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《山东大学学报(理学版)》 ›› 2019, Vol. 54 ›› Issue (12): 38-45.doi: 10.6040/j.issn.1671-9352.0.2019.461

• • 上一篇    

一类奇异二阶阻尼差分方程周期边值问题正解的存在性

苏肖肖   

  1. 西北师范大学数学与统计学院, 甘肃 兰州 730070
  • 发布日期:2019-12-11
  • 作者简介:苏肖肖(1995— ), 女, 硕士研究生, 研究方向为常微分方程边值问题. E-mail:suxiaoxiao2856@163.com
  • 基金资助:
    国家自然科学基金资助项目(11671322)

Existence of positive solutions for periodic boundary conditions of singular second-order damped difference equations

SU Xiao-xiao   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Published:2019-12-11

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摘要： 研究了一类奇异二阶阻尼差分方程周期边值问题{Δ2x(t-1)+αΔx(t-1)+βx(t)=f(t,x(t), Δx(t-1)), t∈［1,T］Z,x(0)=x(T), Δx(0)=Δx(T)正解的存在性,其中T >2是一个整数, α、 β均为常数, f(t,x,y):［1,T］Z×(0,∞)×R→R关于(x,y)∈(0,∞)×R连续且允许f在x=0处奇异即limx→0+ f(t,x,y)=+∞,(t,y)∈［1,T］Z×R。主要结果的证明基于Leray-Schauder非线性抉择。

关键词: 差分方程, 正解, 奇异性, 格林函数, Leray-Schauder非线性抉择

Abstract: This paper studies the existence of positive solutions for periodic boundary value problems of second order damped difference equations{Δ2x(t-1)+αΔx(t-1)+βx(t)=f(t,x(t), Δx(t-1)), t∈［1,T］Z,x(0)=x(T), Δx(0)=Δx(T)where T >2 is an integer, α, β are constants, f(t,x,y):［1,T］Z×(0,∞)×R→R is continuous with respect to (x,y)∈(0,∞)×R, f may be singular at x=0, which means that limx→0+ f(t,x,y)=+∞,(t,y)∈［1,T］Z×R. The proof of main results is based on nonlinear alternative of Leray-Schauder.

Key words: difference equation, positive solution, singular, Greens function, nonlinear alternative of Leray-Schauder

中图分类号: 

  • O175.8
[1] MA Ruyun, MA Huili. Existence of sign-changing periodic solutions of second order difference equations[J]. Applied Mathematics and Computation, 2008, 203(2):463-470.
[2] HE Tieshan, XU Yuantong. Positive solutions for nonlinear discrete second-order boundary value problems with parameter dependence[J]. Journal of Mathematical Analysis and Applications, 2011, 379(2):627-636.
[3] CABADA A, OTERO-ESPINAR V. Fixed sign solutions of second-order difference equations with Neumann boundary conditions[J]. Computers Mathematics with Applications, 2003, 45(6/7/8/9):1125-1136.
[4] BEREANU C, MAWHIN J. Existence and multiplicity results for periodic solutions of nonlinear difference equations[J]. Journal of Difference Equations and Applications, 2006, 12(7):677-695.
[5] ATICI F M, GUSEINOV G S. Positive periodic solutions for nonlinear difference equations with periodic coefficients[J]. Journal of Mathematical Analysis and Applications, 1999, 232(1):166-182.
[6] ATICI F M, CABADA A. Existence and uniqueness results for discrete second-order periodic boundary value problems[J]. Computers Mathematics with Applications, 2003, 45(6/7/8/9):1417-1427.
[7] MA Ruyun, LU Yanqiong, CHEN Tianlan. Existence of one-signed solutions of discrete second-order periodic boundary value problems[J/OL]. Abstract and Applied Analysis, 2012, 2012: 1-13[2019-04-10]. http://dx.doi.org/10.1155/2012/437912.
[8] KELLEY W G, PETERSON A C. Difference eauations: an introduction with applications[M]. San Diego: Academic Press, 2001.
[9] TORRES P J. Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem[J]. Journal of Differential Equations, 2003, 190(2):643-662.
[10] GRANAS A, DUGUNDJI J. Fixed point theory[M]. New York: Springer-Verlag, 2003.
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