The solution sets for second order semilinear impulsive multivalued boundary value problems

A second order semilinear differential inclusion with some boundary continuous and impulse characteristics in a separable Banach space is considered. Some existence theorems for mild solutions are given, when the multivalued nonlinearity of the inclusion is only a locally integrably bounded upper-Carathéodory map with convex and weakly compact values. Then the compactness of the set of all mild solutions for the problem is proved. The results are obtained by using the theory of continuous cosine families of bounded linear operators and a fixed point theorem for multivalued maps due to Agarwal, Meehan and O’Regan. A corresponding result for closed graph of composition is extended.