Nonlinear evolution inclusions with one-sided perron right-hand side

In a Banach space X with uniformly convex dual, we study the evolution inclusion of the form $ {x}^{prime}(t)in Ax(t)+Fleft( {x(t)} right) $ , where A is an m-dissipative operator and F is an upper hemicontinuous multifunction with nonempty convex and weakly compact values. If X^* is uniformly convex and F is one-sided Perron with sublinear growth, then, we prove a variant of the well known Filippov-Pli? theorem. Afterward, sufficient conditions for near viability and (strong) invariance of a set $ Ksubseteq overline{D(A)} $ are established. As applications, we derive ε - δ lower semicontinuity of the solution map and, consequently, the propagation of continuity of the minimum time function associated with the null controllability problem.