Multiplicities of focal points for discrete symplectic systems: revisited
| Authors | |
|---|---|
| Year of publication | 2009 |
| Type | Article in Periodical |
| Magazine / Source | Journal of Difference Equations and Applications |
| MU Faculty or unit | |
| Citation | |
| 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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