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Báo cáo hóa học: " Research Article Fixed Points and Stability in Nonlinear Equations with Variable 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 Fixed Points and Stability in Nonlinear Equations with Variable Delays | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 195916 14 pages doi 10.1155 2010 195916 Research Article Fixed Points and Stability in Nonlinear Equations with Variable Delays Liming Ding 1 2 Xiang Li 1 and Zhixiang Li1 1 Department of Mathematics and System Science College of Science National University of Defence Technology Changsha 410073 China 2 Air Force Radar Academy Wuhan 430010 China Correspondence should be addressed to Liming Ding limingding@sohu.com Received 9 July 2010 Accepted 18 October 2010 Academic Editor Hichem Ben-El-Mechaiekh Copyright 2010 Liming Ding et al. 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. We consider two nonlinear scalar delay differential equations with variable delays and give some new conditions for the boundedness and stability by means of the contraction mapping principle. We obtain the differences of the two equations about the stability of the zero solution. Previous results are improved and generalized. An example is given to illustrate our theory. 1. Introduction Fixed point theory has been used to deal with stability problems for several years. It has conquered many difficulties which Liapunov method cannot. While Liapunov s direct method usually requires pointwise conditions fixed point theory needs average conditions. In this paper we consider the nonlinear delay differential equations xff -a f x t - r1 t b t g x t - r2 t 1.1 tX t -a f f xt - n t b f g x f - rW 1.2 where r1 t r2 t 0 to 0 to r max r1 0 r2 0 a b 0 to R f g R R are continuous functions. We assume the following A1 r1 t is differentiable A2 the functions t - r1 t t - r2 t 0 to -r to is strictly increasing A3 t r1 t t - r2 t to as t to. 2 Fixed Point Theory and Applications Many authors have investigated the special cases of 1.1 and 1.2 . Since Burton 1 .