Generalized Differential Inclusions in Banach Spaces

Abstract

We study a new type of solutions to differential inclusions in Banach spaces, which we call directional solutions. The idea is based on the observation that for a differentiable function \(u\) and a closed set \(V\) $$u\prime {\left( t \right)} \in V\,{\text{iff}}\,{\mathop {\lim }\limits_{h \to 0} }d{\left( {\frac{{u{\left( {t + h} \right)} - u{\left( t \right)}}}{h},V} \right)} = 0.$$