A Hybrid Fixed-Point Theorem for Set-Valued Maps
B. D. Gel'manab
a Voronezh State University
b Peoples Friendship University of Russia, Moscow
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.
set-valued map, Hausdorff metric, contraction, surjective operator, operator inclusion.
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Mathematical Notes, 2017, 101:6, 951–959
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
\paper A Hybrid Fixed-Point Theorem for Set-Valued Maps
\jour Mat. Zametki
\jour Math. Notes
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