Fixed Point Principles for Cones of a Linear Normed Space

Fournier, Gilles

Canadian journal of mathematics, Tome 32 (1980) no. 6, pp. 1372-1381

Résumé

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).

Détail

DOI : 10.4153/CJM-1980-107-8

Bibliographie

[1] 1. Borsuk, K., Theory of retracts (Polish Scientific Publishers, Warszawa PWN, 1967). Google Scholar

[2] 2. Dold, A., Fixed point index and fixed point theorem for Euclidean neighbourhood retracts, Topology, Oxford 4 (1965), 1–8. Google Scholar

[3] 3. Dugundji, J., An extension of Tietze's theorem, Pacific J. Math. 1 (1951), 353–367. Google Scholar

[4] 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] 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] 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] 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] 8. Krasnosel'skiĭ, M. A., Fixed points of cone-compressing or cone-extending operators, Soviet Math. Dokl. 1 (1960), 1285–1288. Google Scholar

[9] 9. Krasnosel'skiĭ, M. A., Positive solutions of operator equations (Groningen, Noordhoff, 1964). Google Scholar

[10] 10. Leray, J., Théorie des points fixes: indice total et nombre de Lefschetz, Bull. Soc. math. France 87 (1959), 221–233. Google Scholar

[11] 11. Nussbaum, R. D., Periodic solutions of some nonlinear, autonomous functional differential equations. II, J. Diff. Eq. 14 (1973), 360–394. Google Scholar

[12] 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] 13. Steinlein, H., Ein Satz ilber den Leray–Schauderschen Abbildungsgrad, Math. Z. 120 (1972), 176–208. Google Scholar

[14] 14. Steinlein, H., Über die verallgemeinerten Fixpunktindizes von Iterierten verdichtender Abbildungen, Manuscripta Math. 8 (1972), 251–266. Google Scholar

[15] 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] 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