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

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 $\mathfrak {V} _U: T \rightrightarrows X,$ $\mathfrak {V} _U (t) = \{x \in U (t): \, g (x) = t \},$ $t \in T.$