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Báo cáo hóa học: " Research Article On Some New Impulsive Integral Inequalities"

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 On Some New Impulsive Integral Inequalities | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 312395 8 pages doi 10.1155 2008 312395 Research Article On Some New Impulsive Integral Inequalities Jianli Li Department of Mathematics Hunan Normal University Changsha Hunan 410081 China Correspondence should be addressed to Jianli Li ljianli@sina.com Received 4 June 2008 Accepted 21 July 2008 Recommended by Wing-Sum Cheung We establish some new impulsive integral inequalities related to certain integral inequalities arising in the theory of differential equalities. The inequalities obtained here can be used as handy tools in the theory of some classes of impulsive differential and integral equations. Copyright 2008 Jianli Li. 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 Differential and integral inequalities play a fundamental role in global existence uniqueness stability and other properties of the solutions of various nonlinear differential equations see 1-4 . A great deal of attention has been given to differential and integral inequalities see 1 2 5-8 and the references given therein. Motivated by the results in 1 5 7 the main purpose of this paper is to establish some new impulsive integral inequalities similar to Bihari s inequalities. Let 0 to t1 t2 limk TOtk TO R 0 to and I c R then we introduce the following spaces of function PC R I u R I u is continuous for t ftk u 0 u fjp and u t- exist and u t- u tk k 1 2 . PC1 R I u e PC R I u is continuously differentiable for t ftk u 0 u t and uft- exist and uft- u tk k 1 2 . . To prove our main results we need the following result see 1 Theorem 1.4.1 . Lemma 1.1. Assume that A0 the sequence tk satisfies 0 t0 t1 t2 with limk TOtk to A1 m e PC1 R R and m f is left-continuous at tk k 1 2 . A2 for k 1 2 . t to m t p fi m fi q t t ftk m tk dkm tk