Viability for Differential Inclusions on Graphs

Let X be a Banach space, I a nonempty, bounded interval and let $K:I\leadsto X$ be a given multi-valued function. We rephrase the concept of tangent set introduced by Cârj? et al. (Trans Amer Math Soc, 2008; Viability, Invariance and Applications. North-Holland Mathematics Studies, vol. 207. Elsevier, 2007) by saying that a bounded set $E\subseteq X$ is right tangent to $\liminf_{h\downarrow0}{1}/{h}\,\text{\rm dist}(\xi+hE;K(\tau+h))=0$ . Next, by using a tangency condition expressed in the terms of this concept, we establish several necessary and sufficient conditions for viability referring to differential inclusions on graphs.