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The Topological Degree of A-Proper Maps

Published online by Cambridge University Press:  20 November 2018

H. Ship Fah Wong
H. Ship Fah Wong*
Affiliation:
University of Warwick, Coventry, England
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Recently several fixed-point theorems have been proved for new classes of non-compact maps between Banach spaces. First, Petryshyn [15] generalized the concept of compact and quasi-compact maps when he introduced the P-compact maps and proved a fixed-point theorem for this class of maps. Then in [6], de Figueiredo defined the notion of G-operator to unify his own work on fixed-point theorems and that of Petryshyn. He also proved that the class of G-operators was a fairly large one.

We notice the following facts: (i) The essential idea in the above cases is that if certain finite-dimensional “approximations” of the map have fixed points, then the map has a fixed point; (ii) One of the tools used in proving fixed-point theorems in the finite-dimensional case is the Brouwer degree and its generalization to maps of the type Identity + Compact in [8].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 3 , June 1971 , pp. 403 - 412
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Bourbaki, N., Éléments de mathématique; Première partie, Fasc. II, Livre III; Topologie générale, chapitre 1: Structures topologiques, Actualités Sci. Indust., No. 1142 (Hermann, Paris, 1961).Google Scholar
2. Browder, F. E. and Petryshyn, W. V., The topological degree and Galerkin approximations for non-compact operators in Banach spaces, Bull. Amer. Math. Soc. 7J+ (1968), 641–646.Google Scholar
3. Browder, F. E. and Petryshyn, W. V., Approximation methods and the generalized topological degree for non-linear mappings in Banach spaces, J. Functional Analysis 8 (1969), 217–245.Google Scholar
4. Cronin, J., Fixed points and topological degree in non-linear analysis, Amer. Math. Soc. Surveys, Vol. 11 (Amer. Math. Soc, Providence, R.I., 1964).Google Scholar
5. Figueiredo, D. G. de, Topics in non-linear functional analysis, University of Maryland Lecture series, No. 48, Chapter IV, pp. 112–153 (University of Maryland, College Park, Maryland).Google Scholar
6. Figueiredo, D. G. de, Fixed-point theorems for nonlinear operators and Galerkin approximations, J. Differential Equations 8 (1967), 271–281.Google Scholar
7. Krasnoselskii, M. A., Topological methods in the theory of non-linear integral equations, Translated by Armstrong, A. H., International series of monographs on Pure and Applied Math., Vol. 45 (a Pergamon Press Book, Macmillan, New York; Pergamon, Oxford, 1964).Google Scholar
8. Leray, J. and Schauder, J., Topologie et équations fonctionnelles, Ann. Sci. École Norm. Sup. 51 (1934), 45–73.Google Scholar
9. Nagumo, M., A theory of degree of mapping based on infinitesimal analysis, Amer. J. Math. 78 (1951), 485–496.Google Scholar
10. Nagumo, M., Degree of mapping in convex linear topological spaces, Amer. J. Math. 73 (1951), 497–511.Google Scholar
11. Petryshyn, W. V., Some examples concerning the distinctive features of bounded linear A-proper mappings and Fredholm mappings, Arch. Rational Mech. Anal. 88 (1969), 331–338.Google Scholar
12. Petryshyn, W. V., On projectional solvability and the Fredholm alternative for equations involving linear A-proper operators, Arch. Rational Mech. Anal. 80 (1968), 270–284.Google Scholar
13. Petryshyn, W. V., Invariance of domain theorem for locally A-proper mappings and its implications, J. Functional Anal. 5 (1970), 137–159.Google Scholar
14. Petryshyn, W. V., Fixed-point theorems involving P-compact, semicontractive, and accretive operators not defined on all of a Banach space, J. Math. Anal. Appl. 28 (1968), 336–354.Google Scholar
15. Petryshyn, W. V., On a fixed point theorem for nonlinear P-compact operators in Banach space, Bull. Amer. Math. Soc. 72 (1966), 329–334.Google Scholar
16. Petryshyn, W. V., On the approximation-solvability of nonlinear equations, Math. Ann. 177 (1968), 156–164.Google Scholar
17. Petryshyn, W. V. and Tucker, T. S., On the functional equations involving non-linear generalised P-compact operators, Trans. Amer. Math. Soc. 185 (1969), 343–373.Google Scholar
18. Robinson, A., Non-standard analysis (North-Holland, Amsterdam, 1966).Google Scholar