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

Abstract

In a Banach space X with uniformly convex dual, we study the evolution inclusion of the form ${x}^{\prime}(t)\in Ax(t)+F\left( {x(t)} \right)$, 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-Pliś theorem. Afterward, sufficient conditions for near viability and (strong) invariance of a set $K\subseteq \overline{D(A)}$ 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.