Renormalized oscillation theory for symplectic eigenvalue problems with nonlinear dependence on the spectral parameter

In this paper, we establish new renormalized oscillation theorems for discrete symplectic eigenvalue problems with Dirichlet boundary conditions. These theorems present the number of finite eigenvalues of the problem in the arbitrary interval using the number of focal points of a transformed conjoin...

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Bibliographic Details
Published in:Journal of difference equations and applications Vol. 26; no. 4; pp. 458 - 487
Main Author: Elyseeva, Julia
Format: Journal Article
Language:English
Published: Abingdon Taylor & Francis 02.04.2020
Taylor & Francis Ltd
Subjects:
Boundary conditions
comparative index
Dirichlet problem
Discrete eigenvalue problem
Eigenvalues
Parameters
renormalized oscillation theory
Spectra
symplectic difference system
Theorems
ISSN:1023-6198, 1563-5120
Online Access:Get full text
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Summary:In this paper, we establish new renormalized oscillation theorems for discrete symplectic eigenvalue problems with Dirichlet boundary conditions. These theorems present the number of finite eigenvalues of the problem in the arbitrary interval using the number of focal points of a transformed conjoined basis associated with Wronskian of two principal solutions of the symplectic system evaluated at the endpoints a and b. We suppose that the symplectic coefficient matrix of the system depends nonlinearly on the spectral parameter and that it satisfies certain natural monotonicity assumptions. In our treatment, we admit possible oscillations in the coefficients of the symplectic system by incorporating their non-constant rank with respect to the spectral parameter.
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ISSN:1023-6198
1563-5120
DOI:10.1080/10236198.2020.1748020