Decompositions of set-valued mappings

Let $X$ be a set, $B_{X}$ denotes the family of all subsets of $X$ and $F\colon X \to B_{X}$ be a set-valued mapping such that $x \in F(x)$, $\sup_{x\in X} |F(x)|< \kappa$, $\sup_{x\in X} |F^{-1}(x)|< \kappa$ for all $x\in X$ and some infinite cardinal $\kappa$. Then there exists a family $\mathcal{F}$ of bijective selectors of $F$ such that $|\mathcal{F}|<\kappa$ and $F(x) = \{ f(x)\colon f\in\mathcal{F}\}$ for each $x\in X$. We apply this result to $G$-space representations of balleans.