Basis properties of a fourth order differential operator with spectral parameter in the boundary condition

Ziyatkhan Aliyev

Displaying similar documents to “Basis properties of a fourth order differential operator with spectral parameter in the boundary condition”

Inequalities among eigenvalues of second-order difference equations with general coupled boundary conditions.

Zhang, Chao, Sun, Shurong (2009)

Advances in Difference Equations [electronic only]

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Asymptotic analysis of non-self-adjoint Hill operators

Oktay Veliev (2013)

Open Mathematics

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We obtain uniform asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators L t (q) with a potential q ∈ L 1[0,1] and t-periodic boundary conditions, t ∈ (−π, π]. Using these formulas, we find sufficient conditions on the potential q such that the number of spectral singularities in the spectrum of the Hill operator L(q) in L 2(−∞,∞) is finite. Then we prove that the operator L(q) has no spectral singularities at infinity and it is an asymptotically...

Spectral properties of some regular boundary value problems for fourth order differential operators

Nazim Kerimov, Ufuk Kaya (2013)

Open Mathematics

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In this paper we consider the problem $\begin{matrix} y^{i v} + p_{2} (x) y^{''} + p_{1} (x) y^{'} + p_{0} (x) y = λ y, 0 < x < 1, \\ y^{(s)} (1) - {(- 1)}^{σ} y^{(s)} (0) + \sum_{l = 0}^{s - 1} α_{s, l} y^{(l)} (0) = 0, s = 1, 2, 3, \\ y (1) - {(- 1)}^{σ} y (0) = 0, \end{matrix}$ where λ is a spectral parameter; p j (x) ∈ L 1(0, 1), j = 0, 1, 2, are complex-valued functions; α s;l, s = 1, 2, 3, $l = \bar{0, s - 1}$ , are arbitrary complex constants; and σ = 0, 1. The boundary conditions of this problem are regular, but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established in the case α 3,2 + α 1,0 ≠ α 2,1. It is proved that the system of root functions of this...

A Sturm-Liouville problem with spectral and large parameters in boundary conditions and the associated Cauchy problem

Jamel Ben Amara (2011)

Colloquium Mathematicae

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We study a Sturm-Liouville problem containing a spectral parameter in the boundary conditions. We associate to this problem a self-adjoint operator in a Pontryagin space Π₁. Using this operator-theoretic formulation and analytic methods, we study the asymptotic behavior of the eigenvalues under the variation of a large physical parameter in the boundary conditions. The spectral analysis is applied to investigate the well-posedness and stability of the wave equation of a string. ...

Some global results for nonlinear fourth order eigenvalue problems

Ziyatkhan Aliyev (2014)

Open Mathematics

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In this paper, we consider the nonlinear fourth order eigenvalue problem. We show the existence of family of unbounded continua of nontrivial solutions bifurcating from the line of trivial solutions. These global continua have properties similar to those found in Rabinowitz and Berestycki well-known global bifurcation theorems.