Comparative index for solutions of symplectic difference systems

Eliseeva, Yu. V.

doi:10.1134/s0012266109030148

Cited by 41 publications

(23 citation statements)

References 6 publications

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“…where l Ã ðkÞ ¼ lðZ k ;Z k Þ is the dual comparative index of Z k ;Z k , which relates this index to the multiplicities of a backward focal point m Ã ðkÞ;m Ã ðkÞ in ½k; k þ 1Þ of Z andZ, respectively, see [15]. Nonoscillation and controllability of (1) implies that both X k andX k are invertible for large k, this means that l Ã…”

Section: Singular Sturmian Separation Theoremsmentioning

confidence: 97%

“…As a starting point of our treatment, let us recall the ''regular'' Sturmian separation theorems for solutions of (1) as established in [8] and independently in [15]. Together with (1) we will consider its matrix version (referred to again as (1))…”

Section: Preliminariesmentioning

confidence: 99%

“…The first statement is a ''reversed'' version of [8, Theorem 1.3] (''reversed'' means that it is Theorem 1.3 of [8] applied to the reversed system (5)), and it is proved using a variational argument (quadratic functional associated with (1) and Picone's identity). The second statement is formulated in [15] and it is proved via the comparative index of conjoined bases of (1). In both statements and later on in our paper, all focal points are counted with their multiplicities.…”

Section: Preliminariesmentioning

confidence: 99%

“…More precisely, letX U be the recessive solution of (1) at þ1, then the (symmetric) matrixQ ¼ŨX À1 is the eventually minimal (sometimes called also distinguished) solution of (8) in the sense that any other symmetric solution Q of (8) which is defined for large k and B T k D k þ B T k Q kþ1 B k P 0, satisfies Q k PQ k for large k, see [9]. We finish this preliminary section by recalling the definition and some properties of the comparative index of conjoined bases as introduced and treated in [13][14][15][16][17]. Let Z ¼ X U…”

Section: Preliminariesmentioning

confidence: 99%

See 3 more Smart Citations

Focal points and recessive solutions of discrete symplectic systems

Došlý

2014

Applied Mathematics and Computation

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Section: Singular Sturmian Separation Theoremsmentioning

confidence: 97%

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

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Focal points and recessive solutions of discrete symplectic systems

Došlý

2014

Applied Mathematics and Computation

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“…(iii) The third research direction is connected with a possible extensions and applications of the classical notion of a backward focal point (see [17,18]). Concerning linear symplectic spectral problems, the number of backward focal points in [0, N + 1) is associated with the number of finite eigenvalues of (1.4), (1.5), (1.8) less than a given λ (see formula (3.9) in [15]).…”

Section: )mentioning

confidence: 99%

Discrete oscillation theorems and weighted focal points for Hamiltonian difference systems with nonlinear dependence on a spectral parameter

Došlý

Elyseeva

2015

Applied Mathematics Letters

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a b s t r a c tIn this paper we generalize oscillation theorems for discrete Hamiltonian eigenvalue problems with nonlinear dependence on the spectral parameter λ. In our version of the discrete oscillation theorems, we incorporate the case when the block B k (λ) of the discrete Hamiltonian H k (λ) has nonconstant rank with respect to λ. We introduce a new notion of weighted focal points for conjoined bases of the Hamiltonian difference systems and we show that the number of weighted focal points plays the role of the classical number of focal points in the discrete oscillation theorems for the Hamiltonian spectral problems with the nonconstant rank of B k (λ).

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Relative oscillation theory for linear Hamiltonian systems with nonlinear dependence on the spectral parameter

Elyseeva

2022

Mathematische Nachrichten

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In this paper, we consider two linear Hamiltonian differential systems that depend in general nonlinearly on the spectral parameter 𝜆 and with Dirichlet boundary conditions. For the Hamiltonian problems, we do not assume any controllability and strict normality assumptions and also omit the classical Legendre condition for their Hamiltonians. The main result of the paper, the relative oscillation theorem, relates the difference of the numbers of finite eigenvalues of the two problems in the intervals (−∞, 𝛽] and (−∞, 𝛼], respectively, with the so-called oscillation numbers associated with the Wronskian of the principal solutions of the systems evaluated for 𝜆 = 𝛼 and 𝜆 = 𝛽. As a corollary to the main result, we prove the renormalized oscillation theorems presenting the number of finite eigenvalues of one single problem in (𝛼, 𝛽]. The consideration is based on the comparative index theory applied to the continuous case.

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Comparative index for solutions of symplectic difference systems

Cited by 41 publications

References 6 publications

Focal points and recessive solutions of discrete symplectic systems

Focal points and recessive solutions of discrete symplectic systems

Discrete oscillation theorems and weighted focal points for Hamiltonian difference systems with nonlinear dependence on a spectral parameter

Relative oscillation theory for linear Hamiltonian systems with nonlinear dependence on the spectral parameter

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