Periodic Solutions of Semilinear Impulsive Periodic System with Time-Varying Generating Operators on Banach Space

A class of semilinear impulsive periodic systems with time-varying generating operators on Banach space is considered. Using impulsive periodic evolution operator given by us, the T0periodic PC-mild solution is introduced and suitable Poincaré operator is constructed. Showing the compactness of Poincaré operator and using a new generalized Gronwall inequality with mixed type integral operators given by us, we utilize Leray-Schauder fixed point theorem to prove the existence of T0-periodic PC-mild solutions. Our method is an innovation and it is much different from methods of other papers. At last, an example is given for demonstration.