PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
1995 | 15 | 2 | 163-185
Tytuł artykułu

Fixed points of set-valued maps with closed proximally ∞-connected values

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Introduction
Many authors have developed the topological degree theory and the fixed point theory for set-valued maps using homological techniques (see for example [19, 28, 27, 16]).
Lately, an elementary technique of single-valued approximation (on the graph) (see [11, 1, 13, 5, 9, 2, 6, 7]) has been used in constructing the fixed point index for set-valued maps with compact values (see [21, 20, 4]).
In [20, 4] authors consider set-valued upper semicontinuous maps with compact proximally ∞-connected values. Making use of their ideas we prove new continuous approximation theorems and present a topological degree theory for u.s.c. maps with closed proximally ∞-connected values in Euclidean spaces. Thus we generalize the earlier known results (see [22, 12, 23]).
One of the main fact which permits us to construct the topological degree is the bijection (Theorem 3.6, Section 3). In Section 4 we define the class of set-valued maps appropriate to a topological degree theory (that is, for which the bijection theorem holds). For this class the definition of a degree can be reduced to the single-valued case (to the Brouwer degree). Some consequences of approximation methods and some remarks are given (Section 5).
Finally, let us note that without the assumption about compactness of values we obtain a large class of set-valued maps. We show (Section 4) that, for example, it contains all u.s.c. set-valued maps φ:P∪Y with closed contractible values, where P is a finite polyhedron and Y is an ANR-space.
Słowa kluczowe
Twórcy
  • Department of Mathematics and Informatics, Nicolas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
Bibliografia
  • [1] G. Anichini, G. Conti, P. Zecca, Approximation of nonconvex set valued mappings, Boll. Un. Mat. Ital. serie VI 1 (1985), 145-153.
  • [2] G. Anichini, G. Conti, P. Zecca, Approximation and selection for nonconvex multifunctions in infinite dimensional spaces, Boll. Un. Mat. Ital. 4-B (7) (1990), 411-422.
  • [3] R. Bader, G. Gabor, W. Kryszewski, On the extension of approximations for set-valued maps and the repulsive fixed points, Boll. Un. Mat. Ital.,(to appear).
  • [4] R. Bader, W. Kryszewski, Fixed point index for compositions of set-valued maps with proximally ∞-connected values on arbitrary ANR's, Set-Valued Analysis 2 (1994), 459-480.
  • [5] G. Beer, On a theorem of Cellina for set valued functions, Rocky Mountain J. Math. 18 (1988), 37-47.
  • [6] H. Ben-El-Mechaiekh, Approximation of nonconvex set-valued maps, C. R. Acad. Sci. Paris, Serie I Math. 312 (1991), 379-384.
  • [7] H. Ben-El-Mechaiekh, P. Deguire, Approchiability and fixed points for nonconvex set-valued maps, J. Math. Anal. Appl. 170 (1992), 477-500.
  • [8] K. Borsuk, Theory of Retracts, PWN, Warszawa 1967.
  • [9] A. Bressan, G. Colombo, Extentions and selections of maps with decomposable values, Studia Math. Univ. South Africa, Pretoria 40 (1988), 69-86.
  • [10] R. Brown, The Lefschetz Fixed Point Theorem, London 1971.
  • [11] A. Cellina, A theorem on the approximation of compact multivalued mappings, Atti. Accad. Naz. Lincei Rend. 8 (1969), 149-153.
  • [12] A. Cellina, A. Lasota, A new approach to the definition of topological degree for multi-valued mappings, Atti. Accad. Naz. Lincei. Rend. 6 (1970), 434-440.
  • [13] F.S. de Blazi, J. Myjak, On continuous approximation for multi-functions, Pacific J. Math. 1 (1986), 9-30.
  • [14] J. Dugundji, Modified Vietoris theorems for homotopy, Fund. Math. 66 (1970), 223-235.
  • [15] J. Dugundji, A. Granas, Fixed point theory, Vol.I, Monografie Matematyczne, 61, Warszawa 1982.
  • [16] Z. Dzedzej, Fixed point index theory for a class of nonacyclic multivalued mappings, Dissertationes Math. (Rozprawy Mat.) 253 (1985), 1-58.
  • [17] M.K. Fort, Essential and nonessential fixed points, Amer. J. Math. 72 (1950), 315-322.
  • [18] G. Gabor, On the classifications of fixed points, Math. Japon. 40 (2) (1994), 361-369.
  • [19] L. Górniewicz, Homological methods in fixed point theory of multi-valued mappings, Dissertationes Math. (Rozprawy Mat.) 126 (1976), 1-71.
  • [20] L. Górniewicz, A. Granas, W. Kryszewski, On the homotopy method in the fixed point index theory of multi-valued mappings of compact absolute neighbourhood retracts, J. Math. Anal. Appl. 161 (1991), 457-473.
  • [21] L. Górniewicz, A. Granas, W. Kryszewski, Sur la méthode de l'homotopie dans la théorie des points fixes pour les applications multivoques; Partie 1: Transversalité topologique, C. R. Acad. Sci. Paris 307 (1988), 489-492; Partie 2: L'indice pour les ANR-s compactes, C. R. Acad. Sci. Paris. 308 (1989), 449-452.
  • [22] A. Granas, Sur la notion du degrée topologique pour une certaine classe de transformations multivalentes dans les espaces de Banach, Bull. Acad. Polon. Sci. 7 (1959), 271-275.
  • [23] J.-M. Lasry, R. Robert, Degré pour les fonctions multivoques et applications, C. R. Acad. Sc. Paris 280 (1975), 1435-1438.
  • [24] A. McLennan, Fixed Points of Contractible Valued Correspondences, International Journal of Game Theory 18 (1989), 175-184.
  • [25] B. O'Neill, Essential sets and fixed points, Amer. J. Math. 75 (1953), 497-509.
  • [26] H.O. Peitgen, Asymptotic fixed point theorems and stability, J. Math. Anal. Appl. 47 (1974), 32-42.
  • [27] G. Skordev, Fixed point index for open sets in Euclidean spaces, Fund. Math. 121 (1984), 41-58.
  • [28] H.W. Siegberg, G. Skordev, Fixed point index and chain approximations, Pacific J. Math. 102 (1982), 455-486.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-div15i2n4bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.