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 Convergence for Hyperbolic Singular Perturbation of Integrodifferential Equations | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 80935 11 pages doi 2007 80935 Research Article Convergence for Hyperbolic Singular Perturbation of Integrodifferential Equations Jin Liang James Liu and Ti-Jun Xiao Received 18 March 2007 Accepted 26 June 2007 Recommended by Marta Garcia-Huidobro By virtue of an operator-theoretical approach we deal with hyperbolic singular perturbation problems for integrodifferential equations. New convergence theorems for such singular perturbation problems are obtained which generalize some previous results by Fattorini 1987 and Liu 1993 . Copyright 2007 Jin Liang 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 Let A and B be linear unbounded operators in a Banach space X let K t be a linear bounded operator for each t 0 in X and let f t e and f t be X-valued functions. We study the convergence of derivatives of solutions of f t e2u t e u t e e2A B u t e j K t - s AA B u s e ds f t e t 0 u 0 e u0 e u 0 e u1 e to derivatives of solutions of r t w t Bw t J K t - s Bw s ds f t t 0 w 0 W0 as e 0. 2 Journal of Inequalities and Applications The notion of hyperbolic singular perturbation problem comes from the work of Fat-torini 1 where the inhomogeneous hyperbolic singular perturbation problem e2u t e u t e e2A B u t e f t e t 0 u 0 e u0 e u 0 e u1 e arising from problems of traffic flow is studied. It was shown in 1 under some conditions on A B and f that as e 0 if u0 e w0 u1 e Bw0 Bu0 e Bw0 f e f and f e f then u t e w t and u t e w t uniformly on compact subsets of t 0 where u t e is the solution of the Cauchy problem and w is the solution of the Cauchy problem w t Bw t f t t 0 w 0 w0. This generalizes his earlier result in 3 about the parabolic singular perturbation problem e2u
