Continuous selections in asymmetric spaces

In asymmetric seminormed and semimetric spaces, sets admitting a continuous $\varepsilon$-selection for any $\varepsilon>0$ are studied. A characterization of closed subsets of a complete symmetrizable asymmetric seminormed space admitting a continuous $\varepsilon$-selection for all $\varepsilon>0$ is obtained. Sufficient conditions for the existence of continuous selections in seminormed linear spaces and semimetric semilinear spaces are put forward. Applications to generalized rational functions in the asymmetric space of continuous functions and in the semilinear space $\mathbf{L}_h$ of all boundedly compact convex sets with Hausdorff metric are found. A metric-topological fixed point theorem for a stable set-valued mapping in the space $\mathbf{L}_h$ is obtained.
Bibliography: 17 titles.