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: OSCILLATION OF SECOND-ORDER NEUTRAL DELAY AND MIXED-TYPE DYNAMIC EQUATIONS ON TIME SCALES | OSCILLATION OF SECOND-ORDER NEUTRAL DELAY AND MIXED-TYPE DYNAMIC EQUATIONS ON TIME SCALES Y. SAHÌNER Received 31 January 2006 Revised 11 May 2006 Accepted 15 May 2006 We consider the equation r t _yA t Y A f t x S i 0 t e T where y t x t p t x T t and Y is a quotient of positive odd integers. We present some sufficient conditions for neutral delay and mixed-type dynamic equations to be oscillatory depending on deviating arguments T t and 8 t t e T. Copyright 2006 Y. Sahiner. 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. Some preliminaries on time scales A time scale T is an arbitrary nonempty closed subset of the real numbers. The theory of time scales was introduced by Hilger 6 in his . thesis in 1988 in order to unify continuous and discrete analysis. Several authors have expounded on various aspects of this new theory see 7 and the monographs by Bohner and Peterson 3 4 and the references cited therein. First we give a short review of the time scales calculus extracted from 3 . For any t e T we define the forward and backward jump operators by Ơ t inf s e T s t p t sup s e T s t respectively. The graininess function p T 0 to is defined by p t Ơ t - t. A point t e T is said to be right dense if t sup T and Ơ t t left dense if t inf T and p t t. Also t is said to be right scattered if Ơ t t left scattered if t p t . A function f T R is called rd-continuous if it is continuous at right dense points in T and its left-sided limit exists finite at left dense points in T. For a function f T R if there exists a number a e R such that for all e 0 there exists a neighborhood U of t with I f ơ t - f s - a ơ t - s I e ơ t - s for all s e U then f is A-differentiable at t and we call a the derivative of f at t and denote Hindawi Publishing Corporation Advances in Difference Equations Volume 2006 Article
