Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Remarks on approximation methods in degree theory


Authors: W. Kryszewski, B. Przeradzki and S. Wereński
Journal: Trans. Amer. Math. Soc. 316 (1989), 97-114
MSC: Primary 47H09; Secondary 58B05, 58C99
DOI: https://doi.org/10.1090/S0002-9947-1989-0929237-X
MathSciNet review: 929237
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An approximative approach to a generalized theory of the topological mapping degree is presented. Some new wide classes of operators acting in Banach spaces, which include $ A$-proper mappings of Petryshyn, are introduced and studied from the viewpoint of the homotopic properties of the topological degree. The results are applied in some existence aspects of abstract nonlinear equations.


References [Enhancements On Off] (What's this?)

  • [1] L. E. J. Brouwer, Über Abbildung von Mannigfaltigkeiten, Math. Ann. 71 (1912), 97-115. MR 1511644
  • [2] F. E. Browder, The degree of mapping, and its generalizations, Contemp. Math. 21 (1983), 15-40. MR 729503 (85e:47086)
  • [3] F. E. Browder and W. V. Petryshyn, The topological degree and Galerkin approximation for noncompact operators in Banach spaces, Bull. Amer. Math. Soc. 74 (1968), 641-646. MR 0229100 (37:4678)
  • [4] -, Approximation methods and the generalized topological degree for nonlinear mappings in Banach spaces, J. Funct. Anal. 3 (1969), 217-245. MR 0244812 (39:6126)
  • [5] J. Dugundji and A. Granas, Fixed point theory, PWN, Warszawa, 1982. MR 660439 (83j:54038)
  • [6] A. I. Fet, Generalization of a theorem of Lusternik-Schnirelman of coverings of spheres and some theorems connected with it, Dokl. Akad. Nauk SSSR 95 (1954), 1149-1151. MR 0063034 (16:61b)
  • [7] P. M. Fitzpatrick, A generalized degree for uniform limits of $ A$-proper mappings, J. Math. Anal. Appl. 35 (1971), 536-552. MR 0281069 (43:6788)
  • [8] J. W. Jaworowski, Involutions of compact spaces and a generalization of Borsuk's theorem on antipodes, Bull. Acad. Polon. Sci. 3 (1955), 289-292. MR 0074826 (17:653e)
  • [9] W. Kryszewski, Some remarks on the nonlinear eigenvalue problems of Birkhoff-Kellogg type, Bull. Acad. Polon. Sci. 32 (1984), 455-462. MR 782762 (86g:47077)
  • [10] -, An approximation method in the theory of nonlinear noncompact operators, C.R. Semi. Math. Sup. Col. A. Granas, Les Presses de l'Université de Montréal, 1986.
  • [11] W. Kryszewski and B. Przeradzki, The topological degree and fixed points of $ DC$-mappings, Fund. Math. 126 (1984). MR 817077 (87e:47076)
  • [12] -, A new approximation method in the theory of nonlinear noncompact operators, Proc. Conf. Classical Analysis, Kazimierz Dolny, 1983.
  • [13] J. Leray and J. Schauder, Topologie et équations fonctionelles, Ann. Sci. École Norm. Sup. 51 (1934), 45-78. MR 1509338
  • [14] N. G. Lloyd, Degree theory, Cambridge Univ. Press, Cambridge, 1978.
  • [15] T. Nirenberg, Topics in nonlinear functional analysis, New York Univ. Press, New York, 1974. MR 0488102 (58:7672)
  • [16] B. Nowak, $ DJ$-odwzorowania i ich homotopie, Acta Univ. Lodziensis, Lódź, 1981.
  • [17] R. D. Nussbaum, Degree theory for local condensing maps, J. Math. Anal. Appl. 37 (1972), 741-766. MR 0306986 (46:6107)
  • [18] W. V. Petryshyn, On the approximation solvability of equations involving $ A$-proper and pseudo $ A$-proper mappings, Bull. Amer. Math. Soc. 81 (1975), 223-312. MR 0388173 (52:9010)
  • [19] -, On the approximation-solvability of nonlinear equations, Math. Ann. 177 (1968), 156-164. MR 0226458 (37:2048)
  • [20] B. Przeradzki, On the homotopical classification of $ DJ$-mappings of infinitely-dimensional spheres, Fund. Math. 120 (1984), 41-49. MR 773788 (86d:55015)
  • [21] B. N. Sadowski, Ultimately compact and condensing operators (Russian), Uspehi Mat. Nauk 27 (1971), 81-146.
  • [22] I. V. Skripnik, Nonlinear elliptic equations of higher order, "Naukova Dumka", Kiev, 1973. MR 0355330 (50:7805)
  • [23] S. Wereński, On the fixed point index of noncompact mappings, Studia Math. 78 (1984), 155-160. MR 766710 (86b:47102)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 47H09, 58B05, 58C99

Retrieve articles in all journals with MSC: 47H09, 58B05, 58C99


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0929237-X
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society