Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article The Generalized Gronwall Inequality and Its Application to Periodic Solutions of Integrodifferential Impulsive Periodic System on Banach Space | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 430521 22 pages doi 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 1 College of Computer Science and Technology Guizhou University Guiyang Guizhou 550025 China 2 College of Science Guizhou University Guiyang Guizhou 550025 China Correspondence should be addressed to JinRong Wang wjr9668@ Received 27 June 2008 Accepted 29 September 2008 Recommended by Ondrej Dosly 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 Poincare operator is constructed. Showing the compactness of Poincare 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 p0-periodic PC-mild solutions. Our method is much different from methods of other papers. At last an example is given for demonstration. 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. 1. Introduction It is well known that impulsive periodic motion is a very important and special phenomenon not only in natural science but also in social science such as climate food supplement insecticide population and sustainable development. Periodic system with applications on finite-dimensional spaces has been extensively studied. Particularly impulsive periodic systems on finite-dimensional spaces are considered and some important results such as the existence and .
