tailieunhanh - Báo cáo hóa học: " Research Article The Existence of Periodic Solutions for Non-Autonomous Differential Delay Equations via Minimax Methods"

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 The Existence of Periodic Solutions for Non-Autonomous Differential Delay Equations via Minimax Methods | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID i37084 15 pages doi 2009 137084 Research Article The Existence of Periodic Solutions for Non-Autonomous Differential Delay Equations via Minimax Methods Rong Cheng1 2 1 College of Mathematics and Physics Nanjing University of Information Science and Technology Nanjing 210044 China 2 Department of Mathematics Southeast University Nanjing 210096 China Correspondence should be addressed to Rong Cheng mathchr@ Received 9 April 2009 Accepted 19 October 2009 Recommended by Ulrich Krause By using variational methods directly we establish the existence of periodic solutions for a class of nonautonomous differential delay equations which are superlinear both at zero and at infinity. Copyright 2009 Rong Cheng. 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 Main Result Many equations arising in nonlinear population growth models 1 communication systems 2 and even in ecology 3 can be written as the following differential delay equation x f -af x f - 1 where f e C R R is odd and a is parameter. Since Jone s work in 4 there has been a great deal of research on problems of existence multiplicity stability bifurcation uniqueness density of periodic solutions to by applying various approaches. See 2 4-23 . But most of those results concern scalar equations and generally slowly oscillating periodic solutions. A periodic solution x f of is called a slowly oscillating periodic solution if there exist numbers p 1 and q p 1 such that x f 0 for 0 f p x f 0 for p f q and x f q x f for all f. In a recent paper 17 Guo and Yu applied variational methods directly to study the following vector equation x f -f x - r 2 Advances in Difference Equations where f e C R R is odd and r 0 is a given .