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Warianty tytułu
Abstrakty
CONTENTS
Introduction................................................................................................................................................. 5
I. General properties of set-valued compact fields............................................................................. 6
1. Upper semicontinuous maps............................................................................................................. 6
2. Generalization of Dugundji's extension theorem............................................................................ 7
3. Set-valued compact fields................................................................................................................... 9
4. Reduction to finite dimensional vector spaces............................................................................... 10
5. Reduction to single-valued compact fields...................................................................................... 12
II. Topological degrees of set-valued compact fields in locally convex spaces............................. 16
6. Basic known facts about Brouwer's degrees................................................................................... 16
7. Definition of topological degree and its homotopy invariance...................................................... 17
8. Sum theorem.......................................................................................................................................... 20
9. The case of odd degrees..................................................................................................................... 22
10. The case of non-vanishing degrees................................................................................................ 25
11. Reduction formula............................................................................................................................... 28
12. Translation invariance and component dependence.................................................................. 29
13. Product of domains............................................................................................................................. 30
14. Generalized Hopf theorem for metrizable locally convex spaces.............................................. 31
15. Product theorem for composite maps............................................................................................ 33
III. Extension of some classical results to set-valued maps............................................................. 38
16. Fixed point theorems and fixed point indices................................................................................ 38
17. Extension of Borsuk's sweeping theorem...................................................................................... 39
18. Extension of Borsuk-Ulam's theorem............................................................................................. 40
19. Extension of Brouwer's invariance of domains............................................................................. 40
References.................................................................................................................................................. 43
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne
tom/nr w serii:
92
Liczba stron
43
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom XCII
Daty
wydano
1972
Twórcy
autor
Bibliografia
- [1] M. Altman, A fixed point theorem in Banach spaces, Bull. Acad. Polon, Sci, 2 (1957), pp. 89-92.
- [2] M. Altman, An extension to locally convex spaces of Borsuk's Theorem on antipodes, Bull. Acad. Polon. Sci. 6 (1958), pp. 293-295.
- [3] M. Altman, Continuous transformations of open sets in locally convex spaces, Bull. Acad. Polon. Sci. 6 (1968), pp. 297-301.
- [4] C. Berge, Topological spaces, MacMillan, New York 1963.
- [5] M. S. Berger and M. S. Berger, Perspectives in non-linearity, Benjamin, New York 1968.
- [6] L. Вers, Topology, New York University, Courant Institute of Mathematics (1956).
- [7] K. Borsuk, Drei Sätze über die n-dimensionale Euklidische Sphäre, Fund. Math. 21 (1933), pp. 177-190.
- [8] F. E. Browder, Non-linear operators and non-linear equations of evolution in Banach spaces, Symposium on Non-linear Functional Analysis, Amer. Math. Soc. 1968.
- [9] J. Cronin, Fixed points and topological degree in non-linear analysis, Amer. Math. Soc. Survey Series, Vol. 11, Providence 1964.
- [10] J. Dugundji, An extension of Tietze's theorem, Pacific J. Math. 1 (1951), pp. 353-367.
- [11] K. Fan, Fixed point and minimax theorems in locally convex topological linear spaces, Proc. Nat. Acad. Sci., U.S. 38 (1952),. pp. 121-126.
- [12] A. Granas, Theorems on antipodes and theorems on fixed points for a certain class of multivalued mappings in Banach spaces, Bull. Acad. Polon. Sci. 8 (1959), pp. 271-275.
- [13] A. Granas, Sur la notion de degré topologique pour une certaine classe de transformations multivalentes dans les espaces de Banach, Bull. Acad. Polon. Sci. 7 (1959), pp. 191-194.
- [14] A. Granas, The theory of compact fields and some of its applications to the topology of functional spaces, I, Rozprawy Mat. 30 (1962), pp. 1-89.
- [15] A. Granas, and J. W. Jaworowski, Some theorems on multivalued mappings of subsets of the Euclidean spaces, Bull. Acad. Polon. Sci. 8 (1959), pp. 277-283.
- [16] I. J. Glicksberg, A further generalization of the Kakutani fixed point theorem with applications to Nash equilibrium points, Proc. Amer. Math. Soc. 3 (1952), pp. 170-174.
- [17] S. T. Hu, Homotopy theory, Academic Press, New York 1959.
- [18] J. W. Jaworowski, Theorems on antipodes for multivalued mappings and a fixed point theorem, Bull. Acad. Polon. Sci. 4 (1956), pp. 187-192.
- [19] S. Kakutani, A generalization of Brouwer's fixed point theorem, Duke Math. J. 8 (1941), pp. 467-459.
- [20] V. L. Klee, Leray-Schauder theory without local convexity, Math, Ann. 141 (1960), pp. 286-296.
- [21] M. A. Krasnoselskii, Topological methods in the theory of non-linear integral equations, English translation, MacMillan, New York 1964.
- [22] J. Leray, La théorie des points fixes et sea applications en analyse, Proc. Int. Congress Math., Cambridge, Mass. 2 (1950), pp. 202-208.
- [23] J. Leray, and J. Schauder, Topologie et équations fonctionnelles, Ann. Sci. École Norm. Super. (3) 51 (1934), pp. 45-78.
- [24] J. H. Michael, Completely continuous movements in topological vector spaces, Proc. Glasgow Math. Ass. 3 (1957), pp. 135-141.
- [25] M. Nagumo, Degree of mapping in convex linear topological spaces, Amer. J. Math. 73 (1951), pp. 497-511.
- [26] E. H. Rothe, Theory of topological order in some linear topological spaces, Iowa State College J. of Sci. 13 (1039), pp. 373-390.
- [27] J. T. Schwartz, Non-linear functional analysis, Gordon and Beach, New York 1969.
- [28] A. Cellina and A. Lasota, A new approach to the definition of topological degree for multi-valued mappings, Lincei-Rend. Sci. Mat. e Nat. 47 (1969), pp. 434-440.
- [29] Ju. G. Borisovic, B. D. Gelman, E. Muhamadiev and V. V. Obuhovskii, On the relation of multi-valued vector fields, Dokl. Akad. Nauk. SSSR, 187 (1969), pp. 956-959.
- [30] T. W. Ma, Non-singular set-valued compact fields in locally convex spaces (to appear in Fund. Math.).
- [31] M. Hukuhara, Sur l'application semi-continue dont la valeur est un compact convexe, Funkcialaj Ekvacioj, 10 (1967), pp. 43-66.
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