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Báo cáo hóa học: " Research Article Oscillation Criteria for Second-Order Forced Dynamic Equations with Mixed Nonlinearities"

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 Criteria for Second-Order Forced Dynamic Equations with Mixed Nonlinearities | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 938706 20 pages doi 10.1155 2009 938706 Research Article Oscillation Criteria for Second-Order Forced Dynamic Equations with Mixed Nonlinearities Ravi P. Agarwal1 and A. Zafer2 1 Department of Mathematical Sciences Florida Institute of Technology Melbourne FL 32901 USA 2 Department of Mathematics Middle East Technical University 06531 Ankara Turkey Correspondence should be addressed to A. Zafer zafer@metu.edu.tr Received 21 January 2009 Accepted 19 April 2009 Recommended by Mariella Cecchi We obtain new oscillation criteria for second-order forced dynamic equations on time scales containing mixed nonlinearities of the form rụỵ a xA A f t xa e t t e to to t with f t x qO afx ỉní-qiO Pt x m m -1m where t0 to t is a time scale interval with t0 e T the functions r q qi e t0 to t R are right-dense continuous with r 0 a is the forward jump operator xa t x a t and p1 pm a pm 1 pn 0. All results obtained are new even for T R and T Z. In the special case when T R and a 1 our theorems reduce to Y. G. Sun and J. S. W. Wong Journal of Mathematical Analysis and Applications. 337 2007 549-560 . Therefore our results in particular extend most of the related existing literature from the continuous case to arbitrary time scale. Copyright 2009 R. P. Agarwal and A. Zafer. 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 Let T be a time scale which is unbounded above and t0 e T a fixed point. For some basic facts on time scale calculus and dynamic equations on time scales one may consult the excellent texts by Bohner and Peterson 1 2 . We consider the second-order forced nonlinear dynamic equations containing mixed nonlinearities of the form r t a xA f t xa e t t e Í0 to t 1.1 with n f t x q if a x y qiiif pt x G m w -1w 1.2