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Báo cáo hóa học: "OSCILLATORY MIXED DIFFERENCE SYSTEMS"

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: OSCILLATORY MIXED DIFFERENCE SYSTEMS | OSCILLATORY MIXED DIFFERENCE SYSTEMS JOSÉ M. FERREIRA AND SANDRA PINELAS Received 2 November 2005 Accepted 21 February 2006 The aim of this paper is to discuss the oscillatory behavior of difference systems of mixed type. Several criteria for oscillations are obtained. Particular results are included in regard to scalar equations. Copyright 2006 J. M. Ferreira and S. Pinelas. 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 The aim of this work is to study the oscillatory behavior of the difference system c m Ax n Pix n - i Qjx n j n 0 1 2 . 1.1 where x n e Rd Ax n x n 1 - x n is the usual difference operator f m e N and for i 1 . f and j 1 . m Pi and Qj are given d X d real matrices. For a particular form of the scalar case of 1.1 the same question is studied in 1 see also 2 Section 1.16 . The system 1.1 is introduced in 9 . In this paper the authors show that the existence of oscillatory or nonoscillatory solutions of that system determines an identical behavior to the differential system with piecewise constant arguments m x t Pix t - i Qjx t j 1.2 i 1 j 1 where for t e R x t e Rd and means the greatest integer function see also 8 Chapter 8 . By a solution of 1.1 we mean any sequence x n of points in Rd with n -Ể . 0 1 . which satisfy 1.1 . In order to guarantee its existence and uniqueness for given Hindawi Publishing Corporation Advances in Difference Equations Volume 2006 Article ID 92923 Pages 1-18 DOI 10.1155 ADÉ 2006 92923 2 Oscillatory mixed difference systems initial values x- . x0 . xm-1 denoting by I the d X d identity matrix we will assume throughout this paper that the matrices Pl . Pe Qi . Qm are such that det I - Q1 0 if m 1 detQm 0 if m 2 1.3 Pi 0 for every i 1 . f with no restrictions in other cases see 8 Chapter 7 and 9 . We will say that a sequence y n satisfies .