tailieunhanh - báo cáo hóa học:" Research Article Oscillation of Second-Order Mixed-Nonlinear Delay Dynamic Equations"

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 of Second-Order Mixed-Nonlinear Delay Dynamic Equations | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 389109 21 pages doi 2010 389109 Research Article Oscillation of Second-Order Mixed-Nonlinear Delay Dynamic Equations M. Unal1 and A. Zafer2 1 Department of Software Engineering Bahce ehir University Beậiktaậ 34538 Istanbul Turkey 2 Department of Mathematics Middle East Technical University 06531 Ankara Turkey Correspondence should be addressed to M. Unal munal@ Received 19 January 2010 Accepted 20 March 2010 Academic Editor Josef Diblik Copyright 2010 M. Unal 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. New oscillation criteria are established for second-order mixed-nonlinear delay dynamic equations on time scales by utilizing an interval averaging technique. No restriction is imposed on the coefficient functions and the forcing term to be nonnegative. 1. Introduction In this paper we are concerned with oscillatory behavior of the second-order nonlinear delay dynamic equation of the form r t xA t pũ t x T0 t Pi t x Ti t ai-1x Ti t e if t Í0 i 1 on an arbitrary time scale T where a1 a2 am 1 am i an 0 n m 1 the functions r Pi e T R are right-dense continuous with r 0 nondecreasing the delay functions Ti T T are nondecreasing right-dense continuous and satisfy Ti f t for t T with Ti f TO as t TO. We assume that the time scale T is unbounded above that is sup T TO and define the time scale interval tũ to t by tũ to t tũ to n T. It is also assumed that the reader is already familiar with the time scale calculus. A comprehensive treatment of calculus on time scales can be found in 1-3 . 2 Advances in Difference Equations By a solution of we mean a nontrivial real valued function x T R such that x e C1d T to t and rxA e C1d T X T for all T e T with T t0 and that x satisfies