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International mathematical journal of analysis and its applications

Editor-in-Chief: Schulz, Friedmar

CiteScore 2017: 0.66

SCImago Journal Rank (SJR) 2017: 0.564
Source Normalized Impact per Paper (SNIP) 2017: 0.674

Mathematical Citation Quotient (MCQ) 2017: 0.38

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Volume 28, Issue 1


Time scale embedding theorem and coercivity of quadratic functionals

Roman Hilscher / Vera Zeidan
Published Online: 2009-09-25 | DOI: https://doi.org/10.1524/anly.2008.0900

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.

Keywords: Time scale; time scale embedding theorem; quadratic functional; positivity; coercivity; Riccati equation; Legendre condition; calculus of variations; weak local extremum

About the article

Received: 2007-05-03

Published Online: 2009-09-25

Published in Print: 2008-01-01

Citation Information: Analysis, Volume 28, Issue 1, Pages 1–28, ISSN (Online) 2196-6753, ISSN (Print) 0174-4747, DOI: https://doi.org/10.1524/anly.2008.0900.

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