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A hybrid fixed-point theorem for set-valued maps

  • Volume 101, Number 6, June, 2017
  • Published:
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Abstract

In 1955, M. A. Krasnosel’skii 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.

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Authors and Affiliations

  1. Voronezh State University, Voronezh, Russia

    B. D. Gel’man

  2. Peoples’ Friendship University of Russia, Moscow, Russia

    B. D. Gel’man

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  1. B. D. Gel’man
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Correspondence to B. D. Gel’man.

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Original Russian Text © B. D. Gel’man, 2017, published in Matematicheskie Zametki, 2017, Vol. 101, No. 6, pp. 832–842.

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Gel’man, B.D. A hybrid fixed-point theorem for set-valued maps. Math Notes 101, 951–959 (2017). https://doi.org/10.1134/S0001434617050212

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  • DOI: https://doi.org/10.1134/S0001434617050212

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