Open Access

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

Journal of Inequalities and Applications20082008:430521

https://doi.org/10.1155/2008/430521

Received: 27 June 2008

Accepted: 29 September 2008

Published: 22 October 2008

Abstract

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

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Authors’ Affiliations

(1)
College of Computer Science and Technology, Guizhou University
(2)
College of Science, Guizhou University

Copyright

© JinRong Wang et al. 2008

This article is published under license to BioMed Central Ltd. 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.