Differentiability Properties of Solutions of a Second-Order Evolution Inclusion

Abstract

We study a certain second-order evolution inclusion defined by a family of linear closed operators which is the generator for an evolution system of operators and by a set-valued map with nonconvex values in a separable Banach space. We provide results concerning the differentiability of mild solutions with respect to the initial conditions of the problem considered. Certain variational inclusions are obtained in terms of set-valued derivatives defined by the contingent cone, the quasitangent cone and Clarke’s tangent cone. Our results may be interpreted as extensions to a special class of second-order differential inclusions of the classical Bendixson–Picard–Lindelőf theorem concerning the differentiability of the maximal flow of a differential equation.