scholarly journals Spectral Properties of the Differential Operators of the Fourth-Order with Eigenvalue Parameter Dependent Boundary Condition

2012 ◽  
Vol 2012 ◽  
pp. 1-28 ◽  
Author(s):  
Ziyatkhan S. Aliyev ◽  
Nazim B. Kerimov
Keyword(s):  
Boundary Condition ◽  
Fourth Order ◽  
Spectral Properties ◽  
Differential Operators ◽  
Eigenvalue Parameter ◽  
Parameter Dependent

scholarly journals Asymptotics of the Eigenvalues of a Self-Adjoint Fourth Order Boundary Value Problem with Four Eigenvalue Parameter Dependent Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Manfred Möller ◽  
Bertin Zinsou
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Imaginary Axis ◽  
Order Ordinary Differential Equation ◽  
Eigenvalue Parameter ◽  
Upper Half Plane ◽  
Parameter Dependent ◽  
Asymptotics Of The Eigenvalues

Considered is a regular fourth order ordinary differential equation which depends quadratically on the eigenvalue parameterλand which has separable boundary conditions depending linearly onλ. It is shown that the eigenvalues lie in the closed upper half plane or on the imaginary axis and are symmetric with respect to the imaginary axis. The first four terms in the asymptotic expansion of the eigenvalues are provided.

Pencils of differential operators containing the eigenvalue parameter in the boundary conditions

2003 ◽  
Vol 133 (4) ◽  
pp. 893-917 ◽  
Author(s):  
Marco Marletta ◽  
Andrei Shkalikov ◽  
Christiane Tretter
Keyword(s):  
Boundary Conditions ◽  
Spectral Properties ◽  
Differential Operators ◽  
Finite Interval ◽  
Linear Operators ◽  
Riesz Bases ◽  
Ordinary Differential Operators ◽  
Associated Functions ◽  
Eigenvalue Parameter

The paper deals with linear pencils N − λP of ordinary differential operators on a finite interval with λ-dependent boundary conditions. Three different problems of this form arising in elasticity and hydrodynamics are considered. So-called linearization pairs (W, T) are constructed for the problems in question. More precisely, functional spaces W densely embedded in L2 and linear operators T acting in W are constructed such that the eigenvalues and the eigen- and associated functions of T coincide with those of the original problems. The spectral properties of the linearized operators T are studied. In particular, it is proved that the eigen- and associated functions of all linearizations (and hence of the corresponding original problems) form Riesz bases in the spaces W and in other spaces which are obtained by interpolation between D(T) and W.

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