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@article{695806, author = {Hilscher, Roman and Zeidan, Vera}, article_location = {San Diego (USA)}, article_number = {1}, keywords = {Time scale; Time scale symplectic system; Linear Hamiltonian system; Quadratic functional; Nonnegativity; Positivity; Conjoined basis; Generalized focal point}, language = {eng}, issn = {0022-0396}, journal = {Journal of Differential Equations}, title = {Time scale symplectic systems without normality}, volume = {230}, year = {2006} }
TY - JOUR ID - 695806 AU - Hilscher, Roman - Zeidan, Vera PY - 2006 TI - Time scale symplectic systems without normality JF - Journal of Differential Equations VL - 230 IS - 1 SP - 140-173 EP - 140-173 PB - Elsevier Science SN - 00220396 KW - Time scale KW - Time scale symplectic system KW - Linear Hamiltonian system KW - Quadratic functional KW - Nonnegativity KW - Positivity KW - Conjoined basis KW - Generalized focal point N2 - 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. ER -
HILSCHER, Roman a Vera ZEIDAN. Time scale symplectic systems without normality. \textit{Journal of Differential Equations}. San Diego (USA): Elsevier Science, 2006, roč.~230, č.~1, s.~140-173. ISSN~0022-0396.
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