Báo cáo sinh học: Homoclinic solutions of some second-order non-periodic discrete systems

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: Homoclinic solutions of some second-order non-periodic discrete systems
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Báo cáo sinh học: "Homoclinic solutions of some second-order non-periodic discrete systems"Advances in DifferenceEquations This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Homoclinic solutions of some second-order non-periodic discrete systems Advances in Difference Equations 2011, 2011:64 doi:10.1186/1687-1847-2011-64 Yuhua Long (longyuhua214@163.com) ISSN 1687-1847 Article type Research Submission date 15 July 2011 Acceptance date 20 December 2011 Publication date 20 December 2011 Article URL http://www.advancesindifferenceequations.com/content/2011/1/64 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). For information about publishing your research in Advances in Difference Equations go to http://www.advancesindifferenceequations.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2011 Long ; licensee Springer.This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Homoclinic solutions of somesecond-order non-periodic discrete systems Yuhua Long College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, P. R. China Email address: longyuhua214@163.com AbstractIn this article, we discuss how to use a standard minimizing argumentin critical point theory to study the existence of non-trivial homoclinicsolutions of the following second-order non-autonomous discrete systems ∆2 xn−1 + A∆xn − L(n)xn + W (n, xn ) = 0, n ∈ Z,without any periodicity assumptions. Adopting some reasonable as-sumptions for A and L, we establish that two new criterions for guaran-teeing above systems have one non-trivial homoclinic solution. Besides 1 that, in some particular case, for the first time the uniqueness of homo- clinic solutions is also obtained. MSC: 39A11. Keywords: homoclinic solution; variational functional; critical point; subquadratic second-order discrete system.1. IntroductionThe theory of nonlinear discrete systems has widely been used to study discretemodels appearing in many fields such as electrical circuit analysis, matrix the-ory, control theory, discrete variational theory, etc., see for example [1, 2]. Sincethe last decade, there have been many literatures on qualitative properties ofdifference equations, those studies cover many branches of difference equations,see [3-7] and references therein. In the theory of differential equations, homo-clinic solutions, namely doubly asymptotic solutions, play an important role inthe study of various models of continuous dynamical systems and frequentlyhave tremendous effects on the dynamics of nonlinear systems. So, homoclinicsolutions have extensively been studied since the time of Poincar´, see [8-13]. eSimilarly, we give the following definition: if xn is a solution of a discrete sys-tem, xn will be called a homoclinic solution emanating from 0 if xn → 0 as|n| → +∞. If xn = 0, xn is called a non-trivial homoclinic solution. For our convenience, let N, Z, and R be the set of all natural numbers, 2integers, and real numbers, respectively. Throughout this article, | · | denotesthe usual norm in RN with N ∈ N, (·, ·) stands for the inner product. Fora, b ∈ Z, define Z(a) = {a, a + 1, . . .}, Z(a, b) = {a, a + 1, . . . , b} when a ≤ b. In this article, we consider the existence of non-trivial homoclinic solutionsfor the following second-order non-autonomous discrete system ∆2 xn−1 + A∆xn − L(n)xn + W (n, xn ) = 0 (1.1)without any periodicity assumptions, where A is an antisymmetric constantmatrix, L(n) ...