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 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

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

Bibliographic databases:

UDC: 517.988.6
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
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