tailieunhanh - báo cáo hóa học:" Research Article Regularly Varying Solutions of Second-Order Difference Equations with Arbitrary Sign Coefficient"

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 Regularly Varying Solutions of Second-Order Difference Equations with Arbitrary Sign Coefficient | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 673761 16 pages doi 2010 673761 Research Article Regularly Varying Solutions of Second-Order Difference Equations with Arbitrary Sign Coefficient Serena Matucci1 and Pavel Rehak2 1 Department of Electronics and Telecommunications University of Florence 50139 Florence Italy 2 Institute of Mathematics Academy of Sciences CR Zizkova 22 61662 Brno Czech Republic Correspondence should be addressed to Pavel Rehak rehak@ Received 15 June 2010 Accepted 25 October 2010 Academic Editor E. Thandapani Copyright 2010 S. Matucci and P. Rehak. 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. Necessary and sufficient conditions for regular or slow variation of all positive solutions of a second-order linear difference equation with arbitrary sign coefficient are established. Relations with the so-called M-classification are also analyzed and a generalization of the results to the half-linear case completes the paper. 1. Introduction We consider the second-order linear difference equation A2yk Pk k t 0 on N where p is an arbitrary sequence. The principal aim of this paper is to study asymptotic behavior of positive solutions to in the framework of discrete regular variation. Our results extend the existing ones for see 1 where the additional condition pk 0 was assumed. We point out that the relaxation of this condition requires a different approach. At the same time our results can be seen as a discrete counterpart to the ones for linear differential equations see for example 2 . As a byproduct we obtain new nonoscillation criterion of Hille-Nehari type. We also examine relations with the so-called M-classification . the classification of monotone solutions with respect to their limit behavior and the .