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Keywords
(5)
Boundary Condition
Eigenvalues
Indexation
sturmliouville equation
sturmliouville problem
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(1)
Inequalities amo...
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SturmLiouville Problems Whose Leading Coefcient Function Changes Sign
SturmLiouville Problems Whose Leading Coefcient Function Changes Sign,Xifang Cao,Qingkai Kong,Hongyou Wu,Anton Zettl
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SturmLiouville Problems Whose Leading Coefcient Function Changes Sign
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Citations: 6
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Xifang Cao
,
Qingkai Kong
,
Hongyou Wu
,
Anton Zettl
For a given
SturmLiouville equation
whose leading coefcient function changes sign, we es tablish inequalities among the
eigenvalues
for any coupled selfadjoint
boundary condition
and those for two corresponding separated selfadjoint boundary conditions. By a recent result of Binding and Volkmer, the
eigenvalues
(unbounded from both below and above) for a separated selfadjoint bound ary condition can be numbered in terms of the Prufer angle; and our inequalities can then be used to index the
eigenvalues
for any coupled selfadjoint boundary condition. Under this indexing scheme, we determine the discontinuities of each eigenvalue as a function on the space of such SturmLiouville problems, and its range as a function on the space of selfadjoint boundary conditions. We also re late this indexing scheme to the number of zeros of eigenfunctions. In addition, we characterize the discontinuities of each eigenvalue under a different indexing scheme. In this paper, we study selfadjoint SturmLiouville problems (SLP's) associated with regular SturmLiouville equations (SLE's) (0.1) (p y0)0 + qy = wy on (a; b);
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Citation Context
(2)
...We believe that this explicit representation of the selfadjoint BCs is an important preparation for an indepth study of the dependence of the eigenvalues of a selfadjoint problem on its BC for the orders greater than or equal to 3 (see, e.g., [
1
, 3, 9, 12, 14 ]f or order 2), and we plan to undertake this task in forthcoming publications...
...For a topological manifold M ,a topological Mbottle is a quotient space N that one obtains from M × [0,
1
] via modeling M ×{ 0} by an equivalence relation on M to form a subset of N, to be called the top of N, and modeling M ×{ 1} by another equivalence relation on M to form a topological submanifold of N, to be called the bottom of N. In this case, M × (0, 1) is called the side of N. With the concept of topological ...
Xifang Cao
,
et al.
Geometric aspects of highorder eigenvalue problems I. Structures on s...
...In this case, our indexing scheme for eigenvalues of the associated regular problems is adopted from the recent paper of Binding and Volkmer [4] for separated BC’s and from [
5
] for coupled BC’s...
...The singular problems with p > 0 were studied in [18], and the regular problems with p changing sign were investigated in [
5
]...
...Consequently, the eigenvalues for coupled BC’s can be indexed according to the eigenvalue inequalities established in [
5
]...
...Although the numbers of zeros of eigenfunctions for this case are dieren t from (i), they can be determined based on the sign changes of p on J, see [
5
] for details...
...The following are analogues of Lemmas 4.1, 4.2 and Remark 4.1 for the case where p changes sign, see Theorem 3.10 in [
5
]...
...Remark 4.2. For the case when p changes sign on J, it has been further proved in [
5
] that for n 2 N0, the nth eigenvalue n = n(a0; b0; [AjB]) of SLP (4.1), (4.2) as a function of the endpoints a0; b0 2 (a; b) and the BC [AjB] 2 B is continuous whenever [AjB] 2 B n J ; and the continuous and discontinuous behavior of n(a0; b0; [AjB]) at a point where [AjB] 2 J is exactly the same as characterized in Lemma 4.6, no matter how a0; b0 change ...
L. KONG
,
et al.
REGULAR APPROXIMATIONS OF SINGULAR STURMLIOUVILLE PROBLEMS WITH LIMIT...
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QINGKAI KONG
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Dependence of Eigenvalues on the Problem
(
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Journal:
Mathematische Nachrichten  MATH NACHR
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Citations
(6)
Computing the indices of Sturm–Liouville eigenvalues for coupled boundary conditions (the EIGENINDSLP codes)
(
Citations: 1
)
Guixia Wang
,
Zhong Wang
,
Hongyou Wu
Journal:
Journal of Computational and Applied Mathematics  J COMPUT APPL MATH
, vol. 220, no. 1, pp. 490507, 2008
Relations among eigenvalues of Sturm–Liouville problems with different types of leading coefficient functions
Guixia Wang
,
Zhong Wang
,
Hongyou Wu
Journal:
Journal of Mathematical Analysis and Applications  J MATH ANAL APPL
, vol. 336, no. 2, pp. 10611072, 2007
Geometric aspects of highorder eigenvalue problems I. Structures on spaces of boundary conditions
(
Citations: 2
)
Xifang Cao
,
Hongyou Wu
Journal:
International Journal of Mathematics and Mathematical Sciences
, vol. 2004, no. 13, pp. 647678, 2004
Multiplicity of Sturm–Liouville eigenvalues
(
Citations: 2
)
Q. Kong
,
H. Wu
,
A. Zettl
Journal:
Journal of Computational and Applied Mathematics  J COMPUT APPL MATH
, vol. 171, no. 1, pp. 291309, 2004
Geometric Aspects of SturmLiouville Problems V. Natural Loops of Boundary Conditions for Monotonicity of Eigenvalues and Their Applications
(
Citations: 2
)
WUJIAN PENG
,
MIHAI RACOVITAN
,
HONGYOU WU