Abstract and Applied Analysis
Volume 2008 (2008), Article ID 939062, 19 pages
Research Article

Existence of Periodic Solutions for Integrodifferential Impulsive Periodic System on Banach Space

JinRong Wang,1 X. Xiang,2 and W. Wei2

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 T0-periodic PC-mild solution is introduced and suitable Poincaré operator is constructed. By virtue of the generalized new Gronwall lemma with impulse and B-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 T0-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.