On the implicit and inverse many-valued functions in topological spaces

E. S. Zhukovskiy ; Zh. Munembe

Vestnik rossijskih universitetov. Matematika, Tome 24 (2019) no. 128, pp. 384-392

Résumé

The conditions of continuity of the implicit set-valued map and the inverse set-valued map acting in topological spaces are proposed. For given mappings $ f: T \times X \to Y, $ $ y: T \to Y, $ where $ T, X, Y $ are topological spaces, the space $ Y $ is Hausdorff, the equation $$ f (t , x) = y (t) $$ with the parameter $ t \in T $ relative to the unknown $ x \in X $ is considered. It is assumed that for some multi-valued map $ U: T \rightrightarrows X $ for all $ t \in T $ the inclusion $ f (t, U (t)) \ni y (t)$ is satisfied. An implicit mapping $ \mathfrak {R} _U: T \rightrightarrows X, $ which associates with each value of the parameter $ t \in T $ the set of solutions $ x (t) \in U (t) $ of this equation. It is proved that $ \mathfrak {R} _U $ is upper semicontinuous at the point $ t_0 \in T, $ if the following conditions are satisfied: for any $ x \in X $ the map $ f $ is continuous at $ (t_0, x), $ the map $ y $ is continuous at $ t_0, $ a multi-valued map $ U $ is upper semicontinuous at the point $ t_0 $ and the set $ U (t_0) \subset X $ is compact. If, in addition, with the value of the parameter $ t_0 $, the solution to the equation is unique, then the map $ \mathfrak {R} _U $ is continuous at $ t_0 $ and any section of this map is also continuous at $ t_0. $ The listed results are applied to the study of a multi-valued inverse mapping. Namely, for a given map $ g: X \to T $ we consider the equation $ g (x) = y $ with respect to the unknown $ x \in X. $ We obtain conditions for upper semicontinuity and continuity of the map $ \mathfrak {V} _U: T \rightrightarrows X, $ $ \mathfrak {V} _U (t) = \{x \in U (t): \, g (x) = t \}, $ $ t \in T. $

Détail

Keywords: implicit function; inverse function; multi-valued mapping; upper semicontinuity; parameter.