Fixed Point Theory and Applications
Volume 2008 (2008), Article ID 401947, 15 pages
doi:10.1155/2008/401947
Bounded and periodic solutions of semilinear impulsive periodic system on Banach spaces
Jinrong Wang1
, X. Xiang1
, W. Wei3
and Qian Chen4
1College of Computer Science and Technology, Guizhou University, Guiyang, Guizhou 550025, China
3College of Science, Guizhou University, Guiyang, Guizhou 550025, China
4College of Electronic Science and Information Technology, Guizhou University, Guiyang, Guizhou 550025, China
Abstract
A class of semilinear impulsive periodic system on Banach spaces is considered. First, we introduce the T0-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 T0-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.