where f E C(I x R”, R”), S c C(Z, R”), consists of finding fixed points of a solution map E for a suitable linearized problem associated to (BV) (see e.g. [2, 5, 6, 71). In these problems the map X is either continuous and single-valued or upper semicontinuous set-valued with convex values. These are rather strong restrictions because the set of solutions of a general boundary value problem may well be neither a singleton nor a convex set. The case in which the solution map is upper semicontinuous but with nonnecessarily convex values is considered in this paper. More precisely we want to consider the case when the values of X form an acyclic set and the case when the values of C are sets with a finite number of elements. An application to the existence of periodic solutions for a LiCnard type equation is also given. Moreover an existence theorem for a boundary value problem of the following form: = f(l, x, x’, x”), teIcR, x E R”