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POSITIVE SEMIDEFINITENESS OF DISCRETE QUADRATIC FUNCTIONALS

Published online by Cambridge University Press:  10 December 2003

Martin Bohner ,
Ondřej Došlý  and
Werner Kratz
Martin Bohner
Affiliation:
University of Missouri-Rolla, Department of Mathematics, Rolla, MO 65401, USA (bohner@umr.edu)
Ondřej Došlý
Affiliation:
Masaryk University, Department of Mathematics, CZ-66295 Brno, Czech Republic (dosly@math.muni.cz)
Werner Kratz
Affiliation:
Universität Ulm, Abteilung Angewandte Analysis, D-89069 Ulm, Germany (kratz@mathematik.uni-ulm.de)
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Abstract

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We consider symplectic difference systems, which contain as special cases linear Hamiltonian difference systems and Sturm–Liouville difference equations of any even order. An associated discrete quadratic functional is important in discrete variational analysis, and while its positive definiteness has been characterized and is well understood, a characterization of its positive semidefiniteness remained an open problem. In this paper we present the solution to this problem and offer necessary and sufficient conditions for such discrete quadratic functionals to be non-negative definite.

AMS 2000 Mathematics subject classification: Primary 39A12; 39A13. Secondary 34B24; 49K99

Keywords

Hamiltonian systemssymplectic systemsdiscrete quadratic functionalsnon-negativity
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 46 , Issue 3 , October 2003 , pp. 627 - 636
Copyright
Copyright © Edinburgh Mathematical Society 2003