On the theory of perturbed inclusions and its applications

Inclusions with right-hand side that is the algebraic sum of the values of a compact-valued operator and a map equal to the product of a linear integral operator and a set-valued operator with values convex with respect to switching are considered. Existence questions for solutions of such inclusions are discussed, and the density principle and the ‘bang-bang’ principle are established. Properties of the solution sets of inclusions with internal and external perturbations are studied. A necessary and sufficient condition ensuring that the intersection of the closures of the sets of approximate solutions coincides with the closure of the set of the original inclusion is obtained. The results are applied to the analysis of boundary-value problems for functional-differential inclusions.