Abstract and Applied Analysis
Volume 2008 (2008), Article ID 939062, 19 pages
Existence of Periodic Solutions for Integrodifferential Impulsive Periodic System on Banach Space
1College of Computer Science and Technology, Guizhou University, Guiyang, Guizhou 550025, China
2College of Science, Guizhou University, Guiyang, Guizhou 550025, China
Received 5 July 2008; Accepted 28 October 2008
Academic Editor: Agacik Zafer
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.
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. By virtue of the generalized new Gronwall lemma with impulse and -norm, the
estimate on the PC-mild solutions is derived. Showing the continuity and compactness of the Poincaré operator, we utilize Horn's fixed point theorem to prove the existence of -periodic PC-mild solutions when the PC-mild solutions
are bounded and ultimate bounded. This extends the study of periodic solutions of integrodifferential periodic system
without impulse to integrodifferential periodic system with impulse on general Banach spaces. At last, an example is
given for demonstration.