Báo cáo sinh học: Periodic solutions for a class of higher order difference equations

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Báo cáo sinh học: " Periodic solutions for a class of higher order difference equations"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. Periodic solutions for a class of higher order difference equations Advances in Difference Equations 2011, 2011:66 doi:10.1186/1687-1847-2011-66 Huantao Zhu (zhu-huan-tao@163.com) Weibing Wang (gfwwbing@yahoo.com.cn) ISSN 1687-1847 Article type Research Submission date 16 September 2011 Acceptance date 23 December 2011 Publication date 23 December 2011 Article URL http://www.advancesindifferenceequations.com/content/2011/1/66 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 Zhu and Wang ; 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. Periodic solutions for a class of higher-order difference equations Huantao Zhu1 and Weibing Wang∗2 1 Hunan College of Information, Changsha, Hunan 410200, P.R. China of Mathematics, Hunan University of Science and Technology, 2 Department Xiangtan, Hunan 411201, P.R. China *Corresponding author: gfwwbing@yahoo.com.cn Email address: HZ: zhu-huan-tao@163.com Abstract In this article, we discuss the existence of periodic solutions for the higher-order difference equation x(n + k ) = g (x(n)) − f (n, x(n − τ (n)).We show the existence of periodic solutions by using Schauder’s fixed point the-orem, and illustrate three examples.MSC 2010: 39A10; 39A12.Keywords: functional difference equation; periodic solution; fixed point theo-rem. 11 Introduction and main resultsLet R denote the set of the real numbers, Z the integers and N the positive integers. Inthis article, we investigate the existence of periodic solutions of the following high-orderfunctional difference equation x(n + k ) = g (x(n)) − f (n, x(n − τ (n)), n ∈ Z, (1.1)where k ∈ N, τ : Z → Z and τ (n + ω ) = τ (n), f (n + ω, u) = f (n, u) for any (n, u) ∈Z × R, ω ∈ N. Difference equations have attracted the interest of many researchers in the last20 years since they provided a natural description of several discrete models, in whichthe periodic solution problem is always a important topic, and the reader can consult[1–7] and the references therein. There are many good results about existence ofperiodic solutions for first-order functional difference equations [8–12]. Only a fewarticle have been published on the same problem for higher-order functional differenceequations. Recently, using coincidence degree theory, Liu [13] studied the second-ordernonlinear functional difference equation ∆2 x(n − 1) = f (n, x(n − τ1 (n)), x(n − τ2 (n)), . . . , x(n − τm (n))), (1.2)and obtain sufficient conditions for the existence of at least one periodic solution ofequation (1.2). By using fixed point theorem in a cone, Wang and Chen [14] discussedthe following higher-order functional difference equation x(n + m + k ) − ax(n + m) − bx(n + k ) + abx(n) = f (n, x(n − τ (n))), (1.3)where a = 1, b = 1 are positive constants, τ : Z → Z and τ (n + ω ) = τ (n), ω, m, k ∈ N,and obtained existence theorem for single and multiple positive periodic solutions of 2(1.3). Our aim of this article is to study the existence of periodic solutions for the higher ...