scholarly journals On Inverse Nodal Problem and Multiplicities of Eigenvalues of a Vectorial Sturm-Liouville Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Xiaoyun Liu
Keyword(s):  
Boundary Conditions ◽  
Infinite Sequence ◽  
Liouville Problem ◽  
Unitary Matrix ◽  
Multiplicity Results ◽  
Inverse Nodal Problem ◽  
Potential Matrix ◽  
Sturm Liouville ◽  
Nodal Problem ◽  
Vectorial Functions

An m-dimensional vectorial inverse nodal Sturm-Liouville problem with eigenparameter-dependent boundary conditions is studied. We show that if there exists an infinite sequence ynj,rx,λnj,r2j=1∞ of eigenfunctions which are all vectorial functions of type (CZ), then the potential matrix Qx and A are simultaneously diagonalizable by the same unitary matrix U. Subsequently, some multiplicity results of eigenvalues are obtained.

scholarly journals INVERSE NODAL PROBLEM FOR A CLASS OF NONLOCAL STURM‐LIOUVILLE OPERATOR

2010 ◽  
Vol 15 (3) ◽  
pp. 383-392 ◽  
Author(s):  
Chuan-Fu Yang
Keyword(s):  
Boundary Condition ◽  
Dense Subset ◽  
Liouville Problem ◽  
Integral Boundary ◽  
Inverse Nodal Problem ◽  
Classical Boundary Condition ◽  
Classical Boundary ◽  
Sturm Liouville ◽  
Nodal Problem ◽  
The Given

Inverse nodal problem consists in constructing operators from the given nodes (zeros) of their eigenfunctions. In this work, the Sturm‐Liouville problem with one classical boundary condition and another nonlocal integral boundary condition is considered. We prove that a dense subset of nodal points uniquely determine the boundary condition parameter and the potential function of the Sturm‐Liouville equation. We also provide a constructive procedure for the solution of the inverse nodal problem.

On the quasi-nodal map for the Sturm–Liouville problem

2006 ◽  
Vol 136 (1) ◽  
pp. 71-86 ◽  
Author(s):  
Y. H. Cheng ◽  
C. K. Law
Keyword(s):  
Liouville Problem ◽  
Well Posedness ◽  
Reconstruction Formula ◽  
Inverse Nodal Problem ◽  
Sturm Liouville ◽  
Nodal Problem ◽  
Definition Of ◽  
Admissible Sequences ◽  
Liouville Operators

We show that the space of Sturm–Liouville operators characterized by H = (q, α, β) ∈ L1 (0, 1) × [0, π)2 such that is homeomorphic to the partition set of the space of all admissible sequences which form sequences that converge to q, α, and β individually. This space, Γ, of quasi-nodal sequences is a superset of, and is more natural than, the space of asymptotically nodal sequences defined in Law and Tsay (On the well-posedness of the inverse nodal problem. Inv. Probl.17 (2001), 1493–1512). The definition of Γ relies on the L1 convergence of the reconstruction formula for q by the exactly nodal sequence.

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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