依赖参数的2$n$阶差分方程边值问题多个非平凡解的存在性

数学物理学报 ›› 2022, Vol. 42 ›› Issue (3): 760-766.

依赖参数的2$n$阶差分方程边值问题多个非平凡解的存在性

王振国()

^{黄淮学院数学与统计学院河南驻马店 463000}

收稿日期:2021-05-20 出版日期:2022-06-26 发布日期:2022-05-09
作者简介:王振国, E-mail: wangzhg123@163.com
基金资助:
国家自然科学基金(11971126);吕梁市科学技术重点研发项目(Rc2020213)

Existence and Multiplicity of Solutions for a 2$n$th-Order Discrete Boundary Value Problems with a Parameter

Zhenguo Wang()

^{School of Mathematics and Statistics, Huanghuai University, Henan Zhumadian 463000}

Received:2021-05-20 Online:2022-06-26 Published:2022-05-09
Supported by:
the NSFC(11971126);the Program for Scientific and Technological Research of Lüliang City(Rc2020213)

摘要/Abstract

摘要：

该文研究了一类具有参数的$2n$阶差分方程边值问题多个非平凡解的存在性问题.当$\lambda\in\left (\frac{p (T)}{2B}, \frac{1}{2A}\right)$时, 运用临界点理论得到这类差分方程边值问题无穷多个非平凡解存在的充分条件.最后, 举例验证文中的主要结论.

关键词: 差分方程, 边值问题, 临界点理论, 非平凡解

Abstract:

In this paper, we consider the existence and multiplicity of solutions for a $2n$th-order discrete boundary value problems depending on a parameter $\lambda$. When $\lambda\in\left(\frac{p(T)}{2B}, \frac{1}{2A}\right)$, we obtain a sufficient condition for the existence of solutions of a discrete boundary value problems by means of critical point theory. Finally, one example is given to illustrate our main result.

Key words: Difference equation, Boundary value problems, Critical point theory, Nontrivial solutions

中图分类号:

O175.8

王振国. 依赖参数的2$n$阶差分方程边值问题多个非平凡解的存在性[J]. 数学物理学报, 2022, 42(3): 760-766.

Zhenguo Wang. Existence and Multiplicity of Solutions for a 2$n$th-Order Discrete Boundary Value Problems with a Parameter[J]. Acta mathematica scientia,Series A, 2022, 42(3): 760-766.

参考文献

1	Kelly W G , Peterson A C . Difference Equations: An Introduction with Applications. San Diego: Academic Press, 1991
2	Agarwal R P . Difference Equations and Inequalities. Theory, Methods, and Applications. New York: Marcel Dekker, 2000
3	Yu J S , Zheng B . Modeling Wolbachia infection in mosquito population via discrete dynamical model. Journal of Difference Equations and Applications, 2019, 25 (11): 1549- 1567
4	徐家发, 罗洪林, 刘立山. 一类具有p-Laplacian算子的分数阶差分方程边值问题的正解. 数学物理学报, 2021, 41A (2): 402- 414
4	Xu J F , Luo H L , Liu L S . Positive solutions for a class of fractional difference equations boundary value problems with p-Laplacian operator. Acta Mathematica Scientia, 2021, 41A (2): 402- 414
5	Zhou Z , Yu J S , Guo Z M . Periodic solutions of higher-dimensional discrete systems. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2004, 134 (5): 1013- 1022
6	Zhou Z , Yu J S , Chen Y M . Periodic solutions of a 2nth-order nonlinear difference equation. Science China Mathematics, 2010, 53 (1): 41- 50
7	D'AguíG , Mawhin J , Sciammetta A . Positive solutions for a discrete two point nonlinear boundary value problem with p-Laplacian. Journal of Mathematical Analysis and Applications, 2017, 447 (1): 383- 397
8	Mei P , Zhou Z . Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations. Open Mathematics, 2018, 16, 1435- 1444
9	Lin G H , Zhou Z . Homoclinic solutions of discrete φ-Laplacian equations with mixed nonlinearities. Communications on Pure and Applied Analysis, 2018, 17 (5): 1723- 1747
10	Zhou Z , Ling J X . Infinitely many positive solutions for a discrete two point nonlinear boundary value problem with φ_c-Laplacian. Applied Mathematics Letters, 2019, 91, 28- 34
11	Guo Z M , Yu J S . Existence of periodic and subharmonic solutions for second order superlinear difference equtions. Science China Mathematics, 2013, 46, 506- 515
12	Bereanu C , Mawhin J . Boundary value problems for second-order nonlinear difference equations with discrete φ-Laplacian and singular φ. Journal of Difference Equations and Applications, 2008, 14, 1099- 1118
13	Obersnel F , Omari P . Positive solutions of the Dirichlet problem for the prescribed mean curvature equation. Journal of Differential Equations, 2010, 249 (7): 1674- 1725
14	Lu Y Q , Ma R Y , Gao H L . Existence and multiplicity of positive solutions for one-dimensional prescribed mean curvature equations. Boundary Value Problems, 2014, 2014, 120
15	Bonanno G , Candito P . Nonlinear difference equations investigated via critical point methods. Nonlinear Analysis, 2009, 70, 3180- 3186
16	Bonanno G , Candito P , D'Aguí G . Variational methods on finite dimensional Banach space and discrete problems. Advanced Nonlinear Studies, 2014, 14 (4): 915- 939
17	Bonanno G , Jebelean P , ?erban C . Superlinear discrete problems. Applied Mathematics Letters, 2016, 52, 162- 168
18	Bonanno G , Molica B G . Infinitely many solutions for a boundary value problem with discontinuous nonlinearities. Boundary Value Problems, 2009, 2009 (2009): 670675

[1]	邓钧元, 张峻皓. 半空间中满足 Cattaneo 定律的双曲系统的稳定性[J]. 数学物理学报, 2025, 45(5): 1444-1462.
[2]	万沪龙, 刘慧芳. 某类非线性微分差分方程亚纯解的结构[J]. 数学物理学报, 2025, 45(5): 1577-1585.
[3]	段誉, 孙歆. 一类 Klein-Gordon-Maxwell 系统解的存在性和多重性[J]. 数学物理学报, 2025, 45(3): 756-766.
[4]	霍会霞, 李永祥. 一类弯曲的弹性梁方程正解的存在性[J]. 数学物理学报, 2024, 44(6): 1476-1484.
[5]	张潇, 张宏武. 分数阶椭圆方程反边值问题的分数 Tikhonov 正则化方法[J]. 数学物理学报, 2024, 44(4): 978-993.
[6]	王振国, 丁廉业. 具有无界势能的 Kirchhoff-型差分方程多个同宿轨的存在性[J]. 数学物理学报, 2024, 44(1): 80-92.
[7]	王文霞. 带有 $p$-Laplacian 算子的分数阶非线性积分边值问题的唯一正解与和算子方法[J]. 数学物理学报, 2023, 43(6): 1731-1743.
[8]	李易娴,张正杰. 一类与 Klein-Gordon-Maxwell 问题有关的方程组的基态解的存在性[J]. 数学物理学报, 2023, 43(3): 680-690.
[9]	廖丹, 张慧萍, 姚旺进. 基于变分方法的脉冲微分方程 Neumann 边值问题多重解的存在性[J]. 数学物理学报, 2023, 43(2): 447-457.
[10]	徐家发, 杨志春. 高阶Riemann-Liouville型分数阶脉冲微分方程积分边值问题的正解[J]. 数学物理学报, 2023, 43(1): 53-68.
[11]	谢亚君,马昌凤. 源于自由边值离散的弱非线性互补问题的m+1阶收敛性算法[J]. 数学物理学报, 2022, 42(5): 1506-1516.
[12]	段磊,陈天兰. 带平均曲率算子的离散混合边值问题凸解的存在性[J]. 数学物理学报, 2022, 42(2): 379-386.
[13]	韩晓玲,蔡蕙泽,杨虎军. 星图上非线性分数阶微分方程边值问题解的存在唯一性[J]. 数学物理学报, 2022, 42(1): 139-156.
[14]	陈敏风,高宗升,黄志波. $f^{n}(z)+f^{m}(z+c)=e^{Az+B}$的有限级亚纯解[J]. 数学物理学报, 2021, 41(4): 913-920.
[15]	张纪凤,张伟,倪晋波,任丹丹. 带p(t)-Laplacian算子的分数阶Langevin方程反周期边值问题解的存在性[J]. 数学物理学报, 2021, 41(4): 1024-1032.