Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T00:56:47.068Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Homotopy Property of Nussbaum's Fixed Point Index

Published online by Cambridge University Press:  20 November 2018

Gilles Fournier  and
Reine Fournier
Gilles Fournier
Affiliation:
Université de Sherbrooke, Sherbrooke, Québec
Reine Fournier
Affiliation:
Université de Sherbrooke, Sherbrooke, Québec
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In [14] R. D. Nussbaum generalized the fixed point index to a class of maps larger than the one in [5]. Unfortunately his homotopy property conditions are more restrictive than the often more readily verifiable ones of Eells-Fournier. In this paper we shall try to find an intermediate class of maps which will contain all the known examples of maps for which the index is defined and for which the condition of Eells-Fournier will imply the homotopy property.

In doing so, we shall give general conditions for which the sum of a compact map and a differentiable map will be a map having a fixed point index and for which the Lefschetz fixed point theorem is true.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 1 , 01 February 1982 , pp. 44 - 62
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Borsuk, K., Theory of retracts, Polish Scientific Publishers, Warszawa PWN, 1967 (Polska Akad. Nauk. Monographie Matem., 44)- Google Scholar
2. Darbo, G., Punti uniti in transformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padova 24 (1955), 84–92.Google Scholar
3. Dieudonné, J., Eléments d'analyse, tome I (Gauthier-Villars, Paris, 1972).Google Scholar
4. Dold, A., Fixed point index and fixed point theorem for euclidean neighbourhood retracts, Topology, Oxford t. 4 (1965), 1–8.Google Scholar
5. Eells, J. and Fournier, G., La théorie des points fixes des applications à itérée condensante, Journées Géom. dimens. infinie (1975-Lyon), Bull. Soc. Math. France, Mémoire 46 (1976), 91–120.Google Scholar
6. Fournier, G. and Peitgen, H.-O., Leray endomorphisms and cone maps, Ann. Scuola Normale Superiore Pisa, Série IV, Vol. V, no. 1, (1978), 149–179.Google Scholar
7. Granas, A., Topics infixed point theory, Lecture Notes (1970).Google Scholar
8. Granas, A., The Leray-Schauder index and the fixed point theory for arbitrary ANR's, Bull. Soc. Math. France 100 (1972), 209–228.Google Scholar
9. Kuratowski, K., Sur les espaces complets, Fund. Math. 15 (1930), 301–309.Google Scholar
10. Kuratowski, K., Topologie I, Warsaw (1948), Topologie II, Warsaw (1950).Google Scholar
11. Leray, J., Théorie des points fixes: indice total et nombre de Lefschetz, Bull. Soc. Math. France 87 (1959), 221–233.Google Scholar
12. Nussbaum, R. D., The fixed point index for local condensing maps, Ann. Mat. Pura App. (IV) 89, 217–258.Google Scholar
13. Nussbaum, R. D., Some asymptotic fixed point theorems, Trans. Amer. Math. Soc. 171 (1972), 349–375.Google Scholar
14. Nussbaum, R. D., Generalizing the fixed point index, Math. Ann. 228 (1977), 259–278.Google Scholar
15. O'Neill, B., Essential sets and fixed points, Amer. Math. J. 75 (1953), 497–509.Google Scholar