scholarly journals DISCONTINUOUS STURM-LIOUVILLE PROBLEMS INVOLVING AN ABSTRACT LINEAR OPERATOR

2020 ◽  
Vol 10 (4) ◽  
pp. 1545-1560
Author(s):  
Oktay Sh. Mukhtarov ◽  
◽  
Kadriye Aydemir ◽  
◽  
Keyword(s):  
Linear Operator ◽  
Sturm Liouville

scholarly journals New Type of Sturm-Liouville Problems in Associated Hilbert Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
O. Sh. Mukhtarov ◽  
K. Aydemir
Keyword(s):  
Linear Operator ◽  
Asymptotic Behaviour ◽  
Spectral Parameter ◽  
Hilbert Spaces ◽  
Resolvent Operator ◽  
Sufficient Conditions ◽  
Transmission Conditions ◽  
Continuous Case ◽  
New Type ◽  
Sturm Liouville

We introduce a new type of discontinuous Sturm-Liouville problems, involving an abstract linear operator in equation. By suggesting own approaches we define some new Hilbert spaces to establish such properties as isomorphism, coerciveness, and maximal decreasing of resolvent operator with respect to spectral parameter. Then we find sufficient conditions for discreteness of the spectrum and examine asymptotic behaviour of eigenvalues. Obtained results are new even for continuous case, that is, without transmission conditions.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

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