Summary: We present some existence results of mild solutions and study the topological structure of solution sets for the following first-order impulsive semilinear differential inclusions with periodic boundary conditions:
where and is the infinitesimal generator of a -semigroup on a separable Banach space and is a multi-valued map. The functions characterize the jump of the solutions at impulse points (. We will have to distinguish between the cases when either or neither 1 lies in the resolvent of . Accordingly, the problem is either formulated as a fixed point problem for an integral operator or for a Poincaré translation operator. Our existence results rely on fixed point theory and on a new nonlinear alternative for compact u.s.c. maps respectively. Then, we present some existence results and investigate the topological structure of the solution set. A continuous version of Filippov’s theorem is provided and the continuous dependence of solutions on parameters in both convex and nonconvex cases are examined.