tailieunhanh - Báo cáo hoa học: " Research Article Oscillation for Second-Order Nonlinear Delay Dynamic Equations on Time Scales"

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 Oscillation for Second-Order Nonlinear Delay Dynamic Equations on Time Scales | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 756171 13 pages doi 2009 756171 Research Article Oscillation for Second-Order Nonlinear Delay Dynamic Equations on Time Scales Zhenlai Han 1 2 Tongxing Li 2 Shurong Sun 2 and Chenghui Zhang1 1 School of Control Science and Engineering Shandong University Jinan Shandong 250061 China 2 School of science University of Jinan Jinan Shandon 250022 China Correspondence should be addressed to Zhenlai Han hanzhenlai@ Received 6 December 2008 Revised 27 February 2009 Accepted 25 May 2009 Recommended by Alberto Cabada By means of Riccati transformation technique we establish some new oscillation criteria for the second-order nonlinear delay dynamic equations r t xA t Y A p t f x r t 0 on a time scale T here Y 0 is a quotient of odd positive integers with r and p real-valued positive rd-continuous functions defined on T. Our results not only extend some results established by Hassan in 2008 but also unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation. Copyright 2009 Zhenlai Han et al. 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 The theory of time scales which has recently received a lot of attention was introduced by Hilger in his . Thesis in 1988 in order to unify continuous and discrete analysis see Hilger 1 . Several authors have expounded on various aspects of this new theory see the survey paper by Agarwal et al. 2 and references cited therein. A book on the subject of time scales by Bohner and Peterson 3 summarizes and organizes much of the time scale calculus. We refer also to the last book by Bohner and Peterson 4 for advances in dynamic equations on time scales. For the notation used hereinafter we .