In this paper we study the existence of solutions for a system of upper semicontinuous, non necessarily convex, multivalued maps defined in abstract spaces. To this aim, we investigate the properties of the solution set of a multivalued equation. In particular we give conditions assuring the admissibility or the acyclicity of this map. The main tool is a technique to investigate fixed points of fiber-preserving maps. Some applications to boundary value problems for multivalued differential equations and for delay equations, are given.