tailieunhanh - Báo cáo hóa học: " The Clark dual and multiple periodic solutions of delay differential equations"

Tuyển tập các báo cáo nghiên cứu về hóa học được đăng trên tạp chí hóa hoc quốc tế đề tài : The Clark dual and multiple periodic solutions of delay differential equations | Xiao et al. Boundary Value Problems 2011 2011 44 http content 2011 1 44 RESEARCH o Boundary Value Problems a SpringerOpen Journal Open Access The Clark dual and multiple periodic solutions of delay differential equations Huafeng Xiao Jianshe Yu and Zhiming Guo Correspondence huafeng@gzhu. College of Mathematics and Information Sciences Guangzhou University Guangzhou 510006 PRC Abstract We study the multiplicity of periodic solutions of a class of non-autonomous delay differential equations. By making full use of the Clark dual the dual variational functional is considered. Some sufficient conditions are obtained to guarantee the existence of multiple periodic solutions. 2000 Mathematics Subject Classification 34K13 34K18. Keywords periodic solution Clark dual Morse-Ekeland index delay differential equation asymptotic linearity 1 Introduction The existence and multiplicity of periodic solutions of delay differential equations have been investigated since 1962. Various methods have been used to study such a problem 1-10 . Among those methods critical point theory is a very important tool. By combining with Kaplan-Yorke method it can be used indirectly to study the existence of periodic solutions of delay differential equation 11-16 . By building the variational frame for some special systems it can be used directly to study the existence of delay differential systems 17 18 . However the variational functionals in the above two cases are strongly indefinite. They are hard to be dealed with. In this article we study multiple periodic solutions of the following non-autonomous delay differential equations x t -f t x t - 2 1 where x t e R f e C R X R R . We assume that f1 ft x is odd with respect to x and n 2-periodic with respect to t . f t x f t x f t n x f t x V t x eR xR f2 There exists a continuous differentiable function F t x which is strictly convex with respect to x uniformly in t such that ft x is the gradient of F t x .