Open Access

Bounded and Periodic Solutions of Semilinear Impulsive Periodic System on Banach Spaces

Fixed Point Theory and Applications20082008:401947

https://doi.org/10.1155/2008/401947

Received: 20 February 2008

Accepted: 7 July 2008

Published: 14 July 2008

Abstract

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.

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Authors’ Affiliations

(1)
College of Computer Science and Technology, Guizhou University
(2)
College of Science, Guizhou University
(3)
College of Electronic Science and Information Technology, 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.