Time scale symplectic systems without normality
|Year of publication
|Article in Periodical
|Magazine / Source
|Journal of Differential Equations
|MU Faculty or unit
|Time scale; Time scale symplectic system; Linear Hamiltonian system; Quadratic functional; Nonnegativity; Positivity; Conjoined basis; Generalized focal point
|We present a theory of the definiteness (nonnegativity and positivity) of a quadratic functional F over a bounded time scale. The results are given in terms of a time scale symplectic system (S), which is a time scale linear system that generalizes and unifies the linear Hamiltonian differential system and discrete symplectic system. The novelty of this paper resides in removing the assumption of normality in the characterization of the positivity of F, and in establishing equivalent conditions for the nonnegativity of F without any normality assumption. To reach this goal, a new notion of generalized focal points for conjoined bases (X,U) of (S) is introduced, results on the piecewise-constant kernel of X(t) are obtained, and various Picone-type identities are derived under the piecewise-constant kernel condition.