Spectral properties of a fourth-order eigenvalue problem with quadratic spectral parameters in a boundary condition
Consider the linear eigenvalue problem of fourth-order$$y^{(4)}(x)-(q(x)y'(x))'=\lambda y(x),\ \ \ 0<x<l,$$$$y(0)=y'(0)=0,$$$$(a_0+a_1\lambda+a_2\lambda^2)y'(l)+(b_0+b_1\lambda+b_2\lambda^2)y''(l)=0,$$$$y(l)\cos\delta-Ty(l)\sin\delta=0,$$where <em>λ&...
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AIMS Press
2020-01-01
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| author | Chenghua Gao Maojun Ran |
| author_facet | Chenghua Gao Maojun Ran |
| author_sort | Chenghua Gao |
| collection | DOAJ |
| description | Consider the linear eigenvalue problem of fourth-order$$y^{(4)}(x)-(q(x)y'(x))'=\lambda y(x),\ \ \ 0<x<l,$$$$y(0)=y'(0)=0,$$$$(a_0+a_1\lambda+a_2\lambda^2)y'(l)+(b_0+b_1\lambda+b_2\lambda^2)y''(l)=0,$$$$y(l)\cos\delta-Ty(l)\sin\delta=0,$$where <em>λ</em> is a spectal parameter, $\delta\in[\frac{\pi}{2},\pi]$, <em>Ty</em> = <em>y</em>''' - <em>qy</em>', <em>q</em>(<em>x</em>) is a positive absolutely continuous function on the interval [0,<em>l</em>], <em>δ</em>, <em>a</em><sub><em>i</em></sub> and <em>b</em><sub><em>i</em></sub> (<em>i</em>=0,1,2) are real constants. We obtain not only the existence, simplicity and interlacing properties of the eigenvalues, the oscillation properties of the eigenfunctions, but also the asymptotic formula of the eigenvalues and the corresponding eigenfunctions for sufficiently large <em>n</em>. Moreover, a new inner Hilbert space and a new sufficient conditions will be given to discuss the basis properties of the system of the eigenfunctions in <em>L</em><sub><em>p</em></sub>(0,<em>l</em>). |
| first_indexed | 2024-12-20T20:32:22Z |
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| id | doaj.art-480f5be6d8e54ec8ad18e3c7c71a6320 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-20T20:32:22Z |
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| spelling | doaj.art-480f5be6d8e54ec8ad18e3c7c71a63202022-12-21T19:27:20ZengAIMS PressAIMS Mathematics2473-69882020-01-015290492210.3934/math.2020062Spectral properties of a fourth-order eigenvalue problem with quadratic spectral parameters in a boundary conditionChenghua Gao0Maojun Ran1Department of Mathematics, Northwest Normal University, Lanzhou, Gansu 730070, People’s Republic of ChinaDepartment of Mathematics, Northwest Normal University, Lanzhou, Gansu 730070, People’s Republic of ChinaConsider the linear eigenvalue problem of fourth-order$$y^{(4)}(x)-(q(x)y'(x))'=\lambda y(x),\ \ \ 0<x<l,$$$$y(0)=y'(0)=0,$$$$(a_0+a_1\lambda+a_2\lambda^2)y'(l)+(b_0+b_1\lambda+b_2\lambda^2)y''(l)=0,$$$$y(l)\cos\delta-Ty(l)\sin\delta=0,$$where <em>λ</em> is a spectal parameter, $\delta\in[\frac{\pi}{2},\pi]$, <em>Ty</em> = <em>y</em>''' - <em>qy</em>', <em>q</em>(<em>x</em>) is a positive absolutely continuous function on the interval [0,<em>l</em>], <em>δ</em>, <em>a</em><sub><em>i</em></sub> and <em>b</em><sub><em>i</em></sub> (<em>i</em>=0,1,2) are real constants. We obtain not only the existence, simplicity and interlacing properties of the eigenvalues, the oscillation properties of the eigenfunctions, but also the asymptotic formula of the eigenvalues and the corresponding eigenfunctions for sufficiently large <em>n</em>. Moreover, a new inner Hilbert space and a new sufficient conditions will be given to discuss the basis properties of the system of the eigenfunctions in <em>L</em><sub><em>p</em></sub>(0,<em>l</em>).https://www.aimspress.com/article/10.3934/math.2020062/fulltext.htmlfourth-order eigenvalue problemquadratic spectral parameterinterlaceoscillationbasis properties |
| spellingShingle | Chenghua Gao Maojun Ran Spectral properties of a fourth-order eigenvalue problem with quadratic spectral parameters in a boundary condition AIMS Mathematics fourth-order eigenvalue problem quadratic spectral parameter interlace oscillation basis properties |
| title | Spectral properties of a fourth-order eigenvalue problem with quadratic spectral parameters in a boundary condition |
| title_full | Spectral properties of a fourth-order eigenvalue problem with quadratic spectral parameters in a boundary condition |
| title_fullStr | Spectral properties of a fourth-order eigenvalue problem with quadratic spectral parameters in a boundary condition |
| title_full_unstemmed | Spectral properties of a fourth-order eigenvalue problem with quadratic spectral parameters in a boundary condition |
| title_short | Spectral properties of a fourth-order eigenvalue problem with quadratic spectral parameters in a boundary condition |
| title_sort | spectral properties of a fourth order eigenvalue problem with quadratic spectral parameters in a boundary condition |
| topic | fourth-order eigenvalue problem quadratic spectral parameter interlace oscillation basis properties |
| url | https://www.aimspress.com/article/10.3934/math.2020062/fulltext.html |
| work_keys_str_mv | AT chenghuagao spectralpropertiesofafourthordereigenvalueproblemwithquadraticspectralparametersinaboundarycondition AT maojunran spectralpropertiesofafourthordereigenvalueproblemwithquadraticspectralparametersinaboundarycondition |