Acyclicity of the solution set of two-point boundary value problems for second order multivalued differential equations

Abstract

The topological and geometrical structure of the set of solutions of two-point boundary value problems for second order differential inclusions in Banach spaces is investigated. It is shown that under the Carath\'eodory-type assumptions the solution set of the periodic boundary value problem is nonempty compact acyclic in the space of continuously differentiable functions as well as in the Bochner-Sobolev space $\mathbb{H}^2$ endowed with the weak topology. The proof relies heavily on the accretivity of the right-hand side of differential inclusion. The Lipschitz case is treated separately. As one might expect the solution set is, in this case, an absolute retract.