michael, Continuous selections I, Ann. of Math. 63 (1956), 361-382.

14.
E. Michael and C. Pixley, A unified theorem on continuous selections, Pacific J. Math. 87 (1980), 187-188.

15.
K. Przeslawski and L.E. Rybinski, Michael selection theorem under weak lower semicontinuity assumption, Proc. Amer. Math. Soc. 109 (1990), 537-543.