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@article{488380, author = {Hilscher, Roman}, article_location = {Berlin}, article_number = {1}, keywords = {time scale; (continuous and discrete) linear Hamiltonian system; quadratic functional; disconjugacy; focal point; principal solution}, language = {eng}, issn = {0025-584X}, journal = {Mathematische Nachrichten}, title = {Positivity of quadratic functionals on time scales: necessity}, volume = {226}, year = {2001} }
TY - JOUR ID - 488380 AU - Hilscher, Roman PY - 2001 TI - Positivity of quadratic functionals on time scales: necessity JF - Mathematische Nachrichten VL - 226 IS - 1 SP - 85-98 EP - 85-98 PB - WILEY-VCH Verlag SN - 0025584X KW - time scale KW - (continuous and discrete) linear Hamiltonian system KW - quadratic functional KW - disconjugacy KW - focal point KW - principal solution N2 - In this work we establish that disconjugacy of a linear Hamiltonian system on time scales is a necessary condition for the positivity of the corresponding quadratic functional. We employ a certain minimal normality (controllability) assumption. Hence, the open problems stated by the author in [15], [16] are solved with the result that positivity of the quadratic functional is equivalent with disconjugacy of the Hamiltonian system on the interval under consideration. The general approach on time scales T involves, as special cases, the well known continuous case for T=R and recently developed discrete one for T=Z, so that they are unified. As applications, Sturmian type separation and comparison theorems on time scales are also provided. ER -
HILSCHER, Roman. Positivity of quadratic functionals on time scales: necessity. \textit{Mathematische Nachrichten}. Berlin: WILEY-VCH Verlag, 2001, vol.~226, No~1, p.~85-98. ISSN~0025-584X.
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