Uniform convergence of spectral expansions for a problem with a boundary condition depending on a spectral parameter
type: Item
instance of: scholarly article
title: Uniform convergence of spectral expansions for a problem with a boundary condition depending on a spectral parameter
DOI: 10.1134/S0012266122090014
author: Ziyatkhan S. Aliyev
author: K. F. Abdullaeva
published in: Differential Equations
publication date: 2022-11-08T00:00:00Z
review text: The authors study the fourth-order eigenvatue problem \[ y^{(4)}(x) - (q(x) y'(x))' = \lambda y(x) \] on a compact interval \([0,l]\) with a positive function \(q \in AC[0,l]\) and equipped with boundary conditions involving the (derivatives of) the function at \(0\) and \(l\). It is the essential feature of the paper that one of the four linear boundary conditions depends linearly on the eigenvalue parameter \(\lambda\). In an earlier paper [\textit{Z. S. Aliyev}, Cent. Eur. J. Math. 8, No. 2, 378--388 (2010; Zbl 1207.34112)] a self-adjoint and lower semibounded operator \(L\) with discrete spectrum in the Hilbert space \(L_2(0,l) \oplus {\mathbb C}\) was constructed such that the eigenvalue problem has the linearization \(L\hat{y} = \lambda \hat{y}\). The current focus is at first on asymptotic estimates for the eigenvalues and eigenfunctions. To this end modified problems are considered without \(\lambda\)-dependency. Then, the location of the original eigenvalues are estimated by those of the simpler problems and also oscillation properties of the corresponding eigenfunctions are obtained. In the next step, an asymptotic result on a simpler problem is proved and transferred to the \(\lambda\)-dependent problem. Finally, the basis property of the eigenfunctions in \(L_p(0,l)\) is obtained for \(1 < p < \infty\) where one of the eigenfunctions is omitted. Furthermore, a condion is formulated such that the corresponding Fourier series (i.e. eigenfunction expansion) for a continuous function converges uniformly.
reviewed by: Andreas Fleige
Mathematics Subject Classification ID: 34B09
Mathematics Subject Classification ID: 34L10
Mathematics Subject Classification ID: 34L15
Mathematics Subject Classification ID: 34L20
zbMATH DE Number: 7613629
zbMATH Keywords: boundary eigenvalue problem
zbMATH Keywords: fourth-order differential equation
zbMATH Keywords: parameter-dependent boundary condition
zbMATH Keywords: eigenvalue asymptotics
zbMATH Keywords: oscillation properties
zbMATH Keywords: Fourier series
Wikidata QID: Q115248684
MaRDI profile type: Publication
full work available at URL: s0012266122090014
OpenAlex ID: W4312494576
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description: scientific article; zbMATH DE number 7613629