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 Some Nonlinear Integral Inequalities on Time Scales | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 70465 15 pages doi 2007 70465 Research Article Some Nonlinear Integral Inequalities on Time Scales Wei Nian Li and Weihong Sheng Received 28 August 2007 Accepted 6 November 2007 Recommended by Alberto Cabada The purpose of this paper is to investigate some nonlinear integral inequalities on time scales. Our results unify and extend some continuous inequalities and their corresponding discrete analogues. The theoretical results are illustrated by a simple example at the end of this paper. Copyright 2007 W. N. Li and W. Sheng. 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 The theory of time scales was introduced by Hilger 1 in his . thesis in 1988 in order to unify continuous and discrete analysis. Recently many authors have extended some fundamental integral inequalities used in the theory of differential and integral equations on time scales. For example we refer the reader to the literatures 2-8 and the references cited therein. In this paper we investigate some nonlinear integral inequalities on time scales which unify and extend some inequalities established by Pachpatte in 9 . The obtained inequalities can be used as important tools in the study of certain properties of dynamic equations on time scales. 2. Preliminaries on time scales We first briefly introduce the time scales calculus which can be found in 4 5 . In what follows R denotes the set of real numbers Z denotes the set of integers N0 denotes the set of nonnegative integers C denotes the set of complex numbers and C M S denotes the class of all continuous functions defined on set M with range in the set S. 2 Journal of Inequalities and Applications We use the usual conventions that empty sums and products are taken to .
