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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 Existence of Periodic Solutions of Linear Hamiltonian Systems with Sublinear Perturbation | Hindawi Publishing Corporation Boundary Value Problems Volume 2010 Article ID 945421 12 pages doi 10.1155 2010 945421 Research Article Existence of Periodic Solutions of Linear Hamiltonian Systems with Sublinear Perturbation Zhiqing Han School of Mathematical Sciences Dalian University of Technology Dalian 116023 Liaoning China Correspondence should be addressed to Zhiqing Han hanzhiq@dlut.edu.cn Received 2 June 2009 Revised 4 February 2010 Accepted 19 March 2010 Academic Editor Ivan T. Kiguradze Copyright 2010 Zhiqing Han. 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. We investigate the existence of periodic solutions of linear Hamiltonian systems with a nonlinear perturbation. Under generalized Ahmad-Lazer-Paul type coercive conditions for the nonlinearity on the kernel of the linear part existence of periodic solutions is obtained by saddle point theorems. A note on a result of Rabinowitz is also given. 1. Introduction For the second-order Hamiltonian system ủ t VF t u tf 0 M 0 - u T ủ 0 - ủ T 0 1.1 the existence of periodic solutions is related to the coercive conditions of F t ù on ủ. This fact is first noticed by Berger and Schechter 1 who use the coercive condition F t ù -TO as ù TO uniformly for a.e. t e 0 T . Subsequently Mawhin and Willem 2 consider it by using more general coercive conditions of an integral form. More precisely they assume that F t ù 0 T X Rn Rn is bounded VF t ù g T for some g T e L1 0 T with some additional technical conditions and satisfies one of the following Ahmad-Lazer-Paul type 3 coercive conditions lim ùh TO ueRN T F t ù đt TO 0 1.2 2 Boundary Value Problems then they obtain the existence of at least one solution. How to relax the boundedness of F is a problem which attracted several authors attention for example see 4 5 and the references therein. In 6 7 the .
