Integral inclusions with nonconvex images, and their applications to boundary value problems for differential inclusions

This paper contains a treatment of an integral inclusion of Hammerstein type generated by the product of a linear integral operator and a multivalued mapping with images convex with respect to switching. This product is not a Volterra operator in general. Estimates of the closeness of a solution of the inclusion to a given function are proved on the basis of the theory of existence of continuous branches of multivalued mappings with images convex with respect to switching. By using these estimates it is proved that the solution set of the original inclusion is dense in the solution set of the convexified inclusion in the space of continuous functions. In the case when the kernel of the linear operator consists solely of the zero element the 'bang-bang' principle is proved for the Hammerstein inclusion. In the second part of the paper the theory is used for investigating boundary value problems for differential inclusions with nonconvex right-hand side.