Linear Hamiltonian difference systems: Disconjugacy and Jacobi-type conditions
type: Item
instance of: scholarly article
title: Linear Hamiltonian difference systems: Disconjugacy and Jacobi-type conditions
zbMATH Open document ID: 0855.39018
published in: Journal of Mathematical Analysis and Applications
publication date: 1997-02-04T00:00:00Z
review text: The author discusses linear Hamiltonian difference systems \[ \Delta x_k = A_k x_{k + 1} + B_k u_k,\quad \Delta u_k = C_k x_{k + 1} - A^T_k u_k, \quad k \in J : = \{0, 1, \dots, N\}, \tag{H} \] where \(x_k\), \(u_k \in \mathbb{R}^n\), \(k \in \overline J : = J \cup \{N + 1\}\), \(A_k\), \(B_k\), \(C_k\) are \(n \times n\)-matrices, \(B_k\), \(C_k\) symmetric, \(A_k\) such that \(\widetilde A_k = (I - A_k)^{-1}\) exist. For the controllable system (H) the extended Reid Roundabout Theorem is proved; that is equivalence of: a) positivity of some quadratic functional, b) disconjugacy, c) absence of focal points in the principal solution, d) Riccati condition. A particular case of this result, with boundary conditions \(x_0 = x_N = 0\), is considered separately. To get the main result, a discrete version of Picone's identity is proved, also several definitions like focal points or generalized zeros of vector-valued functions are introduced. Without assumption of nonsingularity of the matrix \(B_k\) the presented theory includes discrete Sturm-Liouville equations of higher order. Various interconnections inside the theory and relations with earlier results are widely discussed. See also e.g. \textit{C. D. Ahlbrandt} [J. Math. Anal. Appl. 180, No. 2, 498-517 (1993; Zbl 0802.39005)], \textit{L. H. Erbe} and \textit{P. Yan} [ibid. 167, No. 2, 355-367 (1992; Zbl 0762.39003)].
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 39A10
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 93C55
zbMATH DE Number: 902576
zbMATH Keywords: Reid roundabout theorem
zbMATH Keywords: linear Hamiltonian difference systems
zbMATH Keywords: controllable system
zbMATH Keywords: disconjugacy
zbMATH Keywords: Riccati condition
zbMATH Keywords: discrete Sturm-Liouville equations
author: Martin Bohner
reviewed by: Jerzy Ryszard Popenda
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0177
OpenAlex ID: W2021740699
DOI: 10.1006/JMAA.1996.0177
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description: scientific article; zbMATH DE number 902576