On the solvability of systems of inclusions involving noncompact operators

We consider the solvability of a system [GRAPHICS] of set-valued maps in two different caws. In the first one, the map (x, y)-o F(x, y)BAR is supposed to be closed graph with convex values and condensing in the second variable and (x, y)-o G(x, y)BAR is supposed to be a permissible map (i.e. composition of an upper semicontinuous map with acyclic values and a continuous, single-valued map), satisfying a condensivity condition in the first variable. In the second case FBAR is as before with compact, not necessarily convex, values and GBAR is an admissible map (i.e. it is composition of upper semicontinuous acyclic maps). In the latter case, in order to apply a fixed point theorem for admissible maps, we have to assume that the solution set x-o S(x) of the first equation is acyclic. Two examples of applications of the abstract results are given. The first is a control problem for a neutral functional differential equation on a finite time interval; the second one deals with a semilinear differential inclusion in a Banach space and sufficient conditions are given to show that it has periodic solutions of a prescribed period.