Time scale embedding theorem and coercivity of quadratic functionals

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Authors

HILSCHER Roman ZEIDAN Vera

Year of publication 2008
Type Article in Periodical
Magazine / Source Analysis (Munich)
MU Faculty or unit

Faculty of Science

Citation
Web http://www.atypon-link.com/OLD/doi/abs/10.1524/anly.2008.0900
Field General mathematics
Keywords Time scale; Time scale embedding theorem; Quadratic functional; Positivity; Coercivity; Riccati equation; Legendre condition; Calculus of variations; Weak local extremum
Description In this paper we study the relation between the coercivity and positivity of a time scale quadratic functional J, which could be a second variation for a nonlinear time scale calculus of variations problem (P). We prove for the case of general jointly varying endpoints that J is coercive if and only if it is positive definite and the time scale version of the strengthened Legendre condition holds. In order to prove this, we establish a time scale embedding theorem and apply it to the Riccati matrix equation associated with the quadratic functional J. Consequently, we obtain sufficiency criteria for the nonlinear problem (P) in terms of the positivity of J or in terms of the time scale Riccati equation. This result is new even for the continuous time case when the endpoints are jointly varying .
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