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

Grzegorz Gabor

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (1995)

  • Volume: 15, Issue: 2, page 163-185
  • ISSN: 1509-9407

Abstract

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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.

How to cite

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Grzegorz Gabor. "Fixed points of set-valued maps with closed proximally ∞-connected values." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 15.2 (1995): 163-185. <http://eudml.org/doc/275854>.

@article{GrzegorzGabor1995,
abstract = { 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. },
author = {Grzegorz Gabor},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
language = {eng},
number = {2},
pages = {163-185},
title = {Fixed points of set-valued maps with closed proximally ∞-connected values},
url = {http://eudml.org/doc/275854},
volume = {15},
year = {1995},
}

TY - JOUR
AU - Grzegorz Gabor
TI - Fixed points of set-valued maps with closed proximally ∞-connected values
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 1995
VL - 15
IS - 2
SP - 163
EP - 185
AB - 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.
LA - eng
UR - http://eudml.org/doc/275854
ER -

References

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  1. [1] G. Anichini, G. Conti, P. Zecca, Approximation of nonconvex set valued mappings, Boll. Un. Mat. Ital. serie VI 1 (1985), 145-153. Zbl0591.54013
  2. [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. Zbl0717.47021
  3. [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). Zbl0848.54013
  4. [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. Zbl0846.55001
  5. [5] G. Beer, On a theorem of Cellina for set valued functions, Rocky Mountain J. Math. 18 (1988), 37-47. Zbl0648.54015
  6. [6] H. Ben-El-Mechaiekh, Approximation of nonconvex set-valued maps, C. R. Acad. Sci. Paris, Serie I Math. 312 (1991), 379-384. Zbl0719.54053
  7. [7] H. Ben-El-Mechaiekh, P. Deguire, Approchiability and fixed points for nonconvex set-valued maps, J. Math. Anal. Appl. 170 (1992), 477-500. Zbl0762.54033
  8. [8] K. Borsuk, Theory of Retracts, PWN, Warszawa 1967. Zbl0153.52905
  9. [9] A. Bressan, G. Colombo, Extentions and selections of maps with decomposable values, Studia Math. Univ. South Africa, Pretoria 40 (1988), 69-86. Zbl0677.54013
  10. [10] R. Brown, The Lefschetz Fixed Point Theorem, London 1971. Zbl0216.19601
  11. [11] A. Cellina, A theorem on the approximation of compact multivalued mappings, Atti. Accad. Naz. Lincei Rend. 8 (1969), 149-153. 
  12. [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. Zbl0194.44801
  13. [13] F.S. de Blazi, J. Myjak, On continuous approximation for multi-functions, Pacific J. Math. 1 (1986), 9-30. 
  14. [14] J. Dugundji, Modified Vietoris theorems for homotopy, Fund. Math. 66 (1970), 223-235. Zbl0196.26801
  15. [15] J. Dugundji, A. Granas, Fixed point theory, Vol.I, Monografie Matematyczne, 61, Warszawa 1982. Zbl0483.47038
  16. [16] Z. Dzedzej, Fixed point index theory for a class of nonacyclic multivalued mappings, Dissertationes Math. (Rozprawy Mat.) 253 (1985), 1-58. 
  17. [17] M.K. Fort, Essential and nonessential fixed points, Amer. J. Math. 72 (1950), 315-322. Zbl0036.13001
  18. [18] G. Gabor, On the classifications of fixed points, Math. Japon. 40 (2) (1994), 361-369. Zbl0812.54047
  19. [19] L. Górniewicz, Homological methods in fixed point theory of multi-valued mappings, Dissertationes Math. (Rozprawy Mat.) 126 (1976), 1-71. Zbl0324.55002
  20. [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. Zbl0757.54019
  21. [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. Zbl0665.54030
  22. [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. [23] J.-M. Lasry, R. Robert, Degré pour les fonctions multivoques et applications, C. R. Acad. Sc. Paris 280 (1975), 1435-1438. Zbl0307.55009
  24. [24] A. McLennan, Fixed Points of Contractible Valued Correspondences, International Journal of Game Theory 18 (1989), 175-184. Zbl0675.54041
  25. [25] B. O'Neill, Essential sets and fixed points, Amer. J. Math. 75 (1953), 497-509. Zbl0050.39202
  26. [26] H.O. Peitgen, Asymptotic fixed point theorems and stability, J. Math. Anal. Appl. 47 (1974), 32-42. Zbl0284.54027
  27. [27] G. Skordev, Fixed point index for open sets in Euclidean spaces, Fund. Math. 121 (1984), 41-58. Zbl0561.55001
  28. [28] H.W. Siegberg, G. Skordev, Fixed point index and chain approximations, Pacific J. Math. 102 (1982), 455-486. Zbl0458.55001

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