tailieunhanh - Báo cáo hóa học: " Research Article Oscillatory Solutions for Second-Order Difference Equations in Hilbert Spaces"

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 Oscillatory Solutions for Second-Order Difference Equations in Hilbert Spaces | Hindawi Publishing Corporation Advances in Difference Equations Volume 2007 Article ID 86925 9 pages doi 2007 86925 Research Article Oscillatory Solutions for Second-Order Difference Equations in Hilbert Spaces Cristobal Gonzalez and Antonio Jimenez-Melado Received 16 March 2007 Revised 24 July 2007 Accepted 27 July 2007 Recommended by Donal O Regan We consider the difference equation A2xn f n Xn T 0 T 0 1 . in the context of a Hilbert space. In this setting we propose a concept of oscillation with respect to a direction and give sufficient conditions so that all its solutions be directionally oscillatory as well as conditions which guarantee the existence of directionally positive monotone increasing solutions. Copyright 2007 C. Gonzalez and A. Jimenez-Melado. 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 study of difference equations has experienced a significant interest in the past years as they arise naturally in the modelling of real-world phenomena see . 1-3 and the references therein . The qualitative properties of solutions of both differential and difference equations have been extensively studied and some of the results obtained in the scalar case for instance the asymptotic behaviour are easily extended to an abstract setting see . 4-10 . In this paper we extend the concept of oscillation to the vector case. Hence in the context of a real Hilbert space we introduce the notion of oscillation with respect to a direction and show that some known results in the scalar case have their analogues in this more general context. The following two difference equations often appear in the literature in the study of oscillation and asymptotic behaviour A2Xn-1 f n Xn 0 A2Xn f n Xn 0 2 Advances in Difference Equations where Axn xn 1 - xn is the forward difference .