PERTURBATION OF A CONVEX-VALUED OPERATOR BY A SET-VALUED MAP OF HAMMERSTEIN TYPE WITH NONCONVEX VALUES, AND BOUNDARY-VALUE-PROBLEMS FOR FUNCTIONAL-DIFFERENTIAL INCLUSIONS

A functional inclusion in the space of continuous vector-valued functi ons on the interval [a,bl is considered, the right-hand side of which is the sum of a convex-valued set-valued map and the product of a line ar integral operator and a set-valued map with images convex with resp ect to switching. Estimates for the distance between a solution of thi s inclusion and a fixed continuous vector-valued function are obtained and the structure of the set of solutions of this inclusion is studie d on the basis of these estimates. A result on the density of the solu tions of this inclusion in the set of solutions of the 'convexized' in clusion is obtained and the 'bang-bang' principle for the original inc lusion is established. This theory is applied to the study of the solu tion sets of boundary-value problems for functional-differential inclu sions with non-convex right-hand sides.