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Fixed Point Principles for Cones of a Linear Normed Space

Published online by Cambridge University Press:  20 November 2018

Gilles Fournier
Gilles Fournier*
Affiliation:
Université de Sherbrooke, Sherbrooke, Québec
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In [8] and [9], Krasnosel'skiĭ proved several fundamental fixed point principles for operators leaving invariant a cone in a Banach space. In [11], Nussbaum extended one of the results, the theorem about compression and expansion of a cone, to condensing maps and he applied this theorem to prove the existence of periodic solutions of nonlinear autonomous functional differential equations.

Nussbaum's proof makes an essential use of the difficult Zabreiko and Krasnosel'skiĭ, and Steinlein (mod p)-theorem for the fixed point index [13 -16]. In [6], Fournier and Peitgen proved two different versions of this theorem for completely continuous maps each one being sufficient for Nussbaum's applications. The proofs of these two theorems are much less involved and, although they are different, they make use of the same easier generalized Lefschetz number calculations (see [12] for (mod p) and [5] for compact attractor).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 6 , 01 December 1980 , pp. 1372 - 1381
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Borsuk, K., Theory of retracts (Polish Scientific Publishers, Warszawa PWN, 1967).Google Scholar
2. Dold, A., Fixed point index and fixed point theorem for Euclidean neighbourhood retracts, Topology, Oxford 4 (1965), 1–8.Google Scholar
3. Dugundji, J., An extension of Tietze's theorem, Pacific J. Math. 1 (1951), 353–367.Google Scholar
4. Fournier, G., Généralisations du théorème de Lefschetz pour des espaces non-compacts I Applications éventuellement compactes, Bull. Acad. Polon. Sci. Ser. Se. math., astr. et phys. 23 (1975), 693–700.Google Scholar
5. Fournier, G., Généralisations du théorème de Lefschetz pour des espaces non-compacts II Applications d'attraction compacte, Bull. Acad. Polon. Sci. Ser. Sci. math., astr. et phys. 23 (1975), 701–706.Google Scholar
6. Fournier, G. and Peitgen, H.-O., On some fixed point principles for cones in linear normed spaces, Math. Ann. 225 (1977), 205–218.Google Scholar
7. Granas, A., The Leray-Schander index and the fixed point theory for arbitrary ANR's, Bull. Soc. math. France 100 (1972), 209–228.Google Scholar
8. Krasnosel'skiĭ, M. A., Fixed points of cone-compressing or cone-extending operators, Soviet Math. Dokl. 1 (1960), 1285–1288.Google Scholar
9. Krasnosel'skiĭ, M. A., Positive solutions of operator equations (Groningen, Noordhoff, 1964).Google Scholar
10. Leray, J., Théorie des points fixes: indice total et nombre de Lefschetz, Bull. Soc. math. France 87 (1959), 221–233.Google Scholar
11. Nussbaum, R. D., Periodic solutions of some nonlinear, autonomous functional differential equations. II, J. Diff. Eq. 14 (1973), 360–394.Google Scholar
12. Peitgen, H.-O., Some applications cf the fixed point index in asymptotic fixed point theory, Proc. Conf. on Fixed Point Theory and its Applications, Halifax (1975).Google Scholar
13. Steinlein, H., Ein Satz ilber den Leray–Schauderschen Abbildungsgrad, Math. Z. 120 (1972), 176–208.Google Scholar
14. Steinlein, H., Über die verallgemeinerten Fixpunktindizes von Iterierten verdichtender Abbildungen, Manuscripta Math. 8 (1972), 251–266.Google Scholar
15. Steinlein, H., A new proof of the (mod p)—theorem in asymptotic fixed point theory, Proc. Conf. on Problems in Nonlinear Functional Analysis, Bonn (1974), 29–42.Google Scholar
16. Zabreiko, P. P. and Krasnosel'skiĭ, M. A., Iterations of operators and the fixed point index, Doklady Akad, Nauk SSSR 196 (1971), 1000–1009.Google Scholar