tailieunhanh - On the oscillation of higher order dynamic equations

This paper present some new criteria for the oscillation of even order dynamic equation on time scale T, where a is the ratio of positive odd integers a and q is a real valued positive rd-continuous functions defined on T. | Journal of Advanced Research 2013 4 201-204 Cairo University Journal of Advanced Research SHORT COMMUNICATION On the oscillation of higher order dynamic equations Said R. Grace Department of Engineering Mathematics Faculty of Engineering Cairo University Orman Giza 12221 Egypt Received 4 March 2012 revised 2 April 2012 accepted 23 April 2012 Available online 2 July 2012 KEYWORDS Oscillation Higher Order Dynamic equations Introduction Abstract We present some new criteria for the oscillation of even order dynamic equation a t xAn 1 i a q t x t f 0 on time scale T where a is the ratio of positive odd integers a and q is a real valued positive rd-continuous functions defined on T. 2012 Cairo University. Production and hosting by Elsevier . All rights reserved. ii a q T fi R 0 1 is a real-valued rd-continuous functions aD t P 0 for t 2 t0 1 T and This paper is concerned with the oscillatory behavior of all solutions of the even order dynamic equation s Ds 1. a t xA 1 i a q t x t 0 on an arbitrary time-scale T c R with Sup T 1 and n P 2 is an even integer. We shall assume that i a P 1 is the ratio of positive odd integers Tel. 20 2 35876998. E-mail address saidgrace@. Peer review under responsibility of Cairo University. Production and hosting by Elsevier We recall that a solution x of Eq. is said to be nonos-cillatory if there exists a t0 2 T. Such that x t x ơ t 0 for all t 2 t0 1 T otherwise it is said to be oscillatory. Eq. is said to be oscillatory if all its solutions are oscillatory. The study of dynamic equations on time-scales which goes back to its founder Hilger 1 as an area of mathematics that has received a lot of attention. It has been created in order to unify the study of differential and difference equations. Recently there has been an increasing interest in studying the oscillatory behavior of first and second order dynamic equations on time-scales see 2-7 . With respect to dynamic equations on time scales it is fairly new .