RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Subscription
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Mat. Zametki:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Mat. Zametki, 2017, Volume 101, Issue 6, Pages 832–842 (Mi mz10742)  

A Hybrid Fixed-Point Theorem for Set-Valued Maps

B. D. Gel'manab

a Voronezh State University
b Peoples Friendship University of Russia, Moscow

Abstract: In 1955, M. A. Krasnoselskii proved a fixed-point theorem for a single-valued map which is a completely continuous contraction (a hybrid theorem). Subsequently, his work was continued in various directions. In particular, it has stimulated the development of the theory of condensing maps (both single-valued and set-valued); the images of such maps are always compact. Various versions of hybrid theorems for set-valued maps with noncompact images have also been proved. The set-valued contraction in these versions was assumed to have closed images and the completely continuous perturbation, to be lower semicontinuous (in a certain sense). In this paper, a new hybrid fixed-point theorem is proved for any set-valued map which is the sum of a set-valued contraction and a compact set-valued map in the case where the compact set-valued perturbation is upper semicontinuous and pseudoacyclic. In conclusion, this hybrid theorem is used to study the solvability of operator inclusions for a new class of operators containing all surjective operators. The obtained result is applied to solve the solvability problem for a certain class of control systems determined by a singular differential equation with feedback.

Keywords: set-valued map, Hausdorff metric, contraction, surjective operator, operator inclusion.

Funding Agency Grant Number
Russian Foundation for Basic Research 17-11-01168
This work was supported by the Russian Science Foundation under grant 17-11-01168.


DOI: https://doi.org/10.4213/mzm10742

Full text: PDF file (479 kB)
First page: PDF file
References: PDF file   HTML file

English version:
Mathematical Notes, 2017, 101:6, 951–959

Bibliographic databases:

UDC: 517.988.6
Received: 30.03.2015
Revised: 15.11.2015

Citation: B. D. Gel'man, “A Hybrid Fixed-Point Theorem for Set-Valued Maps”, Mat. Zametki, 101:6 (2017), 832–842; Math. Notes, 101:6 (2017), 951–959

Citation in format AMSBIB
\Bibitem{Gel17}
\by B.~D.~Gel'man
\paper A Hybrid Fixed-Point Theorem for Set-Valued Maps
\jour Mat. Zametki
\yr 2017
\vol 101
\issue 6
\pages 832--842
\mathnet{http://mi.mathnet.ru/mz10742}
\crossref{https://doi.org/10.4213/mzm10742}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3659555}
\elib{http://elibrary.ru/item.asp?id=29255095}
\transl
\jour Math. Notes
\yr 2017
\vol 101
\issue 6
\pages 951--959
\crossref{https://doi.org/10.1134/S0001434617050212}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000404236900021}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85021286721}


Linking options:
  • http://mi.mathnet.ru/eng/mz10742
  • https://doi.org/10.4213/mzm10742
  • http://mi.mathnet.ru/eng/mz/v101/i6/p832

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Математические заметки Mathematical Notes
    Number of views:
    This page:229
    References:37
    First page:27

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020