Fixed Point Theory and Applications
Volume 2008 (2008), Article ID 401947, 15 pages
Bounded and Periodic Solutions of Semilinear Impulsive Periodic System on Banach Spaces
1College of Computer Science and Technology, Guizhou University, Guiyang, Guizhou 550025, China
2College of Science, Guizhou University, Guiyang, Guizhou 550025, China
3College of Electronic Science and Information Technology, Guizhou University, Guiyang, Guizhou 550025, China
Received 20 February 2008; Revised 6 April 2008; Accepted 7 July 2008
Academic Editor: Jean Mawhin
Copyright © 2008 JinRong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A class of semilinear impulsive periodic system on Banach spaces is considered. First, we introduce
the -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 -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.