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Zbl 0894.39005
Bohner, Martin; Došlý, Ondřej
Disconjugacy and transformations for symplectic systems.
(English)
[J] Rocky Mt. J. Math. 27, No.3, 707-743 (1997). ISSN 0035-7596

Let $I$ be the $n\times n$ identity matrix, and let $J=\left (\smallmatrix 0 & I \\ -I & 0 \endsmallmatrix \right)$. The $2n\times 2n$ matrix $S=\left( \smallmatrix A & B \\ C & D \endsmallmatrix \right)$ is called symplectic if $S^TJS=J$ holds, and the system $z_{k+1} =S_kz_k$, $0\le k\le N$, is symplectic if the matrix $S$ is symplectic. A solution $z= {x\choose u}$ to the system has a generalized zero in $[k,k+1]$ if $x_k\ne 0$, $x_{k+1} \in\operatorname{Im} B_k$ and $x^T_k B^+_kx_{k+1} =0$, where $B^+_k$ denotes the Moore-Penrose inverse of the matrix $B_k$. The system is called disconjugate on ${\mathcal J}=[0,N]\cap\mathbb{Z}$ if no solution of the system has more than one generalized zero (if $x_0=0$ no zero).\par Theorem 1 shows a lot of equivalent facts about disconjugacy, possession of generalized zeros and other similar properties of the solution to the symplectic system. Similar results are obtained relative to the reciprocal symplectic system $z_{k+1}= S^{-1}_k z_k$. The problems of eventually disconjugate solutions and disconjugacy preserving transformations are considered as well.
[D.Bobrowski (Poznań)]
MSC 2000:
*39A12 Discrete version of topics in analysis
39A10 Difference equations

Keywords: Sturm-Liouville difference equation; Hamiltonian difference equation; symplectic system; disconjugacy; reciprocal system

Cited in: Zbl 1178.39002 Zbl 0913.39010

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