Impulsive differential inclusions involving evolution operators in separable Banach spaces

Abstract

We present some results on the existence of mild solutions and study the topological structure of the sets of solutions for the following first-order impulsive semilinear differential inclusions with initial and boundary conditions: $$y'(t) − A(t)y(t) \in F(t, y(t)) \text{for a.e.} t \in J\ \{t1,..., tm,...\},$$ $$y(t^+_k) − y(t^−_k) = I_k(y(t^−_k)),\quad k = 1,...,$$ $$y(0) = a$$ and $$y'(t) − A(t)y(t) \in F(t, y(t)) \text{for a.e.} t \in J\ \{t1,..., tm,...\},$$ $$y(t^+_k) − y(t^−_k) = I_k(y(t^−_k)),\quad k = 1,...,$$ $$Ly = a,$$ where $J = IR_+,\; 0 = t_0 < t_1 <...< t_m < ...;\; (m \in N), \lim_{k→∞} t_k = ∞,\; A(t)$ is the infinitesimal generator of a family of evolution operator $U(t, s)$ on a separable Banach space $E$, and $F$ is a set-valued mapping. The functions $I_k$ characterize the jump of solutions at the impulse points $t_k,\; k = 1,... .$ The mapping $L: P C_b → E$ is a bounded linear operator. We also investigate the compactness of the set of solutions, some regularity properties of the operator solutions, and the absolute retractness.