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Báo cáo hóa học: " Research Article A Note on the Integral Inequalities with Two Dependent Limits"

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 A Note on the Integral Inequalities with Two Dependent Limits | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 430512 18 pages doi 10.1155 2010 430512 Research Article A Note on the Integral Inequalities with Two Dependent Limits Allaberen Ashyralyev 1 2 Emine Misirli 3 and Ozlem Mogol3 1 Department of Mathematics Fatih University Buyukcekmece 34500 Istanbul Turkey 2 Department of Mathematics ITTU 74200 Ashgabat Turkmenistan 3 Department of Mathematics Ege University 35100 Bornova-Izmir Turkey Correspondence should be addressed to Emine Misirli emine.misirli@gmail.com Received 4 October 2009 Revised 28 April 2010 Accepted 5 July 2010 Academic Editor Martin Bohner Copyright 2010 Allaberen Ashyralyev 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. The theorem on the Gronwall s type integral inequalities with two dependent limits is established. In application the boundedness of the solutions of nonlinear differential equations is presented. 1. Introduction Integral inequalities play a significant role in the study of qualitative properties of solutions of integral differential and integro-differential equations see e.g. 1-4 and the references given therein . One of the most useful inequalities in the development of the theory of differential equations is given in the following lemma see 5 . Lemma 1.1. Let u f and f f be real-valued nonnegative continuous functions for all t 0.If t u2 tf c 2 f s u s ds 0 for all t 0 where c 0 is a real constant then u f c f s ds 0 1.1 1.2 for all t 0. 2 Journal of Inequalities and Applications Note that the generalization of this integral inequality and its discrete analogies are given in papers 5-8 . In paper 9 the following useful inequality with two dependent limits was established. Lemma 1.2. Let u t be a real-valued nonnegative continuous function defined on -T T and let c and a