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Báo cáo: Existence of Solutions for a Weighted p t -Laplacian Impulsive Integrodifferential System with Multipoint and Integral Boundary Value Conditions

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 392545, 31 pages doi:10.1155/2010/392545 Research Article Existence of Solutions for a Weighted p t -Laplacian Impulsive Integrodifferential System with Multipoint and Integral Boundary Value Conditions Rong Dong,1 Yunrui Guo,2 Yuanzhang Zhao,3 and Qihu Zhang1 1 Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China 2 Department of Mathematics, Henan Institute of Science and Technology, Xinxiang, Henan 453003, China 3 Department of Mathematics, Ocean University of China, Qingdao, Shandong 266071, China Correspondence should be addressed to Qihu Zhang, zhangqh1999@yahoo.com.cn Received 5 October 2010; Accepted 15 December 2010. | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 392545 31 pages doi 10.1155 2010 392545 Research Article Existence of Solutions for a Weighted p t -Laplacian Impulsive Integrodifferential System with Multipoint and Integral Boundary Value Conditions Rong Dong 1 Yunrui Guo 2 Yuanzhang Zhao 3 and Qihu Zhang1 1 Department of Mathematics and Information Science Zhengzhou University of Light Industry Zhengzhou Henan 450002 China 2 Department of Mathematics Henan Institute of Science and Technology Xinxiang Henan 453003 China 3 Department of Mathematics Ocean University of China Qingdao Shandong 266071 China Correspondence should be addressed to Qihu Zhang zhangqh1999@yahoo.com.cn Received 5 October 2010 Accepted 15 December 2010 Academic Editor Jewgeni Dshalalow Copyright 2010 Rong Dong 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. By the Leray-Schauder s degree the existence of solutions for a weighted p t -Laplacian impulsive integro-differential system with multi-point and integral boundary value conditions is considered. The sufficient results for the existence are given under the resonance and nonresonance cases respectively. Moreover we get the existence of nonnegative solutions at nonresonance. 1. Introduction In this paper we consider the existence of solutions for the following weighted p t -Laplacian integrodifferential system -àp t U f t u w t 1 p t -1 u zS u zT u 0 t e 0 1 t ti 1.1 where u 0 1 RN f 0 1 XRN XRN XRN XRN RN ti e 0 1 i 1 . k with the following impulsive boundary value conditions limu f - limu J Ai limu t lim w t 1 p t 1 u t i 1 . k 1.2 t t t t- t t- t t- 2 Journal of Inequalities and Applications limw f u p 2u f - limujtju j 2u t t t t t- Bi limu t lim w f 1 p t -1 u f J i 1 . k t t- t t- lim w t u pt 2ujt Va lim w t lu lpt 2u f u