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

Abstract

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 λ is a spectal parameter, $\delta\in[\frac{\pi}{2},\pi]$, Ty = y''' - qy', q(x) is a positive absolutely continuous function on the interval [0,l], δ, ai and bi (i=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 n. 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 Lp(0,l).

Keywords

• fourth-order eigenvalue problem
• quadratic spectral parameter
• interlace
• oscillation
• basis properties