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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 Neutral Stochastic Differential Equations with Variable Delays | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008 Article ID 407352 11 pages doi 10.1155 2008 407352 Research Article Fixed Points and Stability in Neutral Stochastic Differential Equations with Variable Delays Meng Wu 1 Nan-jing Huang 1 and Chang-Wen Zhao2 1 Department of Mathematics Sichuan University Chengdu Sichuan 610064 China 2 College of Business and Management Sichuan University Chengdu Sichuan 610064 China Correspondence should be addressed to Nan-jing Huang nanjinghuang@hotmail.com Received 4 April 2008 Accepted 9 June 2008 Recommended by Tomas Dominguez Benavides We consider the mean square asymptotic stability of a generalized linear neutral stochastic differential equation with variable delays by using the fixed point theory. An asymptotic mean square stability theorem with a necessary and sufficient condition is proved which improves and generalizes some results due to Burton Zhang and Luo. Two examples are also given to illustrate our results. Copyright 2008 Meng Wu 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. 1. Introduction Liapunov s direct method has been successfully used to investigate stability properties of a wide variety of differential equations. However there are many difficulties encountered in the study of stability by means of Liapunov s direct method. Recently Burton 1-4 Jung 5 Luo 6 and Zhang 7 studied the stability by using the fixed point theory which solved the difficulties encountered in the study of stability by means of Liapunov s direct method. Up till now the fixed point theory is almost used to deal with the stability for deterministic differential equations not for stochastic differential equations. Very recently Luo 6 studied the mean square asymptotic stability for a class of linear scalar neutral stochastic differential .
