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@article{575741,
author = {Hilscher, Roman and Zeidan, Vera},
article_number = {9},
keywords = {Second variation; Euler-Lagrange difference equation; Discrete quadratic functional; Nonnegativity; Positivity; Linear Hamiltonian difference system; Conjugate interval; Coupled interval; Conjoined basis; Riccati difference equation},
language = {eng},
issn = {1023-6198},
journal = {Journal of Difference Equations and Applications},
title = {Nonnegativity and positivity of quadratic functionals in the discrete calculus of variations: Survey},
url = {http://journalsonline.tandf.co.uk/app/home/contribution.asp?wasp=567302076cf74cd5bc178e8900c08c9c&referrer=parent&backto=issue,6,7;journal,2,45;linkingpublicationresults,1:103361,1},
volume = {11},
year = {2005}
}
TY - JOUR
ID - 575741
AU - Hilscher, Roman - Zeidan, Vera
PY - 2005
TI - Nonnegativity and positivity of quadratic functionals in the discrete calculus of variations: Survey
JF - Journal of Difference Equations and Applications
VL - 11
IS - 9
SP - 857-875
EP - 857-875
PB - Taylor and Francis
SN - 10236198
KW - Second variation
KW - Euler-Lagrange difference equation
KW - Discrete quadratic functional
KW - Nonnegativity
KW - Positivity
KW - Linear Hamiltonian difference system
KW - Conjugate interval
KW - Coupled interval
KW - Conjoined basis
KW - Riccati difference equation
UR - http://journalsonline.tandf.co.uk/app/home/contribution.asp?wasp=567302076cf74cd5bc178e8900c08c9c&referrer=parent&backto=issue,6,7;journal,2,45;linkingpublicationresults,1:103361,1
N2 - In this paper we provide a survey of characterizations of the nonnegativity and positivity of discrete quadratic functionals which arise as the second variation for nonlinear discrete calculus of variations problems. These characterizations are in terms of (i) (strict) conjugate and (strict) coupled intervals, (ii) the conjoined bases of the associated Jacobi difference equation, and (iii) the solution of the corresponding Riccati difference equation. The results depend on the form of the boundary conditions of the quadratic functional and, basically, we distinguish three types: (a) separable endpoints with zero right endpoint (this of course includes the simplest case of both zero endpoints), (b) separable endpoints, and (c) jointly varying endpoints.
ER -
HILSCHER, Roman and Vera ZEIDAN. Nonnegativity and positivity of quadratic functionals in the discrete calculus of variations: Survey. \textit{Journal of Difference Equations and Applications}. Taylor and Francis, 2005, vol.~11, No~9, p.~857-875. ISSN~1023-6198.
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