Forward-Backward Bounded Solutions to Differential Inclusions

It is shown that some class of differential inclusions has solutions that are defined and bounded for all real values of independent variable. Applications to dynamics are considered. We show that in some sense unstable system with Coulomb friction has solutions defined for all real time and these solutions are bounded in both directions of time. Herewith the Coulomb friction mollifies the system in the direction of positive time and destabilizes it in the negative time direction.