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Spectral Problem for a Polynomial Pencil of the Sturm-Liouville Equations: On the Completeness of the System of Eigenfunctions and Associated Eigenfunction, Asymptotic Formula
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Author(s): Anar Adiloğlu-Nabiev (Süleyman Demirel University, Turkey)
Copyright: 2020
Pages: 19
Source title:
Emerging Applications of Differential Equations and Game Theory
Source Author(s)/Editor(s): Sırma Zeynep Alparslan Gök (Süleyman Demirel University, Turkey)and Duygu Aruğaslan Çinçin (Süleyman Demirel University, Turkey)
DOI: 10.4018/978-1-7998-0134-4.ch008
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Abstract
A boundary value problem for the second order differential equation -y′′+∑_{m=0}N−1λ^{m}q_{m}(x)y=λ2Ny with two boundary conditions a_{i1}y(0)+a_{i2}y′(0)+a_{i3}y(π)+a_{i4}y′(π)=0, i=1,2 is considered. Here n>1, λ is a complex parameter, q0(x),q1(x),...,q_{n-1}(x) are summable complex-valued functions, a_{ik} (i=1,2; k=1,2,3,4) are arbitrary complex numbers. It is proved that the system of eigenfunctions and associated eigenfunctions is complete in the space and using elementary asymptotical metods asymptotic formulas for the eigenvalues are obtained.
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