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báo cáo hóa học:" Research Article Nonoscillation of First-Order Dynamic Equations with Several Delays"

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 Nonoscillation of First-Order Dynamic Equations with Several Delays | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 873459 22 pages doi 10.1155 2010 873459 Research Article Nonoscillation of First-Order Dynamic Equations with Several Delays Elena Braverman1 and Basak Karpuz2 1 Department of Mathematics and Statistics University of Calgary 2500 University Drive N. w. Calgary AB Canada T2N1N4 2 Department of Mathematics Faculty of Science and Arts ANS Campus Afyon Kocatepe University 03200 Afyonkarahisar Turkey Correspondence should be addressed to Elena Braverman maelena@math.ucalgary.ca Received 18 February 2010 Accepted 21 July 2010 Academic Editor John Graef Copyright 2010 E. Braverman and B. Karpuz. 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. For dynamic equations on time scales with positive variable coefficients and several delays we prove that nonoscillation is equivalent to the existence of a positive solution for the generalized characteristic inequality and to the positivity of the fundamental function. Based on this result comparison tests are developed. The nonoscillation criterion is illustrated by examples which are neither delay-differential nor classical difference equations. 1. Introduction Oscillation of first-order delay-difference and differential equations has been extensively studied in the last two decades. As is well known most results for delay differential equations have their analogues for delay difference equations. In 1 Hilger revealed this interesting connection and initiated studies on a new time-scale theory. With this new theory it is now possible to unify most of the results in the discrete and the continuous calculus for instance some results obtained separately for delay difference equations and delay-differential equations can be incorporated in the general type of equations called dynamic .