Positivity of quadratic functionals on time scales: necessity
| Authors | |
|---|---|
| Year of publication | 2001 |
| Type | Article in Periodical |
| Magazine / Source | Mathematische Nachrichten |
| MU Faculty or unit | |
| Citation | |
| Field | General mathematics |
| Keywords | time scale; (continuous and discrete) linear Hamiltonian system; quadratic functional; disconjugacy; focal point; principal solution |
| Description | 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. |
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