# Báo cáo hóa học: "Research Article Variational Method to the Impulsive Equation with Neumann Boundary Conditions"

## 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 Variational Method to the Impulsive Equation with Neumann Boundary Conditions | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 316812 17 pages doi 2009 316812 Research Article Variational Method to the Impulsive Equation with Neumann Boundary Conditions Juntao Sun and Haibo Chen Department of Mathematics Central South University Changsha 410075 Hunan China Correspondence should be addressed to Juntao Sun sunjuntao2008@ Received 28 August 2009 Accepted 28 September 2009 Recommended by Pavel Drabek We study the existence and multiplicity of classical solutions for second-order impulsive Sturm-Liouville equation with Neumann boundary conditions. By using the variational method and critical point theory we give some new criteria to guarantee that the impulsive problem has at least one solution two solutions and infinitely many solutions under some different conditions respectively. Some examples are also given in this paper to illustrate the main results. Copyright 2009 J. Sun and H. Chen. 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 In this paper we consider the boundary value problem of second-order Sturm-Liouville equation with impulsive effects - p f u f r ìfu f qttyuft g t u ty t ftk . t e 0 1 -A p tk u tk Ik u tk k 1 2 . p - 1 u 0 u 1- 0 where 0 to t1 t2 tp-1 tp 1 p e c1 0 1 r q e C 0 1 with p and q positive functions g 0 1 X R R is a continuous function Ik R R 1 k p - 1 are continuous -A p tk u tk -pttkffuftk - u t- u tp and uft- denote the right and the left limits respectively of u f at t tk u 0 is the right limit of u 0 and u 1- is the left limit of u 1 . In the recent years a great deal of work has been done in the study of the existence of solutions for impulsive boundary value problems IBVPs by which a number 2 Boundary Value Problems of chemotherapy population dynamics optimal control ecology industrial

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