Continuous selection of the solution map for one-sided Lipschitz differential inclusions

In the paper we deal with differential inclusions with one-sided Lipschitz (OSL) continuous right-hand sides, and prove the existence of a continuous selection of the solution map which assigns to any point the set of solutions to the multivalued Cauchy problem. As an application we study the problem of the existence of viable trajectories in a prescribed closed subset of a Banach space via the generalized Ważewski retract method.