Publication details

Multiplicities of focal points for discrete symplectic systems: revisited

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Year of publication 2009
Type Article in Periodical
Magazine / Source Journal of Difference Equations and Applications
MU Faculty or unit

Faculty of Science

Field General mathematics
Keywords Discrete symplectic system; Focal point; Multiplicity; Conjoined basis; Sturmian separation theorem; Sturmian comparison theorem; Moore-Penrose generalized inverse
Description In this note we define a notion of multiplicity of focal points for conjoined bases of discrete symplectic systems. We show that this definition is equivalent to the one given by Kratz in [Discrete oscillation, J. Difference Equ. Appl. 9 (2003), no. 1, 135--147] and, furthermore, it has a natural connection to the newly developed continuous time theory on linear Hamiltonian differential systems. Many results obtained recently by Bohner, Došlý, and Kratz regarding the nonnegativity of the corresponding discrete quadratic functionals, Sturmian separation and comparison theorems, and oscillation theorems relating the number of focal points of a certain special conjoined basis with the number of eigenvalues of the associated discrete symplectic eigenvalue problem, are now formulated in terms of this alternative definition of multiplicities.
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