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Matematicheskie Zametki, 2017, Volume 101, Issue 6, Pages 832–842
DOI: https://doi.org/10.4213/mzm10742
(Mi mzm10742)
 

This article is cited in 1 scientific paper (total in 1 paper)

A Hybrid Fixed-Point Theorem for Set-Valued Maps

B. D. Gel'manab

a Voronezh State University
b Peoples Friendship University of Russia, Moscow
Full-text PDF (479 kB) Citations (1)
References:
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.
Received: 30.03.2015
Revised: 15.11.2015
English version:
Mathematical Notes, 2017, Volume 101, Issue 6, Pages 951–959
DOI: https://doi.org/10.1134/S0001434617050212
Bibliographic databases:
Document Type: Article
UDC: 517.988.6
Language: Russian
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
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  • https://doi.org/10.4213/mzm10742
  • https://www.mathnet.ru/eng/mzm/v101/i6/p832
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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