﻿ 一类四阶不定微分算子的非实特征值 Non-Real Eigenvalues of a Class of Fourth Order Indefinite Differential Operators

Vol.06 No.05(2017), Article ID:21579,6 pages
10.12677/AAM.2017.65078

Non-Real Eigenvalues of a Class of Fourth Order Indefinite Differential Operators

Xin Zhao, Yunlan Gao*, Xiaojuan Qin

College of Sciences, Inner Mongolia University of Technology, Hohhot Inner Mongolia

*通讯作者。

Received: Jul. 14th, 2017; accepted: Jul. 29th, 2017; published: Aug. 2nd, 2017

ABSTRACT

The present paper deals with non-real eigenvalues of regular fourth order indefinite differential operators. Bounds of non-real eigenvalues are obtained under mild integrable conditions of coefficients when weighted function’s sign changes one or any time.

Keywords:Indefinite Differential Operator, Fourth Order Differential Operator, Non-Real Eigenvalue

1. 引言

(1.1)

(1.2)

，其中为实值函数，满足条件

(1.3)

(1.4)

2. 主要定理及其证明

(2.1)

(2.2)

. (2.3)

(2.4)

(2.5)

(2.6)

，上述等式可推出，因此

(2.7)

(2.8)

(2.9)

(2.10)

(2.11)

(2.12)

(2.13)

(2.14)

(2.15)

(2.16)

(2.17)

(2.18)

(2.19)

(2.20)

(2.21)

(2.22)

Non-Real Eigenvalues of a Class of Fourth Order Indefinite Differential Operators[J]. 应用数学进展, 2017, 06(05): 664-669. http://dx.doi.org/10.12677/AAM.2017.65078

1. 1. Richardson, R.G.D. (1918) Contributions to the Study of Oscillatory Properties of the Solutions of Linear Differential Equations of the Second Order. American Journal of Mathematics, 40, 283-316. https://doi.org/10.2307/2370485

2. 2. Turyn, L. (1980) Sturm-Liouville Problems with Several Parameters. Journal of Differential Equations, 38, 239-259. https://doi.org/10.1016/0022-0396(80)90007-8

3. 3. Binding, P. and Volkmer, H. (1996) Eigencurves for Two-Parameter Sturm-Liouville Equations. SIAM Review, 38, 27-48. https://doi.org/10.1137/1038002

4. 4. Binding, P. and Browne, P.J. (1988) Applications of Two Parameter Spectral Theory to Symmetric Generalised Eigenvalue Problem. Applicable Analysis, 29, 107-142. https://doi.org/10.1080/00036818808839776

5. 5. Xie, B. and Qi, J. (2013) Non-Real Eigenvalues of Indefinite Sturm-Liouville Problems. Journal of Differential Equations, 255, 2291-2301. https://doi.org/10.1016/j.jde.2013.06.013

6. 6. Behrndt, J., Chen, S., Philipp, F. and Qi, J. (2014) Estimates on the Non-Real Eigenvalues of Regular Indefinite Sturm-Liouville Problems. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, A144, 1113-1126. https://doi.org/10.1017/S0308210513001212

7. 7. Han, X. and Gao, T. (2016) A Priori Bounds and Existence of Non-Real Eigenvalues of Fourth-Order Boundary Value Problem with Indefinite Weight Function. Journal of Differential Equations, 82, 1-9.