Journal of Inequalities and Applications
Volume 2008 (2008), Article ID 430521, 22 pages
doi:10.1155/2008/430521
Research Article

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

JinRong Wang,1 X. Xiang,1,2 W. Wei,2 and Qian Chen2

1College of Computer Science and Technology, Guizhou University, Guiyang, Guizhou 550025, China
2College of Science, Guizhou University, Guiyang, Guizhou 550025, China

Received 27 June 2008; Accepted 29 September 2008

Academic Editor: Ondřej Došlý

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.

Abstract

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. Showing the compactness of Poincaré operator and using a new generalized Gronwall's inequality with impulse, mixed type integral operators and B-norm given by us, we utilize Leray-Schauder fixed point theorem to prove the existence of T0-periodic PC-mild solutions. Our method is much different from methods of other papers. At last, an example is given for demonstration.