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 Integral Equations and Exponential Trichotomy of Skew-Product Flows | Hindawi Publishing Corporation Advances in Difference Equations Volume 2011 Article ID 918274 18 pages doi 2011 918274 Research Article Integral Equations and Exponential Trichotomy of Skew-Product Flows Adina Luminita Sasu and Bogdan Sasu Department of Mathematics Faculty of Mathematics and Computer Science West University of Timisoara V Parvan Boulevard no. 4 300223 Timisoara Romania Correspondence should be addressed to Adina Luminita Sasu sasu@ Received 24 November 2010 Accepted 1 March 2011 Academic Editor Toka Diagana Copyright 2011 A. L. Sasu and B. Sasu. 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 are interested in an open problem concerning the integral characterizations of the uniform exponential trichotomy of skew-product flows. We introduce a new admissibility concept which relies on a double solvability of an associated integral equation and prove that this provides several interesting asymptotic properties. The main results will establish the connections between this new admissibility concept and the existence of the most general case of exponential trichotomy. We obtain for the first time necessary and sufficient characterizations for the uniform exponential trichotomy of skew-product flows in infinite-dimensional spaces using integral equations. Our techniques also provide a nice link between the asymptotic methods in the theory of difference equations the qualitative theory of dynamical systems in continuous time and certain related control problems. 1. Introduction Exponential trichotomy is the most complex asymptotic property of evolution equations being firmly rooted in bifurcation theory of dynamical systems. The concept proceeds from the central manifold theorem and mainly relies on the decomposition of the state space into a direct sum of three invariant .
