Summary
We define below a fixed point index for local condensing maps f defined on open subset of «nice» metricANR’s. We prove that all the properties of classical fixed point index for continuous maps defined in compact polyhedra have appropriate generalizations. If our map is compact (a special case of a condensing map) and defined on an open subset of a Banach space, we prove that our fixed point index agrees with Leray-Schauder degree.
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P. Alexandroff andH. Hopf, Topologie, Berlin, 1935.
A. Ambrosetti,Un teorema di esistenza per le equazioni differenziali sugli spazi di Banach, Rend. Sem. Mat. Univ. Padua, 39 (1967), 349–361.
Ju. G. Borisovic andJu. I. Sapronov,A contribution to the topological theory of condensing operators, Soviet Math. Dokladi, 9 (1968), 1304–1307.
L. Brouwer,Uber abbildungen vom mannigfaltigkeiton, Math. Annalen., 71 (1912), 97–115.
F. E. Browder,The topological fixed point theory and its applications to functional analysis, Unpublished Ph.D. Dissertation, Princeton University, 1948.
—— ——,On the fixed point index for continuous mappings of locally connected spaces, Summa Brasil. Math., 4 (1960), 253–293.
—— ——,On the spectral theory of elliptic differential operators, Math. Annalen., 142 (1961), 22–130.
—— ——,Asymptotic fixed point theorems, Math. Annalen., 185 (1970), 38–61.
—— ——,Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proceedings of the Symposium on Nonlinear Functional Analysis, Amer. Math. Soc., Chicago, April, 1968 (to appear).
—— —— andR. D. Nussbaum,The topological degree for noncompact nonlinear mappings in Banach spaces, Bull. Amer. Math., Soc., 74 (1968), 641–646.
—— —— andW. V. Petryshyn,Approximation methods and the generalized topological degree for nonlinear mappings in Banach spaces, Jour. of Functional Analysis, 3 (1969), 217–245.
G. Darbo,Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padua, 24 (1955), 84–92.
A. Deleanu,Theorie des points fixes sur les retractes de voisinage des espaces convexoides, Bull. Soc. Math. France, 89 (1959), 235–243.
A. Dold,Fixed point index and fixed point theorems for Euclidean neighborhood retracts, Topology, 4 (1965), 1–8.
J. Dugundji,An extension of Tietze’s theorem, Pac. Jour. Math., 1 (1951), 353–367.
D. G. de Figureido andL. A. Karlovitz,On the radial projection in normed spaccs, Bull. Amer. Math. Soc., 73 (1936), 364–367.
A. Granas,Introduction to Topology of Functional Spaces, University of Chicago Mathematics Lecture Notes, Spring, 1961.
O. Hanner,Some theorems on absolute neighborhood retracts, Arkiv for Matematik, 1 (1951), 389–408.
S. Hu,Theory of Retracts, Detroit (Wayne State University Press), 1965.
C. Kuratowski,Sur les espaces complets, Fund. Math., 15 (1930), 301–309.
J. Leray,Sur la position d’un ensemble ferme de points d’un espace topologique, Jour. de Math. Pures et Appl., 24 (1945), 169–199.
—— ——,Sur les equations et les transformations, Jour. de Math. Pures et Appl., 24 (1945), 201–248.
—— ——,Theorie des points fixed, indice total, et nombre de Lefschetz, Bull. Soc. Math. France, 87 (1959), 221–233.
J. Leray andJ. Schauder,Topologie et equations fonctionelles, Ann. Sci. Ec. Norm. Sup., 51 (1934), 45–78.
M. Nagumo,A theory of degree of mapping based on infinitesimal analysis, Amer. Jour. Math., 73 (1951), 485–496.
—— ——,Degree of mapping in convex linear topological spaces, Amer. Jour. Math., 73 (1951), 497–511.
M. Z. Nashed andJ. S. Wong,Some variants of a fixed point theorem of Krasnoselskii and applications to nonlinear integral equations, Jour. of Math. and Mech., 18 (1969), 767–777.
R. D. Nussbaum,The fixed point index and asymptotic fixed point theorems for k-set-contractions, Bull. Amer. Math. Soc, 75 (1969), 490–495.
—— ——,The radius of the essential spectrum, Duke Math. Jour., 38 (1970), pp. 473–478.
—— ——,A generalization of the Ascoli theorem and an application to functional differential equations, Jour. Math. Analysis and its Appl., 35 (1971), pp. 600–610.
-- --,The fixed point index and fixed point theorems for k-set-contractions, Unpublished Ph D. dissertation (University of Chicago, 1969).
—— ——,Asymptotic fixed point theorems for local condensing mops, Math. Annalen, 191 (1971), pp. 181–195.
-- --,Degree theory for local condensing maps, Jour. Math. Analysis and its Appl. (to appear).
R. S. Palais,Homotopy theory of infinite dimensional manifolds, Topology, 5 (1966), 1–16.
W. V. Petryshyn,On the approximation-solvability of nonlinear equations, Math. Annalen., 177 (1968), 156–164.
B. N. Sadovskii,On a fixed point principle, Functional Analysis and Appl., 1 (1967), 74–76.
R. B. Thompson,A unified approach to local and global fixed point indices, Advances in Math., 3 (1969), 1–72.
M. Furi andA. Vignoli,A Fixed point theorem in complete metric spaces, Boll. Unione Matem. Ital., serie IV (N. 4–5), (1969), 505–506.
-- --,On α-nonexpansive mappings and fixed points, Rend. Acc. Naz. Lincei 48 (N. 2), (1970).
—— ——,On a property of the unit sphere in a linear normed space, Boll. Acad. Pol. Sci., 18 (2), (1970), 115–116.
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Entrata in Redazione il 12 ottobre 1970.
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Nussbaum, R.D. The fixed point index for local condensing maps. Annali di Matematica 89, 217–258 (1971). https://doi.org/10.1007/BF02414948
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DOI: https://doi.org/10.1007/BF02414948