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Linear Hamiltonian systems on time scales: Transformations. (English) Zbl 0997.34005

This paper is devoted to study the effect of the transformation of dependent variables \[ \begin{pmatrix} x\\ u\end{pmatrix}= R_t\begin{pmatrix} y\\ z\end{pmatrix}\tag{1} \] on the linear Hamiltonian system \[ x^\Delta= A_t x^\sigma+ B_tu,\quad u^\Delta= -C_t x^\sigma- A^T_t u,\quad t\in I^k,\tag{2} \] where \(I\subseteq T\) is a subinterval of an arbitrary time scale \(T\). The author proves that, under suitable assumptions, the linear Hamiltonian system (2) is transformed into a system of the same form, which includes the corresponding continuous \((T=\mathbb{R})\) and discrete \((T=\mathbb{Z})\) results as special cases. As an application of the main result the author gives a transformation theorem for disconjugate linear Hamiltonian systems and for Sturm-Liouville equations of higher-order.

MSC:

34A30 Linear ordinary differential equations and systems
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
39A10 Additive difference equations
93C70 Time-scale analysis and singular perturbations in control/observation systems
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