×

Eigenvalue and oscillation theorems for time scale symplectic systems. (English) Zbl 1218.34039

This is a very comprehensive paper dealing with oscillation and spectral theory of symplectic dynamic systems on a time scale
\[ x^{\Delta}=A(t)x+B(t)u,\quad u^{\Delta}=C(t)x+D(t)u, \tag{*} \]
where the matrices \(A,B,C,D\) satisfy identities implying that the fundamental matrix of (*) is symplectic. The main result of the paper is the so-called oscillation theorem which relates the number of finite eigenvalues of a certain eigenvalue problem associated with (*) and the number of generalized focal points of a conjoined basis of this eigenvalue problem. To prove this result, the authors develop systematically the general theory of symplectic eigenvalue problems without any controllability assumption. The results of the paper cover as a particular case oscillation theorems for Hamiltonian differential systems (continuous case) and symplectic difference systems (discrete case).

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34N05 Dynamic equations on time scales or measure chains
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
34L05 General spectral theory of ordinary differential operators
34A30 Linear ordinary differential equations and systems
PDFBibTeX XMLCite
Full Text: DOI