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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 216146, 11 pages doi:10.1155/2011/216146 Research Article Generalized Lefschetz Sets ´ Mirosław Slosarski ´ Department of Electronics, Technical University of Koszalin, Sniadeckich 2, 75-453 Koszalin, Poland ´ Correspondence should be addressed to Mirosław Slosarski, slosmiro@gmail.com Received 5 January 2011; Accepted 2 March 2011 Academic Editor: Marl` ne Frigon e ´ Copyright q 2011 Mirosław Slosarski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We generalize and modify Lefschetz sets defined. | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011 Article ID 216146 11 pages doi 10.1155 2011 216146 Research Article Generalized Lefschetz Sets Miroslaw Slosarski Department of Electronics Technical University ofKoszalin ổniadeckich 2 75-453 Koszalin Poland Correspondence should be addressed to Miroslaw ốlosarski slosmiro@gmail.com Received 5 January 2011 Accepted 2 March 2011 Academic Editor Marlene Frigon Copyright 2011 Miroslaw ốlosarski. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. We generalize and modify Lefschetz sets defined in 1976 by L. Gorniewicz which leads to more general results in fixed point theory. 1. Introduction In 1976 L. Gorniewicz introduced a notion of a Lefschetz set for multivalued admissible maps. The paper attempts at showing that Lefschetz sets can be defined on a broader class of multivalued maps than admissible maps. This definition can be presented in many ways and each time it is the generalization of the definition from 1976. These generalizations essentially broaden the class of admissible maps that have a fixed point. Also they are a homologic tool for examining fixed points for a class of multivalued maps broader than just admissible maps. 2. Preliminaries Throughout this paper all topological spaces are assumed to be metric. Let H be the Cech homology functor with compact carriers and coefficients in the field of rational numbers Q from the category of Hausdorff topological spaces and continuous maps to the category of graded vector spaces and linear maps of degree zero. Thus H X Hq X is a graded vector space Hq X being the q-dimensional Cech homology group with compact carriers of X. For a continuous map f X Y H f is the induced linear map f fq where fq Hq X Hq Y see 1 2 . A space X is acyclic if i X is nonempty ii Hq Xf 0 for every q
