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 18

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 doi: 10.1080/10236198.2019.1669578
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 doi: 10.3969/j.issn.1003-3998.2021.02.012
	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 doi: 10.3969/j.issn.1003-3998.2021.02.012
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 doi: 10.1017/S0308210500003607
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 doi: 10.1007/s11425-009-0167-7
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 doi: 10.1016/j.jmaa.2016.10.023
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 doi: 10.1515/math-2018-0123
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 doi: 10.3934/cpaa.2018082
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 doi: 10.1016/j.aml.2018.11.016
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 doi: 10.1080/10236190802332290
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 doi: 10.1016/j.jde.2010.07.001
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 doi: 10.1186/1687-2770-2014-120
15	Bonanno G , Candito P . Nonlinear difference equations investigated via critical point methods. Nonlinear Analysis, 2009, 70, 3180- 3186 doi: 10.1016/j.na.2008.04.021
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 doi: 10.1515/ans-2014-0406
17	Bonanno G , Jebelean P , Şerban C . Superlinear discrete problems. Applied Mathematics Letters, 2016, 52, 162- 168 doi: 10.1016/j.aml.2015.09.005
18	Bonanno G , Molica B G . Infinitely many solutions for a boundary value problem with discontinuous nonlinearities. Boundary Value Problems, 2009, 2009 (2009): 670675

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