Spectral properties of a beam equation with eigenvalue parameter occurring linearly in the boundary conditions

2021 ◽  
pp. 1-22
Author(s):  
Ziyatkhan S. Aliyev ◽  
Gunay T. Mamedova
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Spectral Properties ◽  
Sufficient Conditions ◽  
Beam Equation ◽  
Asymptotic Formulas ◽  
Eigenvalues And Eigenfunctions ◽  
Root Functions ◽  
Eigenvalue Parameter ◽  
Oscillation Properties

In this paper, we consider an eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter in the boundary conditions. The location of eigenvalues on real axis, the structure of root subspaces and the oscillation properties of eigenfunctions of this problem are investigated, and asymptotic formulas for the eigenvalues and eigenfunctions are found. Next, by the use of these properties, we establish sufficient conditions for subsystems of root functions of the considered problem to form a basis in the space $L_p,1 < p < \infty$ .

scholarly journals Transmission problem for the Sturm-Liouville equation involving a retarded argument

Filomat ◽  
2021 ◽  
Vol 35 (6) ◽  
pp. 2071-2080
Author(s):  
Erdoğan Şen
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Spectral Parameter ◽  
Liouville Equation ◽  
Spectral Properties ◽  
Asymptotic Formulas ◽  
Retarded Argument ◽  
Eigenvalues And Eigenfunctions ◽  
Discontinuous Boundary ◽  
Sturm Liouville

In this work, spectral properties of a discontinuous boundary-value problem with retarded argument which contains a spectral parameter in the boundary conditions and in the transmission conditions at the point of discontinuity are investigated. To this aim, asymptotic formulas for the eigenvalues and eigenfunctions are obtained.

scholarly journals On the numerical solution for nonlinear elliptic equations with variable weight coefficients in an integral boundary conditions

2021 ◽  
Vol 26 (4) ◽  
pp. 738-758
Author(s):  
Regimantas Čiupaila ◽  
Kristina Pupalaigė ◽  
Mifodijus Sapagovas
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Sufficient Conditions ◽  
Integral Boundary Conditions ◽  
Nonlinear Elliptic Equations ◽  
Integral Boundary ◽  
Nonlinear Elliptic ◽  
Variable Weight ◽  
The Difference ◽  
Convergence Of Iterative Method

In the paper the two-dimensional elliptic equation with integral boundary conditions is solved by finite difference method. The main aim of the paper is to investigate the conditions for the convergence of the iterative methods for the solution of system of nonlinear difference equations. With this purpose, we investigated the structure of the spectrum of the difference eigenvalue problem. Some sufficient conditions are proposed such that the real parts of all eigenvalues of the corresponding difference eigenvalue problem are positive. The proof of convergence of iterative method is based on the properties of the M-matrices not requiring the symmetry or diagonal dominance of the matrices. The theoretical statements are supported by the results of the numerical experiment.

scholarly journals Indefinite eigenvalue problem with eigenparameter in the two boundary conditions

1998 ◽  
Vol 21 (4) ◽  
pp. 775-784
Author(s):  
S. F. M. Ibrahim
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Eigenfunction Expansion ◽  
Selfadjoint Operator ◽  
Order Differential Equation ◽  
Expansion Theorem ◽  
Compact Resolvent ◽  
Second Order Differential Equation ◽  
Eigenvalue Parameter

The object of this paper is to establish an expansion theorem for a regular indefinite eigenvalue problem of second order differential equation with an eigenvalue parameter,λin the two boundary conditions. We associated with this problem aJ-selfadjoint operator with compact resolvent defined in a suitable Krein space and then we develop an associated eigenfunction expansion theorem.

scholarly journals Positive Solutions for the Eigenvalue Problem of Semipositone Fractional Order Differential Equation with Multipoint Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Ge Dong
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Boundary Conditions ◽  
Positive Solution ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Fractional Order Differential Equation ◽  
Multipoint Boundary ◽  
Multipoint Boundary Conditions

We study the existence of positive solution for the eigenvalue problem of semipositone fractional order differential equation with multipoint boundary conditions by using known Krasnosel'skii's fixed point theorem. Some sufficient conditions that guarantee the existence of at least one positive solution for eigenvalues  λ>0sufficiently small andλ>0sufficiently large are established.

scholarly journals POSITIVE SOLUTIONS OF HIGHER-ORDER BOUNDARY-VALUE PROBLEMS

2005 ◽  
Vol 48 (2) ◽  
pp. 445-464 ◽  
Author(s):  
Lingju Kong ◽  
Qingkai Kong
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Nonlinear Boundary Conditions ◽  
Existence Of Positive Solutions ◽  
Eigenvalue Parameter ◽  
Even Order

AbstractWe consider a class of even-order boundary-value problems with nonlinear boundary conditions and an eigenvalue parameter $\lambda$ in the equations. Sufficient conditions are obtained for the existence and non-existence of positive solutions of the problems for different values of $\lambda$.

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.

scholarly journals Inverse problems for the Sturm–Liouville equation with a spectral parameter in the boundary condition

2017 ◽  
Author(s):  
Namig J. Guliyev
Keyword(s):  
Boundary Condition ◽  
Inverse Problems ◽  
Boundary Conditions ◽  
Spectral Parameter ◽  
Liouville Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Eigenvalue Parameter ◽  
Sturm Liouville ◽  
Necessary And Sufficient

Inverse problems of recovering the coefficients of Sturm--Liouville problems with the eigenvalue parameter linearly contained in one of the boundary conditions are studied: (1) from the sequences of eigenvalues and norming constants; (2) from two spectra. Necessary and sufficient conditions for the solvability of these inverse problems are obtained.

Spectral properties of ordinary differential operators with involuton

2019 ◽  
Vol 484 (1) ◽  
pp. 12-17 ◽  
Author(s):  
V. E. Vladykina ◽  
A. A. Shkalikov
Keyword(s):  
Boundary Conditions ◽  
Unconditional Basis ◽  
Spectral Properties ◽  
Differential Operators ◽  
Root Functions ◽  
Ordinary Differential Operators ◽  
Normal Boundary ◽  
The Space L2

Let P and Q be ordinary differential operators of order n and m generated by s = max{n; m} boundary conditions on a nite interval [a; b]. We study operators of the form L = JP + Q, where J is the involution operator in the space L2[a; b]. We consider three cases n > m, n < m, and n = m, for which we dene concepts of regular, almost regular, and normal boundary conditions. We announce theorems on unconditional basis and completeness of the root functions of operator L depending on the type of boundary conditions from selected classes.

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