Skip to main content
Log in

On the Relative Fixed Point Index for a Class of Noncompact Multivalued Maps

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

We define a topological characteristic, the fixed point index with respect to a convex closed subset of a Banach space for a class of completely fundamentally restrictible multivalued maps which can be represented as a composition of maps with aspheric values. This class includes, in particular, maps which are condensing with respect to a monotone nonsingular measure of noncompactness. Maps of this type naturally arise in the study of nonlinear systems with impulse effects. Applications of the index to some fixed point theorems are considered.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

REFERENCES

  1. Borisovič, Ju.G. "An Application of the Concept of Rotation of a Vector Field", Dokl. Akad. Nauk SSSR 153 (1), 12-15 (1963).

    MathSciNet  Google Scholar 

  2. Borisovič, Ju.G. "On the Relative Rotation of Compact Vector Fields in Linear Spaces", Works of Seminar on Functional Analysis, Voronezhskii Universitet 12, 3-27 (1969).

    Google Scholar 

  3. Borisovich, Ju.G., Sapronov, Yu.I. "On the Topology Theory of Compactly Restrictible Mappings", Works of Seminar on Functional Analysis, Voronezhskii Universitet 12, 43-68 (1969).

    Google Scholar 

  4. Borisovič, Ju.G., Gel'man, B.D., Myškis, A.D., Obuhovski˘i, V.V. "Limit-compact and Condensing Operators", Uspekhi Mat. Nauk 27 (1(163)), 81-146 (1972).

    MathSciNet  Google Scholar 

  5. Borisovič, Ju.G., Gel'man, B.D., Myškis, A.D., Obuhovski˘i, V.V. “Topological Methods in the Theory of Fixed Points of Multivalued Mappings”, Uspekhi Mat. Nauk 35 (1), 59–126, 255 (1980).

  6. Kamenskii, M., Obukhovskii, V., Zecca, P. Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, de Gruyter Ser. in Nonlinear Anal. and Appl. 7 (Walter de Gruyter & Co., Berlin–New York, 2001 ).

    Book  Google Scholar 

  7. Liou, Y.C., Obukhovskii, V., Yao, J.C. "Application of a Coincidence Index to Some Classes of Impulsive Control Systems", Nonlinear Anal. 69 (12), 4392-4411 (2008).

    Article  MathSciNet  Google Scholar 

  8. Obukhovskii, V.V., Kornev, S.V., Getmanova, E.N. "On Topological Characteristics for Some Classes of Multivalued Mappings", Chebyshevski˘i Sb. 21 (2), 301-319 (2020).

    Article  MathSciNet  Google Scholar 

  9. Bader, R., Kryszewski, W. "Fixed-point Index for Compositions of Set-valued Maps with Proximally \(\infty\)-connected Values on Arbitrary ANR's", Set-Valued Anal. 2, 459-480 (1994).

    Article  MathSciNet  Google Scholar 

  10. Borisovich, Yu.G., Gel'man, B.D., Myshkis, A.D., Obukhovski˘i, V.V. Introduction to the Theory of Multivalued Mappings and Differential Inclusions, 2nd edition (LIBROKOM, Moscow, 2011 ) [in Russian].

    Google Scholar 

  11. Górniewicz, L. Topological Fixed Point Theory of Multivalued Mappings, 2nd edition, Topological Fixed Point Theory and Its Appl. 4 (Springer, Dordrecht, 2006 ).

    MATH  Google Scholar 

  12. Hu, S., Papageorgiou, N. Handbook of Multivalued Analysis. Vol. I. Theory (Kluwer, Dordrecht, 1997 ).

    Book  Google Scholar 

  13. Myškis, A.D. "Generalizations of the Theorem on a Fixed Point of a Dynamical System Inside of a Closed Trajectory", Mat. Sbornik N.S. 34 (76), 525-540 (1954).

    MathSciNet  Google Scholar 

  14. Borsuk, K. Theory of Retracts (Państwowe Wydawn. Naukowe, 1967).

  15. Girolo, J. "Approximating Compact Sets in Normed Linear Spaces", Pacific J. Math. 98, 81-89 (1982).

    Article  MathSciNet  Google Scholar 

  16. Hyman, D.M. "On Decreasing Sequences of Compact Absolute Retracts", Fund. Math. 64, 91-97 (1969).

    Article  MathSciNet  Google Scholar 

  17. Górniewicz, L., Granas, A., Kryszewski, W. "On the Homotopy Method in the Fixed Point Index Theory of Multi-valued Mappings of Compact Absolute Neighborhood Retracts", J. Math. Anal. Appl. 161, 457-473 (1991).

    Article  MathSciNet  Google Scholar 

Download references

Funding

The work of E.N. Getmanova and S.V. Kornev was financially supported by the Ministry of Science and Higher Education of the Russian Federation within the framework of a state assignment in the field of science (project no. FZGF-2020-0009). The results in paragraph 4 were obtained by V.V. Obukhovsky with the support of the Russian Science Foundation (project no. 20-11-20131) at the Institute for Control Problems named after V.A. Trapeznikov of RAS.

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to V. V. Obukhovskii, S. V. Kornev or E. N. Getmanova.

Additional information

Dedicated to the 90th anniversary of the birth of Yuri Grigorievich Borisovich

Russian Text © The Author(s), 2021, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2021, No. 5, pp. 64–77.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Obukhovskii, V.V., Kornev, S.V. & Getmanova, E.N. On the Relative Fixed Point Index for a Class of Noncompact Multivalued Maps. Russ Math. 65, 48–59 (2021). https://doi.org/10.3103/S1066369X2105008X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X2105008X

Keywords

Navigation