First-order periodic impulsive semilinear differential inclusions: Existence and structure of solution sets

In this paper, 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: { ( y ′ − A y ) ( t ) ∈ F ( t , y ( t ) ) ,  a.e.  t ∈ J ∖ { t 1 , … , t m } , y ( t k + ) − y ( t k − ) = I k ( y ( t k − ) ) , k = 1 , … , m , y ( 0 ) = y ( b ) where J = [ 0 , b ] and 0 = t 0 < t 1 < ⋯ < t m < t m + 1 = b ( m ∈ N ∗ ) A is the infinitesimal generator of a C 0 -semigroup T on a separable Banach space E and F is a multi-valued map. The functions I k characterize the jump of the solutions at impulse points t k ( k = 1 , … , m ). We will have to distinguish between the cases when either or neither 1 lies in the resolvent of T ( b ) . 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.