二阶差分方程周期边值问题正解存在的最优条件

王晶晶; 路艳琼

doi:10.3969/j.issn.1000-5641.201811039

二阶差分方程周期边值问题正解存在的最优条件

doi: 10.3969/j.issn.1000-5641.201811039

王晶晶,
路艳琼^,

西北师范大学数学与统计学院, 兰州　730070

基金项目: 国家自然科学基金青年基金(11801453, 11901464); 甘肃省青年科技基金计划项目(1606RJYA232); 西北师范大学青年教师科研能力提升计划一般项目(NWNU-LKQN-15-16)

详细信息

通讯作者:
路艳琼, 女, 副教授, 研究方向为差分方程及其应用. E-mail: luyg8610@126.com

中图分类号: O175.8
计量
- 文章访问数: 151
- HTML全文浏览量: 96
- PDF下载量: 3
- 被引次数: 0
出版历程
- 收稿日期: 2018-09-30
- 刊出日期: 2020-03-01

Optimal conditions for the existence of positive solutions to periodic boundary value problems with second order difference equations

WANG Jingjing,
LU Yanqiong^,

College of Mathematics and Statistics, Northwest Normal University, Lanzhou　730070, China

摘要

摘要: 运用锥上的不动点指数理论, 获得了格林函数非负时二阶离散周期边值问题　　　　　　　　　　　　　　　　　　　　　　 $\left\{ {\begin{array}{*{20}{l}} {\Delta^2 y(n-1)+a(n)y(n)=g(n)f(y(n)),}&{n\in[1,N]_{\mathbb{Z}},}\\ {y(0)=y(N), \;\;\;\Delta y(0)=\Delta y(N)}&{} \end{array}} \right.$正解存在的最优条件, 其中$[1,N]_{\mathbb{Z}}=\{1,2,\cdot\cdot\cdot,N\}, $$\,f:[1,N]_{\mathbb{Z}}\times\mathbb{R}^+\rightarrow\mathbb{R}^+$连续, $a: [1,N]_{\mathbb{Z}}\rightarrow(0,+\infty)$且$\mathop {\max }\limits_{n \in {{[1,N]}_{\mathbb{Z}}}} a(n )\leqslant4\sin^2\left(\frac\pi{2N}\right),\,g\in C([1,N]_{\mathbb{Z}},\mathbb{R}^+),\mathbb{R}^+:=[0,\infty).$
- 周期边值问题 /
- 正解 /
- 非负格林函数 /
- 不动点指数
Abstract: By using the fixed-point index theory of cone mapping, we show the optimal conditions for the existence of positive solutions for second order discrete periodic boundary value problems　　　　　　　　　　　　　　　　　$\left\{ {\begin{array}{*{20}{l}} {\Delta^2 y(n-1)+a(n)y(n)=g(n)f(y(n)),}&{n\in[1,N]_{\mathbb{Z}},}\\ {y(0)=y(N), \;\;\;\Delta y(0)=\Delta y(N)}&{} \end{array}} \right.$with vanishing Green’s function, where $[1,N]_{\mathbb{Z}}=\{1,2,\cdot\cdot\cdot, $$N\},\,f:[1,N]_{\mathbb{Z}}\times\mathbb{R}^+\rightarrow\mathbb{R}^+$ is continuous, $a: [1,N]_{\mathbb{Z}}\rightarrow(0,+\infty),$ and $\mathop {\max }\limits_{n \in {{[1,N]}_{\mathbb{Z}}}} a(n)\leqslant4\sin^2(\frac\pi{2N}),\,g\in C([1,N]_{\mathbb{Z}},\mathbb{R}^+), $$\mathbb{R}^+:=[0,\infty)$.
- periodic boundary value problem /
- positive solution /
- nonnegative Green’s function /
- fixed-point index

HTML全文

参考文献(14)

[1]	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. DOI: 10.1006/jmaa.1998.6257.
[2]	ATICI F M, CABADA A. Existence and uniqueness results for discrete second-order periodic boundary value problems [J]. Computers and Mathematics with Applications, 2003, 45(6/7/8/9): 1417-1427. DOI: 10.1016/S0898-1221(03)00097-X.
[3]	王丽颖, 张丽颖, 李晓月. 二阶离散周期边值问题的正解 [J]. 东北师大学报(自然科学版), 2007(2): 11-15. DOI: 10.3321/j.issn:1000-1832.2007.02.003.
[4]	李晓月, 王丽颖. 二阶离散周期边值问题的单个和多个正解 [J]. 数学物理学报, 2009, 29(5): 1187-1195.
[5]	MA R Y, LU Y Q, CHEN T. Existence of one-signed solutions of discrete second-order periodic boundary value problems [J]. Abstract and Applied Analysis, 2012(2): 160-176.
[6]	蒋玲芳. 二阶奇异离散周期边值问题正解的存在性和多解性 [J]. 内蒙古大学学报(自然科学版), 2013, 44(4): 345-351.
[7]	姚庆六. 变系数非线性二阶周期边值问题的正解 [J]. 应用数学学报, 2008(3): 564-573. DOI: 10.3321/j.issn:0254-3079.2008.03.018.
[8]	陈彬. 格林函数变号的三阶周期边值问题 [J]. 山东大学学报(理学版), 2016, 51(8): 79-83.
[9]	GAO C H, ZHANG F, MA R Y. Existence of positive solutions of second-order periodic boundary value problems with sign-changing Green’s function [J]. 应用数学学报(英文版), 2017, 33(2): 263-268.
[10]	CABADA A, ENGUICA R, LÓPEZ-SOMOZA L. Positive solutions for second-order boundary value problems with sign changing Green’s functions [J]. Electronic Journal of Differential Equations, 2017, 2017(245): 1-17.
[11]	ZHANG G W, SUN J X. Positive solutions of m-point boundary value problems [J]. Journal of Mathematical Analysis and Applications, 2004, 291(2): 406-418. DOI: 10.1016/j.jmaa.2003.11.034.
[12]	CUI Y J, ZOU Y M. Nontrivial solutions of singular superlinear m-point boundary value problems [J]. Applied Mathematics and Computation, 2007, 187(2): 1256-1264. DOI: 10.1016/j.amc.2006.09.036.
[13]	WANG F, ZHANG F. Positive solutions for a periodic boundary value problem without assumptions of monotonicity and convexity [J]. Bulletin of Mathematical Analysis and Applications, 2011(2): 261-268.
[14]	GUO D J, LAKSHMIKANTHAM V. Nonlinear Problems in Abstract Cones [M]. New York: Academic Press, 1988.