On the existence problem for a fixed point of a generalized contracting multivalued mapping

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Abstract

We discuss the still unresolved question, posed in [S.~Reich, Some Fixed Point Problems, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 57:8 (1974), 194--198], of existence in a complete metric space $X$ of a fixed point for a generalized contracting multivalued map $\Phi: X \rightrightarrows X $ having closed values $ \Phi (x) \subset X$ for all $ x \in X. $ Generalized contraction is understood as a natural extension of the Browder--Krasnoselsky definition of this property to multivalued maps:
 \begin{equation*}
\forall x, u \in X \ \ h \bigl(\varphi(x), \varphi(u) \bigr) \leq \eta \bigl(\rho(x, u) \bigr),
 \end{equation*}
 where the function $ \eta: \mathbb {R}_+\to\mathbb{R}_+$ is increasing, right continuous, and for all $d>0,$\linebreak $\eta(d)

About the authors

Evgeny S. Zhukovskiy

Derzhavin Tambov State University; V. A. Trapeznikov Institute of Control Sciences of RAS

Author for correspondence.
Email: zukovskys@mail.ru
ORCID iD: 0000-0003-4460-7608

Doctor of Physics and Mathematics, Professor, Director of the Research Institute of Mathematics, Physics and Informatics; Leading Researche

Russian Federation, 33 Internatsionalnaya St., Tambov 392000, Russian Federation; 65 Profsoyuznaya St., Moscow 117997, Russian Federation

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