Journal ArticleDOI
Transformations and the number of focal points for conjoined bases of symplectic difference systems
TLDR
In this article, the authors derived new relations between the number of focal points of Y i and P i Y i, where P i is an arbitrary symplectic transformation and formulated conditions which guarantee that a given transformation preserves oscillatory properties of transformed systems.Abstract:
This paper studies symplectic transformations for conjoined bases of the symplectic difference systems where the matrix W i is symplectic for We derive new relations between the number of focal points of Y i and P i Y i , where P i is an arbitrary symplectic transformation. We formulate conditions which guarantee that a given transformation preserves oscillatory properties of transformed systems. In particular, for the case and , we establish duality between eventual disconjugacy of general symplectic systems and their reciprocals.read more
Citations
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Journal ArticleDOI
Comparison theorems for symplectic systems of difference equations
TL;DR: In this article, the authors generalize comparison theorems for difference analogs of canonical systems of differential equations, and obtain general relations between the number of focal points of conjoined bases of two symplectic systems of difference equations.
Journal ArticleDOI
On relative oscillation theory for symplectic eigenvalue problems
TL;DR: An analog of classical oscillation theory for discrete symplectic eigenvalue problems with Dirichlet boundary conditions is developed which, rather than measuring the spectrum of one single problem, measures the difference between the spectra of two different problems.
Journal ArticleDOI
Oscillation theorems for discrete symplectic systems with nonlinear dependence in spectral parameter
TL;DR: In this article, the authors introduced nonlinear dependence in the spectral parameter of discrete symplectic systems and Sturm-Liouville difference equations and proved the corresponding oscillation theorem for Dirichlet boundary conditions.
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On symplectic transformations of linear Hamiltonian differential systems without normality
TL;DR: The main tool of the paper is the comparative index theory for discrete symplectic systems generalized to the continuous case and formulates the generalized reciprocity principle for the Hamiltonian systems without normality.
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A note on relative oscillation theory for symplectic difference systems with general boundary conditions
TL;DR: An analog of classical oscillation theory for discrete symplectic eigenvalue problems with general self-adjoint boundary conditions which, rather than measuring of the spectrum of one single problem, measures the difference between the spectra of two different problems.
References
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Book
Sturmian Theory for Ordinary Differential Equations
TL;DR: In this article, Sturmian Theory for Real Linear Homogeneous Second Order Ordinary Differential Equations on a Compact Interval is presented, and a survey of recent literature is presented.
Journal ArticleDOI
Disconjugacy and Transformations for Symplectic Systems
Martin Bohner,Ondřej Došlý +1 more
TL;DR: In this paper, the Riccati-type matrix difference equation and a certain quadratic functional play the same role in this theory and their scalar counterparts, and the basic oscillation and transformation properties of symplectic difference systems are established.
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Linear Hamiltonian Difference Systems: Disconjugacy and Jacobi-Type Conditions
TL;DR: In this paper, the authors considered a linear Hamiltonian Difference System for the singular case and proved a Reid Roundabout Theorem which gives conditions equivalent to positive definiteness of a certain discrete quadratic functional, among them the strengthened Jacobi's Condition and a condition on a Riccati Difference Equation.
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An Oscillation Theorem for Discrete Eigenvalue Problems
TL;DR: The main result of as mentioned in this paper relates the number of eigenvalues to the number normalized generalized zeros of solutions of the eigenvalue problems for SDFs, which is an oscillation-theorem.
Journal ArticleDOI
Sturmian and spectral theory for discrete symplectic systems
TL;DR: In this paper, the Rayleigh principle is used to compare the number of focal points of two conjoined bases of two different configurations of a pair of symplectic difference systems, and it is shown that the numbers differ by at most n.