A Class of (ω,T)-Periodic Solutions for Impulsive Evolution Equations of Sobolev Type

In this paper, we study a new class of Sobolev type (ω,T)-periodic linear and semilinear impulsive evolution equations, where T denotes a linear isomorphism from Banach space X to itself. We give a sufficient and necessary condition depending on the initial value, periodic boundary value and linear isomorphism to guarantee that the homogeneous linear impulsive problem has a (ω,T)-periodic solution. Next, we give the explicit expression of (ω,T)-periodic solutions for nonhomogeneous linear impulsive problem and derive two important estimations for the certain sum and integration including the Green function. Further, we show the existence and uniqueness of solutions to semilinear impulsive problem, where we remove the compactness of AB−1 and use the compactness of a mapping depending on the nonlinear term. Finally, examples are provided to illustrate the theoretical results.