边界条件含有特征参数的四阶微分算子的自伴性和特征值的依赖性

数学物理学报 ›› 2022, Vol. 42 ›› Issue (3): 671-693.

边界条件含有特征参数的四阶微分算子的自伴性和特征值的依赖性

闫文文,许美珍*()

^{内蒙古工业大学理学院呼和浩特 010051}

收稿日期:2021-08-12 出版日期:2022-06-26 发布日期:2022-05-09
通讯作者: 许美珍 E-mail:xumeizhen1969@163.com
基金资助:
国家自然科学基金(11561051);内蒙古自然科学基金(2021MS01020)

The Self-Adjointness and Dependence of Eigenvalues of Fourth-Order Differential Operator with Eigenparameters in the Boundary Conditions

Wenwen Yan,Meizhen Xu*()

^{College of Sciences, Inner Mongolia University of Technology, Hohhot 010051}

Received:2021-08-12 Online:2022-06-26 Published:2022-05-09
Contact: Meizhen Xu E-mail:xumeizhen1969@163.com
Supported by:
the NSFC(11561051);the NSF of Inner Mongolia(2021MS01020)

摘要/Abstract

摘要：

该文考虑了一类边界条件一端含有特征参数且具有转移条件的四阶微分算子的自共轭性及特征值的依赖性. 通过在适当的Hilbert空间上定义一个与问题相关的线性算子T, 将上述问题的研究转化为对此空间中算子的研究, 并证明算子T是自共轭的. 另外, 在算子T自伴的基础上证明特征值不仅连续依赖而且可微依赖于问题的各个参数, 并给出相应的微分表达式. 特别地, 给出特征值关于特征参数依赖的边界条件系数矩阵的Fréchet导数及关于内部不连续点c左右两侧的一阶导数.

关键词: 四阶微分算子, 转移条件, 自共轭性, 特征值的依赖性, Fréchet导数

Abstract:

In this paper we consider the self-adjointness and the dependence of eigenvalues of a class of discontinuous fourth-order differential operator with eigenparameters in the boundary conditions of one endpoint. By constructing a linear operator T associated with problem in a suitable Hilbert space, the study of the above problem is transformed into the research of the operator in this space, and the self-adjointness of this operator T is proved. In addition, on the basis of the self-adjointness of the operator T, we show that the eigenvalues are not only continuously but also smoothly dependent on the parameters of the problem, and give the corresponding differential expressions. In particular, giving the Fréchet derivative of the eigenvalue with respect to the eigenparameter-dependent boundary condition coefficient matrix, and the first-order derivatives of the eigenvalue with respect to the left and right sides of the inner discontinuity point c.

Key words: Fourth-order differential operator, Transmission condition, Self-adjointness, Dependence of eigenvalue, Fréchet derivative

中图分类号:

O175.3

闫文文,许美珍. 边界条件含有特征参数的四阶微分算子的自伴性和特征值的依赖性[J]. 数学物理学报, 2022, 42(3): 671-693.

Wenwen Yan,Meizhen Xu. The Self-Adjointness and Dependence of Eigenvalues of Fourth-Order Differential Operator with Eigenparameters in the Boundary Conditions[J]. Acta mathematica scientia,Series A, 2022, 42(3): 671-693.

参考文献 34

1	Pöeschel J , Trubowitz E . Inverse Spectral Theory. New York: Academic Press, 1987
2	Dauge M , Helffer B . Eigenvalues variation, I. Neumann problem for Sturm-Liouville operators. J Differ Equ, 1993, 104 (2): 243- 262 doi: 10.1006/jdeq.1993.1071
3	Dauge M , Helffer B . Eigenvalues variation, Ⅱ. Multidimensional Problems. J Differ Equ, 1993, 104 (2): 263- 297 doi: 10.1006/jdeq.1993.1072
4	Kong Q K , Zettl A . Eigenvalues of regular Sturm-Liouville problems. J Differ Equ, 1996, 131, 1- 19 doi: 10.1006/jdeq.1996.0154
5	Kong Q K , Zettl A . Dependence of eigenvalues of Sturm-Liouville problems on the boundary. J Differ Equ, 1996, 126, 389- 407 doi: 10.1006/jdeq.1996.0056
6	Suo J Q , Wang W Y . Eigenvalues of a class of regular fourth-order Sturm-Liouville problems. Appl Math Comput, 2012, 218 (19): 9716- 9729
7	Ge S Q , Wang W Y , Suo J Q . Dependence of eigenvalues of a class of fourth-order Sturm-Liouville problems on the boundary. Appl Math Comput, 2013, 220, 268- 276
8	Kong Q K , Wu H Y , Zettl A . Dependence of eigenvalues on the problems. Math Nachr, 1997, 188, 173- 201 doi: 10.1002/mana.19971880111
9	Zheng Z W , Ma Y J . Dependence of eigenvalues of 2mth-order spectral problems. Bound Value Probl, 2017, 2017 (1): 126 doi: 10.1186/s13661-017-0857-y
10	Yang Q X , Wang W Y , Gao X C , Rajendran S . Dependence of eigenvalues of a class of higher-order Sturm-Liouville problems on the boundary. Math Probl Eng, 2015, doi: 10.1155/2015/686102
11	李昆, 郑召文. 一类具有转移条件的Sturm-Liouville方程的谱性质. 数学物理学报, 2015, 35A (5): 910- 926 doi: 10.3969/j.issn.1003-3998.2015.05.008
	Li K , Zheng Z W . Spectral properties for Sturm-Liouville equation with transmission conditions. Acta Math Sci, 2015, 35A (5): 910- 926 doi: 10.3969/j.issn.1003-3998.2015.05.008
12	Zhang X Y , Sun J . A class of fourth-order differential operator with eigenparameter-dependent boundary and transmission conditions. Mathematica Applicata, 2013, 26 (1): 205- 219
13	Yang Q X , Wang W Y . A class of fourth order differential operators with transmission conditions. Iran J Sci Technol, 2011, 35 (A4): 323- 332
14	Zhang M Z , Wang Y C . Dependence of eigenvalues of Sturm-Liouville problems with interface conditions. Appl Math Comput, 2015, 265, 31- 39
15	Li K , Sun J , Hao X L . Eigenvalues of regular fourth-order Sturm-Liouville problems with transmission conditions. Math Meth Appl Sci, 2017, 40, 3538- 3551 doi: 10.1002/mma.4243
16	Lv X X , Ao J J , Zettl A . Dependence of eigenvalues of fourth-order differential equations with discontinuous boundary conditions on the problem. J Math Anal Appl, 2017, 456, 671- 685 doi: 10.1016/j.jmaa.2017.07.021
17	Zinsou B . Dependence of eigenvalues of fourth-order boundary value problems with transmission conditions. Rocky Mt J Math, 2020, 50 (1): 369- 381
18	Li K , Sun J , Hao X L . Dependence of eigenvalues of 2nth order boundary value transmission problems. Bound Value Probl, 2017, 2017 (1): 143 doi: 10.1186/s13661-017-0876-8
19	Uǧurlu E . Regular third-order boundary value problems. Appl Math Comput, 2019, 343, 247- 257
20	Zhu H , Shi Y M . Dependence of eigenvalues on the boundary conditions of Sturm-Liouville problems with one singular endpoint. J Differ Equ, 2017, 263, 5582- 5609 doi: 10.1016/j.jde.2017.06.026
21	Kong Q K , Wu H Y , Zettl A . Left-definite Sturm-Liouville problems. J Differ Equ, 2001, 177, 1- 26 doi: 10.1006/jdeq.2001.3997
22	玉林. 两类微分算子与Riesz基的研究[D]. 呼和浩特: 内蒙古大学, 2021: 9-33
	Yu L. Research on Two Classes of Differential Operators and the Construction of Riesz Basis[D]. Hohhot: Inner Mongolia University, 2021: 9-33
23	Walter J . Regular eigenvalue problems with eigenvalue parameter in the boundary condition. Math Z, 1973, 133, 301- 312 doi: 10.1007/BF01177870
24	Fulton C T . Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc R Soc Edinb Sect A-Math, 1977, 77, 293- 308 doi: 10.1017/S030821050002521X
25	Mukhtarov O S , Aydemir K . Eigenfunction expansion for Sturm-Liouville problems with transmission conditions at one interior point. Acta Math Sci, 2015, 35B (3): 639- 649
26	Code W J. Sturm-Liouville Problems with Eigenparameter Dependent Boundary Conditions[D]. Saskatoon: University of Saskatchewan, 2003
27	Altnisikn N , Kadakal M . Eigenvalues and eigenfunctions of discontinuous Sturm-Liouville problems with eigenparameter-dependent boundary conditions. Acta Math Hung, 2004, 102, 159- 175 doi: 10.1023/B:AMHU.0000023214.99631.52
28	Birkhoff G D . On the asymptotic character of solution of the certain linear differential equations containting a parameter. Trans Amer Math Soc, 1908, 9, 219- 231 doi: 10.1090/S0002-9947-1908-1500810-1
29	Binding P A , Browne P J , Watson B A . Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter Ⅱ. J Comput Appl Math, 2002, 148, 147- 169 doi: 10.1016/S0377-0427(02)00579-4
30	Binding P A , Browne P J , Watson B A . Transformation between Sturm-Liouville problems with eigenvalue dependent and independent boundary conditions. Bull London Math Soc, 2001, 33, 749- 757 doi: 10.1112/S0024609301008177
31	Zhang M Z , Li K . Dependence of eigenvalues of Sturm-Liouville problems with eigenparameter dependent boundary conditions. Appl Math Comput, 2020, 378, 125214
32	Naimark M A. Linear Differential Operators. London: Harrap, 1968
33	Kong Q K, Zettl A. Linear ordinary differential equations//Agarwal R P. Inequalities and Applications. Singapore: World Scientific, 1994, 3: 381-397
34	Dieudonné J . Foundations of Modern Analysis. New York: Academis Press, 1969

[1]	孙康,高云兰. 一类边界条件含谱参数的微分算子[J]. 数学物理学报, 2022, 42(3): 661-670.
[2]	赵迎春,孙炯,姚斯琴,布仁满都拉. 一类内部具有不连续性的不定Strum-Liouville算子的非实特征值问题[J]. 数学物理学报, 2021, 41(6): 1643-1656.
[3]	李昆, 郑召文. 一类具有转移条件的Sturm-Liouville方程的谱性质[J]. 数学物理学报, 2015, 35(5): 910-926.