tailieunhanh - Báo cáo hóa học: "Research Article Existence of Four Solutions of Some Nonlinear Hamiltonian System"

RTuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: esearch Article Existence of Four Solutions of Some Nonlinear Hamiltonian System | Hindawi Publishing Corporation Boundary Value Problems Volume 2008 Article ID 293987 17 pages doi 2008 293987 Research Article Existence of Four Solutions of Some Nonlinear Hamiltonian System Tacksun Jung1 and Q-Heung Choi2 1 Department of Mathematics Kunsan National University Kunsan 573-701 South Korea 2 Department of Mathematics Education Inha University Incheon 402-751 South Korea Correspondence should be addressed to Q-Heung Choi qheung@ Received 25 August 2007 Accepted 3 December 2007 Recommended by Kanishka Perera We show the existence of four 2n-periodic solutions of the nonlinear Hamiltonian system with some conditions. We prove this problem by investigating the geometry of the sublevels of the functional and two pairs of sphere-torus variational linking inequalities of the functional and applying the critical point theory induced from the limit relative category. Copyright 2008 T. Jung and . Choi. 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 and statements of main results Let H t z be a c2 function defined on R1 X R2n which is 2n-periodic with respect to the first variable this paper we investigate the number of 2n-periodic nontrivial solutions of the following nonlinear Hamiltonian system z j Hz t z t where z R R2n z dz dt J 0 In -In 0 In is the identity matrix on Rn H R1 X R2n R and Hz is the gradient of H. Let z p q p z1 . zn q zn 1 . z2nf G Rn. Then can be rewritten as p -Hq t p q q Hp t p q . We assume that H G C2 R1 X R2n R1 satisfies the following conditions. 2 Boundary Value Problems H1 There exist constants a ft such that al d22 H ạ z ftl v fi z G R1 X R2n. H2 Let j1 j2 j1 1 and j3 j2 1 be integers and a ft be any numbers without loss of generality we may assume a ft G Z such that j1 - 1 a j1 j2 ft j2 1 j3. Suppose that .