Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí sinh học đề tài :Existence of solutions for weighted p(r)-Laplacian impulsive system mixed type boundary value problems | Yin et al. Boundary Value Problems 2011 2011 42 http content 2011 1 42 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access Existence of solutions for weighted p r -Laplacian impulsive system mixed type boundary value problems Li Yin 1 Yunrui Guo2 Guizhen Zhi1 and Qihu Zhang1 Correspondence zhangqh1999@ department of Mathematics and Information Science Zhengzhou University of Light Industry Zhengzhou Henan 450002 China Full list of author information is available at the end of the article Springer Abstract This paper investigates the existence of solutions for weighted p r -Laplacian impulsive system mixed type boundary value problems. The proof of our main result is based upon Gaines and Mawhin s coincidence degree theory. Moreover we obtain the existence of nonnegative solutions. Keywords Weighted p r -Laplacian impulsive system coincidence degree 1 Introduction In this paper we mainly consider the existence of solutions for the weighted p r -Laplacian system w r u p r 2u r f r u r w r p r 1 u r 0 r e 0 T r n 1 where u 0 T RN with the following impulsive boundary conditions 1 lim u r lim u r Ai lim u r lim w r p r 1 u r i 1 . k 2 r r r r r lim w r lu lp r 2u r lim w r lu lp r 2u r r r T ri .21 . 3 Bi lim u r lim w r p r 1 u r i 1 . k r r- r r 1 au 0 b lim w r p r 1 u r 0 and cu T d lim w r u p r 2u r 0 4 where p e C 0 T R and p r 1 -Ap r u - w r u p r -2 u r is called weighted Ji r -Laplacian 0 r1 r2 . rk T Ai Bi e C RN X RN RN a b c d e 0 ad bc 0. Throughout the paper o 1 means functions which uniformly convergent to 0 as n for any v e RN v will denote the j-th component of v the inner product in Rn will be denoted by - - will denote the absolute value and the Euclidean norm on Rn. Denote J 0 T J 0 T r0 ri . rk 1 0 r0 ri Ji ri ri 1 i 1 . k where r0 0 rk 1 T. Denote JO the interior of Ji i 0 1 . k. Let PC J RN x J 2011 Yin et al licensee Springer. This is an Open Access article distributed under the terms
