tailieunhanh - Báo cáo hóa học: "PERIODIC SOLUTIONS OF SECOND-ORDER NONAUTONOMOUS DYNAMICAL SYSTEMS"

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: PERIODIC SOLUTIONS OF SECOND-ORDER NONAUTONOMOUS DYNAMICAL SYSTEMS | PERIODIC SOLUTIONS OF SECOND-ORDER NONAUTONOMOUS DYNAMICAL SYSTEMS MARTIN SCHECHTER Received 13 March 2006 Revised 10 May 2006 Accepted 15 May 2006 We study the existence of periodic solutions for second-order nonautonomous dynamical systems. We give four sets of hypotheses which guarantee the existence of solutions. We were able to weaken the hypotheses considerably from those used previously for such systems. We employ a new saddle point theorem using linking methods. Copyright 2006 Martin Schechter. 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 We consider the following problem. One wishes to solve -x t VxV t x t where x t X1 t . xn t is a map from I 0 T to Rn such that each component Xj t is a periodic function in H1 with period T and the function V t x V t x1 . xn is continuous from Rn 1 to R with VxV t x . V e C Rn 1 Rn . dx1 ờxn Here H1 represents the Hilbert space of periodic functions in L2 I with generalized derivatives in L2 I . The scalar product is given by u v H1 u v u v . For each x e Rn the function V t x is periodic in t with period T. Hindawi Publishing Corporation Boundary Value Problems Volume 2006 Article ID 25104 Pages 1-9 DOI BVP 2006 25104 2 Periodic solutions of second-order nonautonomous dynamical systems We will study this problem under the following assumptions 1 V t x 0 t e I x e R 2 there are constants m 0 a 6m2 T2 such that V t x a x m t e I x e R 3 there is a constant p 2 such that Hp t x w t e L1 I x C t e I x e R x 2 Hp t x limsup 2 0 x TO x where Hp t x pV t x -VxV t x x 4 there is a subset e c I of positive measure such that liminf y f x 0 t e e. x rc x 2 We have the following theorem. Theorem . Under the above hypotheses the system has a solution. As a variant of Theorem we have the following