An index for set-valued maps in infinite-dimensional spaces

Author:
Stephen A. Williams

Journal:
Proc. Amer. Math. Soc. **31** (1972), 557-563

MathSciNet review:
0287535

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Abstract | References | Additional Information

Abstract: Previous fixed point indexes defined for a set-valued map in an infinite-dimensional space have required the values of this map to be convex sets. The corresponding assumption of this paper is that the values be (co-)acyclic sets, i.e., that the reduced Alexander cohomology group of each of these sets be trivial in each dimension.

Other assumptions are that the space is locally convex and that the map is compact and upper semicontinuous with no fixed points on the boundary of its domain.

The index is defined, proved to be homotopy invariant, and proved to vanish in case there are no fixed points. The main methods used are finite-dimensional approximation and the Vietoris-Begle mapping theorem.

**[1]**Arrigo Cellina,*A theorem on the approximation of compact multivalued mappings*, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)**47**(1969), 429–433 (1970) (English, with Italian summary). MR**0276936****[2]**Arrigo Cellina and Andrzej Lasota,*A new approach to the definition of topological degree for multi-valued mappings*, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)**47**(1969), 434–440 (1970) (English, with Italian summary). MR**0276937****[3]**A. Granas,*Theorem on antipodes and theorems on fixed points for a certain class of multi-valued mappings in Banach spaces*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys.**7**(1959), 271–275 (unbound insert) (English, with Russian summary). MR**0117588****[4]**A. Granas and J. W. Jaworowski,*Some theorems on multi-valued mappings of subsets of the Euclidean space*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys.**7**(1959), 277–283 (English, with Russian summary). MR**0120627****[5]**J. W. Jaworowski,*Some consequences of the Vietoris mapping theorem*, Fund. Math. 45 (1958), 261-272; correction**46**(1958), 359. MR**0132542****[6]**Tsoy-Wo Ma,*Topological degrees of set-valued compact fields in locally convex spaces*, Thesis under Ky Fan, University of California, Santa Barbara, Calif., 1970, 77 pp.**[7]**Edwin H. Spanier,*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112**

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0287535-2

Keywords:
Index,
degree,
set-valued,
multi-valued,
acyclic,
fixed point,
Vietoris mapping theorem

Article copyright:
© Copyright 1972
American Mathematical Society