Abstract and Applied Analysis

Abstract and Applied Analysis / 2006 / Article
Special Issue

Topological and variational methods of nonlinear analysis and their applications

View this Special Issue

Open Access

Volume 2006 |Article ID 086173 | https://doi.org/10.1155/AAA/2006/86173

A. V. Guminskaya, P. P. Zabreiko, "On calculation of the relative index of a fixed point in the nondegenerate case", Abstract and Applied Analysis, vol. 2006, Article ID 086173, 11 pages, 2006. https://doi.org/10.1155/AAA/2006/86173

On calculation of the relative index of a fixed point in the nondegenerate case

Received26 Jun 2005
Accepted01 Jul 2005
Published11 Apr 2006

Abstract

The paper is devoted to the calculation of the index of a zero and the asymptotic index of a linear completely continuous nonnegative operator. Also the case of a nonlinear completely continuous operator A whose domain and image are situated in a closed convex set Q of a Banach space is considered. For this case, we formulate the rules for calculating the index of an arbitrary fixed point and the asymptotic index under the assumption that the corresponding linearizations exist and the operators of derivative do not have eigenvectors with eigenvalue 1 in some wedges.

References

  1. E. N. Dancer, “On the indices of fixed points of mappings in cones and applications,” Journal of Mathematical Analysis and Applications, vol. 91, no. 1, pp. 131–151, 1983. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  2. E. N. Dancer, “Positivity of maps and applications,” in Topological Nonlinear Analysis: Degree, Singularity, and Variations, vol. 15 of Progr. Nonlinear Differential Equations Appl., pp. 303–340, Birkhäuser Boston, Massachusetts, 1995. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  3. Y. Y. Isaenko, “The fixed point index of linear positive operators,” Trudy Aspirantov Voronezhskogo Universiteta (Proceedings of Postgraduate Students of Voronezh State University), pp. 54–58, 1969 (Russian). View at: Google Scholar
  4. M. A. Krasnosel'skiĭ and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Nauka, Moscow, 1975, translated into English: Springer, Berlin, 1984. View at: Google Scholar
  5. E. M. Mukhamadiev and T. Sabirov, “On the computation of the relative index of the zero of a vector field,” Trudy Seminara po Funktional'nomu Analizn (Proceedings of Seminar on Functional Analysis), vol. 12, pp. 116–121, 1969 (Russian). View at: Google Scholar | Zentralblatt MATH
  6. Y. V. Pokornyi, “Relative indices of positive operators,” Trudy Matematicheskogo Fakul'teta Voronezhskogo Universiteta (Proceedings of Department of Mathematics of Voronezh State University), vol. 4, pp. 79–89, 1970 (Russian). View at: Google Scholar | MathSciNet
  7. Y. V. Pokornyi and I. N. Astaf'eva, “On the index at zero and at infinity for positive nonlinear operators,” Uspehi Natematicheskih Nauk, vol. 26, no. 3, p. 189, 1971 (Russian). View at: Google Scholar
  8. P. P. Zabrejko, “Rotation of vector fields: definition, basic properties, and calculation,” in Topological Nonlinear Analysis, II: Degree, Singularity, and Variations (Frascati, 1995), vol. 27 of Progr. Nonlinear Differential Equations Appl., pp. 445–601, Birkhäuser Boston, Massachusetts, 1997. View at: Google Scholar | Zentralblatt MATH | MathSciNet

Copyright © 2006 A. V. Guminskaya and P. P. Zabreiko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

 PDF Download Citation Citation
 Order printed copiesOrder
Views118
Downloads480
Citations

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.