Spectral properties of a fourth-order eigenvalue problem with quadratic spectral parameters in a boundary condition

Chenghua Gao; Maojun Ran; Chenghua Gao; Maojun Ran

doi:10.3934/math.2020062

AIMS Mathematics

2020, Volume 5, Issue 2: 904-922. doi: 10.3934/math.2020062

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Research article

Spectral properties of a fourth-order eigenvalue problem with quadratic spectral parameters in a boundary condition

Chenghua Gao ^,,
Maojun Ran

Department of Mathematics, Northwest Normal University, Lanzhou, Gansu 730070, People’s Republic of China

Received: 25 October 2019 Accepted: 01 January 2020 Published: 08 January 2020
MSC : 34B05, 34B09

Consider the linear eigenvalue problem of fourth-order $ y^{(4)}(x)-(q(x)y'(x))' = \lambda y(x),\ \ \ 0 \lt x \lt 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 \lt i \gt λ \lt /i \gt is a spectal parameter, $\delta\in[\frac{\pi}{2}, \pi]$, \lt i \gt Ty \lt /i \gt = \lt i \gt y \lt /i \gt ''' - \lt i \gt qy \lt /i \gt ', \lt i \gt q \lt /i \gt ( \lt i \gt x \lt /i \gt ) is a positive absolutely continuous function on the interval [0, \lt i \gt l \lt /i \gt ], \lt i \gt δ \lt /i \gt , \lt i \gt a \lt /i \gt \lt sub \gt \lt i \gt i \lt /i \gt \lt /sub \gt and \lt i \gt b \lt /i \gt \lt sub \gt \lt i \gt i \lt /i \gt \lt /sub \gt ( \lt i \gt i \lt /i \gt = 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 \lt i \gt n \lt /i \gt . 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 \lt i \gt L \lt /i \gt \lt sub \gt \lt i \gt p \lt /i \gt \lt /sub \gt (0, \lt i \gt l \lt /i \gt ).
- fourth-order eigenvalue problem,
- quadratic spectral parameter,
- interlace,
- oscillation,
- basis properties
Citation: Chenghua Gao, Maojun Ran. Spectral properties of a fourth-order eigenvalue problem with quadratic spectral parameters in a boundary condition[J]. AIMS Mathematics, 2020, 5(2): 904-922. doi: 10.3934/math.2020062

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Abstract

Consider the linear eigenvalue problem of fourth-order $ y^{(4)}(x)-(q(x)y'(x))' = \lambda y(x),\ \ \ 0 \lt x \lt 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 \lt i \gt λ \lt /i \gt is a spectal parameter, $\delta\in[\frac{\pi}{2}, \pi]$, \lt i \gt Ty \lt /i \gt = \lt i \gt y \lt /i \gt ''' - \lt i \gt qy \lt /i \gt ', \lt i \gt q \lt /i \gt ( \lt i \gt x \lt /i \gt ) is a positive absolutely continuous function on the interval [0, \lt i \gt l \lt /i \gt ], \lt i \gt δ \lt /i \gt , \lt i \gt a \lt /i \gt \lt sub \gt \lt i \gt i \lt /i \gt \lt /sub \gt and \lt i \gt b \lt /i \gt \lt sub \gt \lt i \gt i \lt /i \gt \lt /sub \gt ( \lt i \gt i \lt /i \gt = 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 \lt i \gt n \lt /i \gt . 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 \lt i \gt L \lt /i \gt \lt sub \gt \lt i \gt p \lt /i \gt \lt /sub \gt (0, \lt i \gt l \lt /i \gt ).

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