非局部奇异二阶微分方程的Weyl分类

数学物理学报 ›› 2020, Vol. 40 ›› Issue (5): 1295-1304.

非局部奇异二阶微分方程的Weyl分类

刘志文(),綦建刚*(),徐亚飞()

^{山东大学(威海)数学与统计学院山东威海 264209}

收稿日期:2018-11-30 出版日期:2020-10-26 发布日期:2020-11-04
通讯作者: 綦建刚 E-mail:sdulzw2018@163.com;qijiangang@sdu.edu.cn;yafeixu6@163.com
作者简介:刘志文, E-mail:sdulzw2018@163.com|徐亚飞, E-mail:yafeixu6@163.com
基金资助:
国家自然科学基金(11771253)

Weyl Classification of Nonlocal Singular Second Order Differential Equations

Zhiwen Liu(),Jiangang Qi*(),Yafei Xu()

^{College of Mathematics and Statistics, Shandong University at Weihai, Shandong Weihai 264209}

Received:2018-11-30 Online:2020-10-26 Published:2020-11-04
Contact: Jiangang Qi E-mail:sdulzw2018@163.com;qijiangang@sdu.edu.cn;yafeixu6@163.com
Supported by:
the NSFC(11771253)

摘要/Abstract

摘要：

该文考虑一类带有非局部项的二阶奇异Sturm-Liouville方程.给出了此类方程极限点（圆）型的定义和这两类划分的充分必要条件.此外，该文也研究了上述方程在实轴上平方可积解的个数，并建立了相应的充分必要条件.结果表明非局部问题的情形与经典局部问题之间有本质区别.

关键词: 极限点(圆), 非局部势, 奇异Sturm-Liouville方程, 平方可积解

Abstract:

The present paper is concerned with the Weyl classification of second order singular Sturm-Liouville equations with nonlocal point potential. We give the Weyl classification for these equations and give sufficient and necessary conditions for the division of these two kinds. Furthermore, the most important part is the situation of square integrable solutions for λ on the real axis, which has essential differences with the classical equation, and corresponding sufficient and necessary conditions are also obtained.

Key words: Limit-point, Limit-cricle, Nonlocal potential, Sturm-Liouville equation, Square integrable solution

中图分类号:

O175

刘志文,綦建刚,徐亚飞. 非局部奇异二阶微分方程的Weyl分类[J]. 数学物理学报, 2020, 40(5): 1295-1304.

Zhiwen Liu,Jiangang Qi,Yafei Xu. Weyl Classification of Nonlocal Singular Second Order Differential Equations[J]. Acta mathematica scientia,Series A, 2020, 40(5): 1295-1304.

参考文献

1	Albeverio S , Nizhnik L P . Sch?dinger operators with nonlocal point interactions. J Math Anal Appl, 2007, 332, 884- 895
2	Albeverio S , Hryniv R O , Nizhnik L P . Inverse spectral problems for non-local Sturm-Liouville operators. Inverse Problems, 2007, 23, 523- 535
3	Budd C , Dold B , Stewart A . Blowup in a partial differential equation with conserved first integral. Siam J Appl Math, 1993, 53 (3): 718- 742
4	曹之江. 常微分算子. 北京: 科学出版社, 2016
4	Cao Z . Ordinary Differential Operators. Beijing: Science Press, 2016
5	Catchpole E A . A Cauchy problem for an ordinary integro-differential equation. Proc Roy Soc Edinburgh, 1974, 72A, 39- 55
6	Catchpole E A . An integro-differential operator. J London Math Soc, 1973, 6 (2): 513- 523
7	Chafee N. The Electric Balast Resistor: Homogeneous and Nonhomogeneous Equilibria//de Mottoni P, Salvadori L. Nonlinear Differential Equations: Invariance, Stablity and Bifurcation. New York: Academic Press, 1981: 97-127
8	Deng K , Kwong M K , Levine H A . The influence of nonlocal nonlinearities on the long time behavior of solutions of Burger's equation. Quart Appl Math, 1992, 50, 173- 200
9	Dunford N , Schwartz J T . Linear Operators(Ⅱ). New York: Wiley, 1963
10	Everitt W N , Knowles I W , Read T T . Limit-point and limit-circle criteria for Sturm-Liouville equations with intermittently negative principal coefficients. Proc Roy Soc Edinburgh, 1986, 103A, 215- 228
11	Freitas P . A nonlocal Sturm-Liouville eigenvalue problem. Proc Roy Soc Edinburgh, 1994, 124A, 169- 188
12	Furter J , Grinfeld M . Local vs.non-local interactions in population dynamics. J Math Biol, 1989, 27, 65- 80
13	Harris B J . The limit circle criteria for second order differential expressions. Quart J Math Oxford, 1984, 35, 415- 427
14	Nizhnik L P . Inverse eigenvalue problems for non-local Sturm-Liouville operators. Methods Funct Anal Topology, 2009, 15, 41- 47
15	Nizhnik L P . Inverse nonlocal Sturm-Liouville problem. Inverse Problems, 2010, 26, 125006
16	Qi J . Eigenvalue problems of the model from nonlocal continuum mechanics. J Math Phys, 2011, 52, 073516
17	Weyl H . über gew?hnliche Differentialgleichungen mit Singularit?ten und die zugeh?rigen Entwicklungen willkürlicher Funktionen. Math Ann, 1910, 68, 220- 269

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