tailieunhanh - Báo cáo hóa học: "OSCILLATION OF HIGHER-ORDER DELAY DIFFERENCE 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: OSCILLATION OF HIGHER-ORDER DELAY DIFFERENCE EQUATIONS | OSCILLATION OF HIGHER-ORDER DELAY DIFFERENCE EQUATIONS YINGGAO ZHOU Received 6 January 2006 Revised 18 April 2006 Accepted 20 April 2006 The oscillation and asymptotic behavior of the higher-order delay difference equation Alxn y 1 pi n xn-ki 0 n 0 1 2 . are investigated. Some sufficient conditions of oscillation and bounded oscillation of the above equation are obtained. Copyright 2006 Yinggao Zhou. 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 Consider the following delay difference equation m Alxn pi n xn-ki 0 n 0 1 2 . and its first-order corresponding inequality m Axn pi n xn-ki 0 n 0 1 2 . where pi n are sequences of nonnegative real numbers and not identically equal to zero and ki is positive integer i 1 2 . and A is the first-order forward difference operator Axn xn 1 - xn and Alxn Al-1 Axn for l 2. By a solution of or inequality we mean a nontrival real sequence xn satisfying or inequality for n 0. A solution xn is said to be oscillatory if it is neither eventually positive nor eventually negative and nonoscillatory otherwise. An equation is said to be oscillatory if its every solution is oscillatory. The oscillatory behavior of difference equations has been intensively studied in recent years. Most of the literature has been concerned with equations of type with l 1 see 1-10 and references cited therein . But very little is known regarding the oscillation of higher-order difference equation similar to . The purpose of this paper is to study the oscillatory properties of . Hindawi Publishing Corporation Advances in Difference Equations Volume 2006 Article ID 65789 Pages 1-7 DOI ADE 2006 65789 2 Oscillation of higher-order delay difference equations 2. Main results We need the following several lemmas in order to prove our results. .