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 Sturm-Picone Comparison Theorem of Second-Order Linear Equations on Time Scales | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 496135 12 pages doi 2009 496135 Research Article Sturm-Picone Comparison Theorem of Second-Order Linear Equations on Time Scales Chao Zhang and Shurong Sun School of Science University of Jinan Jinan Shandong 250022 China Correspondence should be addressed to Chao Zhang ss_zhangc@ Received 29 December 2008 Revised 13 March 2009 Accepted 28 May 2009 Recommended by Alberto Cabada This paper studies Sturm-Picone comparison theorem of second-order linear equations on time scales. We first establish Picone identity on time scales and obtain our main result by using it. Also our result unifies the existing ones of second-order differential and difference equations. Copyright 2009 C. Zhang and S. Sun. 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 In this paper we consider the following second-order linear equations p1 t xA t qi f xơ t 0 M p2 t yA t q2 f yơ t 0 L2 where f e a ft n T pA f pA t q1 t and q2 t are real and rd-continuous functions in a ft n T. Let T be a time scale Ơ t be the forward jump operator in T yA be the delta derivative and yơ t y ơ t . First we briefly recall some existing results about differential and difference equations. As we well know in 1909 Picone 1 established the following identity. Picone Identity If x f and y f are the nontrivial solutions of p1 t x t q1 t x t 0 MOyV q2 t y t 0 2 Advances in Difference Equations where t e a ft p1 t p 2 t q1 t and q2 t are real and continuous functions in a ft . If y t Ỷ0 for t e a ft then mA pVx yV - P2 t y t x t y t p1 t - P2 t x 2 t 42 0 - q1 t x2 t P2 t x tyt y tJ - X f . By one can easily obtain the Sturm comparison theorem of second-order linear differential equations . Sturm-Picone Comparison Theorem .