PC-mild Solutions Research Articles

The Generalized Gronwall Inequality and Its Application to Periodic Solutions of Integrodifferential Impulsive Periodic System on Banach Space

This paper deals with a class of integrodifferential impulsive periodic systems on Banach space. Using impulsive periodic evolution operator given by us, the -periodic PC-mild solution is introduced and suitable Poincaré operator is constructed. Showing the compactness of Poincaré operator and using a new generalized Gronwall's inequality with impulse, mixed type integral operators and -norm given by us, we utilize Leray-Schauder fixed point theorem to prove the existence of -periodic PC-mild solutions. Our method is much different from methods of other papers. At last, an example is given for demonstration.

Bounded and Periodic Solutions of Semilinear Impulsive Periodic System on Banach Spaces

AbstractA class of semilinear impulsive periodic system on Banach spaces is considered. First, we introduce the "Equation missing"-periodic PC-mild solution of semilinear impulsive periodic system. By virtue of Gronwall lemma with impulse, the estimate on the PC-mild solutions is derived. The continuity and compactness of the new constructed Poincaré operator determined by impulsive evolution operator corresponding to homogenous linear impulsive periodic system are shown. This allows us to apply Horn's fixed-point theorem to prove the existence of "Equation missing"-periodic PC-mild solutions when PC-mild solutions are ultimate bounded. This extends the study on periodic solutions of periodic system without impulse to periodic system with impulse on general Banach spaces. At last, an example is given for demonstration.