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Báo cáo hóa học: " Research Article A Generalized Halanay Inequality for Stability of Nonlinear Neutral Functional Differential "

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 A Generalized Halanay Inequality for Stability of Nonlinear Neutral Functional Differential | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 475019 16 pages doi 10.1155 2010 475019 Research Article A Generalized Halanay Inequality for Stability of Nonlinear Neutral Functional Differential Equations Wansheng Wang School of Mathematics and Computational Science Changsha University of Science and Technology Changsha 410114 China Correspondence should be addressed to Wansheng Wang w.s.wang@163.com Received 22 March 2010 Accepted 18 July 2010 Academic Editor Kun quan Q. Lan Copyright 2010 Wansheng Wang. 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. This paper is devoted to generalize Halanay s inequality which plays an important rule in study of stability of differential equations. By applying the generalized Halanay inequality the stability results of nonlinear neutral functional differential equations NFDEs and nonlinear neutral delay integrodifferential equations NDIDEs are obtained. 1. Introduction In 1966 in order to discuss the stability of the zero solution of u t -Au f Bu t - T T 0 1.1 Halanay used the inequality as follows. Lemma 1.1 Halanay s inequality see 1 . If v f -Av t B sup v s for t to 1.2 t-T s t where A B 0 then there exist c 0 and K 0 such that v t ce Kti for t t0 1.3 and hence v t 0 as t TO. 2 Journal of Inequalities and Applications In 1996 in order to investigate analytical and numerical stability of an equation of the type uft flt u t u n tf K t s u sf ds t t0 1.4 y t ộ tỵ t t0 Ộ bounded and continuous for t t0 Baker and Tang 2 give a generalization of Halanay inequality as Lemma 1.2 which can be used for discussing the stability of solutions of some general Volterra functional differential equations. Lemma 1.2 see 2 . Suppose v t 0 t e -TO to and v t -A t v t B t sup v s t t0 v t y t t t0 1.5 t-T t s t where y t is bounded and .