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: PERIODIC SOLUTIONS OF NONLINEAR VECTOR DIFFERENCE EQUATIONS | PERIODIC SOLUTIONS OF NONLINEAR VECTOR DIFFERENCE EQUATIONS M. I. GIL Received 31 January 2005 Accepted 7 September 2005 Essentially nonlinear difference equations in a Euclidean space are considered. Conditions for the existence of periodic solutions and solution estimates are derived. Our main tool is a combined usage of the recent estimates for matrix-valued functions with the method of majorants. Copyright 2006 M. I. Gil . 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 and notation Periodic solutions of difference equations in Euclidean and Banach spaces have been considered by many authors see for example 1-3 5-10 12 and the references therein. Mainly equations with separated linear parts and scalar equations were investigated. In this paper we consider essentially nonlinear systems in a Euclidean space. We prove the existence of periodic solutions and derive the estimates for their norms. Let Cn be the set of all complex n-vectors with an arbitrary norm II II I is the unit matrix Rs A denotes the spectral radius of a matrix A and n r z e Cn z r . Consider in Cn the equation x t 1 B x t t x t F x t t t 0 1 2 . where F t continuously maps O r into Cn and B z t are n X n-matrices continuous in z e O r and dependent on t 0 1 . In addition F v t and B v t are periodic in t F z t F z t T z e Q r t 0 1 . B z t B z t T z e O r t 0 1 . Hindawi Publishing Corporation Advances in Difference Equations Volume 2006 Article ID 39419 Pages 1-8 DOI ADE 2006 39419 2 Periodic solutions of nonlinear vector difference equations for some positive integer T. It is also assumed that there are nonnegative constants V and p such that F z t v z p z e Q r t 0 1 2 . T - 1 . Denote by tt r T the set of the finite sequences h v k Ị 0ì whose elements v k belong to Q r . For an h v k T 0 e w
