Download PDF - The coincidence index for fundamentally contractible multivalued maps with nonconvex valuesAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

The coincidence index for fundamentally contractible multivalued maps with nonconvex values

Authors 1

Affiliations

  1. Faculty of Mathematics and Computer Science, Nicholas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

Abstract

We study a coincidence problem of the form A(x) ∈ ϕ (x), where A is a linear Fredholm operator with nonnegative index between Banach spaces and ϕ is a multivalued A-fundamentally contractible map (in particular, it is not necessarily compact). The main tool is a coincidence index, which becomes the well known Leray-Schauder fixed point index when A=id and ϕ is a compact singlevalued map. An application to boundary value problems for differential equations in Banach spaces is given.

Keywords

ENG
condensing map, Fredholm operator, boundary value problem in Banach spaces, fixed point index, coincidence points

Bibliography

  1. R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina and B. N. Sadovskiĭ, Measures of Noncompactness and Condensing Operators, Birkhäuser, Basel, 1992.
  2. S. Armentrout and T. M. Price, Decomposition into compact sets with UV properties, Trans. Amer. Math. Soc. 141 (1969), 433-442.
  3. Yu. G. Borisovich, B. D. Gel'man, A. D. Myshkis and V. V. Obukhovskiĭ, Topological methods in the fixed point theory of multivalued mappings, Russian Math. Surveys 35 (1980), 65-143.
  4. Yu. G. Borisovich, V. G. Zvyagin and Yu. I. Sapronov, Nonlinear Fredholm mappings and Leray-Schauder theory, ibid. 32 (1977), 3-54.
  5. K. Gęba, Algebraic topology methods in the theory of compact fields in Banach spaces, Fund. Math. 46 (1964), 177-209.
  6. K. Gęba, I. Massabò and A. Vignoli, Generalized topological degree and bifurcation, in: Nonlinear Functional Analysis and its Applications (Maratea, 1985), Reidel, 1986, 55-73.
  7. S. Goldberg, Unbounded Linear Operators. Theory and Applications, McGraw-Hill, 1966.
  8. L. Górniewicz, Homological methods in fixed-point theory of multi-valued maps, Dissertationes Math. 129 (1976).
  9. L. Górniewicz and W. Kryszewski, Bifurcation invariants for acyclic mappings, Rep. Math. Phys. 31 (1992), 217-239.
  10. J. Izé, I. Massabò and A. Vignoli, Degree theory for equivariant maps I, Trans. Amer. Math. Soc. 315 (1989), 433-510.
  11. J. L. Kelley, General Topology, Van Nostrand, New York, 1955.
  12. W. Kryszewski, Homotopy properties of set-valued mappings, Wyd. Uniwersytetu Mikołaja Kopernika, Toruń, 1997.
  13. W. Kryszewski, Topological and approximation methods in the degree theory of set-valued maps, Dissertationes Math. 336 (1994).
  14. W. Kryszewski, The fixed-point index for the class of compositions of acyclic set-valued maps on ANR's, Bull. Sci. Math. 120 (1996), 129-151.
  15. W. Kryszewski, Remarks to the Vietoris Theorem, Topol. Methods Nonlinear Anal. 8 (1996), 383-405.
  16. R. C. Lacher, Cell-like mappings and their generalizations, Bull. Amer. Math. Soc. 83 (1977), 336-552.
  17. J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Regional Conf. Ser. in Math. 40, Amer. Math. Soc., Providence, 1979.
  18. T. Pruszko, Some applications of the topological degree theory to multi-valued boundary value problems, Dissertationes Math. 229 (1984).
Annales Polonici Mathematici
2000/75/2
Export to PBN, Opens in new tab
Pages:
143-166
Main language of publication
English
Received
1999-12-22
Accepted
2000-05-31
Published
2000
Exact and natural sciences